From 19d7f4d83d2666b7d068c838bfc526ed8c3490d0 Mon Sep 17 00:00:00 2001 From: George Tankersley Date: Mon, 22 May 2017 00:00:00 +0000 Subject: [PATCH 01/46] Implement an elliptic.Curve for ed25519 --- const.go | 22 + ed25519.go | 242 ++++ ed25519_test.go | 419 ++++++ internal/LICENSE | 27 + internal/edwards25519/const.go | 1422 ++++++++++++++++++++ internal/edwards25519/edwards25519.go | 1771 +++++++++++++++++++++++++ 6 files changed, 3903 insertions(+) create mode 100644 const.go create mode 100644 ed25519.go create mode 100644 ed25519_test.go create mode 100644 internal/LICENSE create mode 100644 internal/edwards25519/const.go create mode 100644 internal/edwards25519/edwards25519.go diff --git a/const.go b/const.go new file mode 100644 index 0000000..a92ff0d --- /dev/null +++ b/const.go @@ -0,0 +1,22 @@ +package ed25519 + +import ( + "math/big" + + "github.com/gtank/ed25519/internal/edwards25519" +) + +var ( + // d2 is 2*d, a curve-specific constant used in our addition formula. + d2 = edwards25519.FieldElement{ + -21827239, -5839606, -30745221, 13898782, 229458, 15978800, -12551817, -6495438, 29715968, 9444199, + } + + // the number 2 as a field element + feTwo = edwards25519.FieldElement{ + 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, + } + + bigZero = big.NewInt(0) + bigOne = big.NewInt(1) +) diff --git a/ed25519.go b/ed25519.go new file mode 100644 index 0000000..af14a95 --- /dev/null +++ b/ed25519.go @@ -0,0 +1,242 @@ +// Implements an elliptic.Curve interface over the ed25519 curve. +package ed25519 + +import ( + "crypto/elliptic" + "math/big" + "sync" + + "github.com/gtank/ed25519/internal/edwards25519" +) + +type ed25519Curve struct { + *elliptic.CurveParams +} + +var once sync.Once +var ed25519Params = &elliptic.CurveParams{Name: "ed25519"} +var ed25519 = ed25519Curve{ed25519Params} + +// Ed25519 uses a twisted Edwards curve -x^2 + y^2 = 1 + dx^2y^2 with the following params: +// The field prime is 2^255 - 19. +// The order of the base point is 2^252 + 27742317777372353535851937790883648493. +// And since B is irrelevant here, we're going to pretend that B is d = -(121665/121666). +func initEd25519Params() { + ed25519Params.P, _ = new(big.Int).SetString("7fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffed", 16) + ed25519Params.N, _ = new(big.Int).SetString("7237005577332262213973186563042994240857116359379907606001950938285454250989", 10) + ed25519Params.B, _ = new(big.Int).SetString("37095705934669439343138083508754565189542113879843219016388785533085940283555", 10) + ed25519Params.Gx, _ = new(big.Int).SetString("15112221349535400772501151409588531511454012693041857206046113283949847762202", 10) + ed25519Params.Gy, _ = new(big.Int).SetString("46316835694926478169428394003475163141307993866256225615783033603165251855960", 10) + ed25519Params.BitSize = 256 +} + +func Ed25519() elliptic.Curve { + once.Do(initEd25519Params) + return ed25519 +} + +// Params returns the parameters for the curve. +func (curve ed25519Curve) Params() *elliptic.CurveParams { + return curve.CurveParams +} + +// IsOnCurve reports whether the given (x,y) lies on the curve by checking that +// -x^2 + y^2 - 1 - dx^2y^2 = 0. +func (curve ed25519Curve) IsOnCurve(x, y *big.Int) bool { + lh := new(big.Int).Mul(x, x) // x^2 + y2 := new(big.Int).Mul(y, y) // y^2 + rh := new(big.Int).Mul(lh, y2) // x^2y^2 + rh.Mul(rh, curve.B) // dx^2y^2 with B repurposed as d + rh.Add(rh, bigOne) // 1 + dx^2y^2 + lh.Neg(lh) // -x^2 + lh.Add(lh, y2) // -x^2 + y^2 + lh.Sub(lh, rh) // -x^2 + y^2 - 1 - dx^2y^2 + lh.Mod(lh, curve.P) // -x^2 + y^2 - 1 - dx^2y^2 mod p + return lh.Cmp(bigZero) == 0 +} + +// Add returns the sum of (x1, y1) and (x2, y2). +func (curve ed25519Curve) Add(x1, y1, x2, y2 *big.Int) (x, y *big.Int) { + var p1, p2, p3 edwards25519.ExtendedGroupElement + + affineToExtended(&p1, x1, y1) + affineToExtended(&p2, x2, y2) + extendedAdd(&p3, &p1, &p2) + + return extendedToAffine(&p3) // 1I + 2M +} + +// The internal edwards25519 package optimizes for a particular sequence of +// operations, so we reimplement addition (add-2008-hwcd-3) here to avoid the +// cost of converting between intermediate representations. +// TODO We know Z1=1 and Z2=1 here, so mmadd-2008-hwcd-3 (6M + 1S + 1*k + 9add) could apply +func extendedAdd(out, p1, p2 *edwards25519.ExtendedGroupElement) { + var tmp1, tmp2, A, B, C, D, E, F, G, H edwards25519.FieldElement + + edwards25519.FeSub(&tmp1, &p1.Y, &p1.X) // tmp1 <-- Y1-X1 + edwards25519.FeSub(&tmp2, &p2.Y, &p2.X) // tmp2 <-- Y2-X2 + edwards25519.FeMul(&A, &tmp1, &tmp2) // A <-- tmp1*tmp2 = (Y1-X1)*(Y2-X2) + edwards25519.FeAdd(&tmp1, &p1.Y, &p1.X) // tmp1 <-- Y1+X1 + edwards25519.FeAdd(&tmp2, &p2.Y, &p2.X) // tmp2 <-- Y2+X2 + edwards25519.FeMul(&B, &tmp1, &tmp2) // B <-- tmp1*tmp2 = (Y1+X1)*(Y2+X2) + edwards25519.FeMul(&tmp1, &p1.T, &p2.T) // tmp1 <-- T1*T2 + edwards25519.FeMul(&C, &tmp1, &d2) // C <-- tmp1*2d = T1*2d*T2 + edwards25519.FeMul(&tmp1, &p1.Z, &p2.Z) // tmp1 <-- Z1*Z2 + edwards25519.FeAdd(&D, &tmp1, &tmp1) // D <-- tmp1 + tmp1 = 2*Z1*Z2 + edwards25519.FeSub(&E, &B, &A) // E <-- B-A + edwards25519.FeSub(&F, &D, &C) // F <-- D-C + edwards25519.FeAdd(&G, &D, &C) // G <-- D+C + edwards25519.FeAdd(&H, &B, &A) // H <-- B+A + edwards25519.FeMul(&out.X, &E, &F) // X3 <-- E*F + edwards25519.FeMul(&out.Y, &G, &H) // Y3 <-- G*H + edwards25519.FeMul(&out.T, &E, &H) // T3 <-- E*H + edwards25519.FeMul(&out.Z, &F, &G) // Z3 <-- F*G +} + +// Double returns 2*(x,y). +// TODO: cheaper to reimplement? the typed path is aff->proj->completed->proj->aff +func (curve ed25519Curve) Double(x1, y1 *big.Int) (x, y *big.Int) { + var p edwards25519.ProjectiveGroupElement + var r edwards25519.CompletedGroupElement + affineToProjective(&p, x1, y1) + p.Double(&r) + r.ToProjective(&p) // 3M + return projectiveToAffine(&p) // 1I + 2M +} + +// ScalarMult returns k*(Bx,By) where k is a number in big-endian form. +func (curve ed25519Curve) ScalarMult(x1, y1 *big.Int, k []byte) (x, y *big.Int) { + // if either coordinate is nil, return the point at infinity + if x1 == nil || y1 == nil { + x = new(big.Int).Set(bigZero) + y = new(big.Int).Set(bigOne) + return + } + + var r0, r1 edwards25519.ExtendedGroupElement + var h edwards25519.CompletedGroupElement + var s [32]byte + + curve.scalarFromBytes(&s, k) + + // Montgomery ladder init: + // R_0 = O, R_1 = P + r0.Zero() + affineToExtended(&r1, x1, y1) + + // Montgomery ladder step: + // R_{1-b} = R_{1-b} + R_{b} + // R_{b} = 2*R_{b} + for i := 255; i >= 0; i-- { + var b = int32((s[i/8] >> uint(i&7)) & 1) + if b == 0 { + extendedAdd(&r1, &r0, &r1) + r0.Double(&h) + h.ToExtended(&r0) + } else { + extendedAdd(&r0, &r0, &r1) + r1.Double(&h) + h.ToExtended(&r1) + } + } + + return extendedToAffine(&r0) +} + +// scalarFromBytes converts a big-endian value to a fixed-size little-endian +// representation. If the value is larger than the scalar group order, we +// reduce it before returning. +func (curve ed25519Curve) scalarFromBytes(out *[32]byte, in []byte) { + scalar := new(big.Int).SetBytes(in) + if scalar.Cmp(curve.N) >= 0 { + scalar.Mod(scalar, curve.N) + } + buf := make([]byte, 32) + scBytes := scalar.Bytes() + copy(buf[32-len(scBytes):], scBytes) + for i := 0; i < 32; i++ { + out[i] = buf[31-i] + } +} + +// ScalarBaseMult returns k*G, where G is the base point of the group and k is +// an integer in big-endian form. +func (curve ed25519Curve) ScalarBaseMult(k []byte) (x, y *big.Int) { + var p edwards25519.ExtendedGroupElement + var scBytes [32]byte + + curve.scalarFromBytes(&scBytes, k) + edwards25519.GeScalarMultBase(&p, &scBytes) + return extendedToAffine(&p) +} + +// Converts (x,y) to (X:Y:T:Z) extended coordinates, or "P3" in ref10. As +// described in "Twisted Edwards Curves Revisited", Hisil-Wong-Carter-Dawson +// 2008, Section 3.1 (https://eprint.iacr.org/2008/522.pdf) +// See also https://hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html#addition-add-2008-hwcd-3 +func affineToExtended(out *edwards25519.ExtendedGroupElement, x, y *big.Int) { + feFromBig(&out.X, x) + feFromBig(&out.Y, y) + edwards25519.FeMul(&out.T, &out.X, &out.Y) + edwards25519.FeOne(&out.Z) +} + +// Extended coordinates are XYZT with x = X/Z, y = Y/Z, or the "P3" +// representation in ref10. Extended->affine is the same operation as moving +// from projective to affine. Per HWCD, it is safe to move from extended to +// projective by simply ignoring T. +func extendedToAffine(in *edwards25519.ExtendedGroupElement) (*big.Int, *big.Int) { + var x, y, zinv edwards25519.FieldElement + var bigX, bigY = new(big.Int), new(big.Int) + + edwards25519.FeInvert(&zinv, &in.Z) + edwards25519.FeMul(&x, &in.X, &zinv) + edwards25519.FeMul(&y, &in.Y, &zinv) + + feToBig(bigX, &x) + feToBig(bigY, &y) + + return bigX, bigY +} + +// Projective coordinates are XYZ with x = X/Z, y = Y/Z, or the "P2" representation in ref10. +func affineToProjective(out *edwards25519.ProjectiveGroupElement, x, y *big.Int) { + feFromBig(&out.X, x) + feFromBig(&out.Y, y) + edwards25519.FeOne(&out.Z) +} + +func projectiveToAffine(in *edwards25519.ProjectiveGroupElement) (*big.Int, *big.Int) { + var x, y, zinv edwards25519.FieldElement + var bigX, bigY = new(big.Int), new(big.Int) + + edwards25519.FeInvert(&zinv, &in.Z) + edwards25519.FeMul(&x, &in.X, &zinv) + edwards25519.FeMul(&y, &in.Y, &zinv) + + feToBig(bigX, &x) + feToBig(bigY, &y) + + return bigX, bigY +} + +func feFromBig(h *edwards25519.FieldElement, in *big.Int) { + tmp := new(big.Int).Mod(in, ed25519.P) + tmpBytes := tmp.Bytes() + var buf, reverse [32]byte + copy(buf[32-len(tmpBytes):], tmpBytes) + for i := 0; i < 32; i++ { + reverse[i] = buf[31-i] + } + edwards25519.FeFromBytes(h, &reverse) +} + +func feToBig(out *big.Int, h *edwards25519.FieldElement) { + var buf, reverse [32]byte + edwards25519.FeToBytes(&buf, h) + for i := 0; i < 32; i++ { + reverse[i] = buf[31-i] + } + out.SetBytes(reverse[:]) + out.Mod(out, ed25519.P) +} diff --git a/ed25519_test.go b/ed25519_test.go new file mode 100644 index 0000000..2734089 --- /dev/null +++ b/ed25519_test.go @@ -0,0 +1,419 @@ +package ed25519 + +import ( + "bytes" + "crypto/elliptic" + "crypto/rand" + "crypto/sha512" + "encoding/hex" + "io" + "math/big" + "testing" + + "github.com/gtank/ed25519/internal/edwards25519" +) + +// TEST MATH + +// d is a constant from the curve equation. it's easy to find the value in multiple formats. +var d_fe = edwards25519.FieldElement{ + -10913610, 13857413, -15372611, 6949391, 114729, -8787816, -6275908, -3247719, -18696448, -12055116, +} +var d_bn, _ = new(big.Int).SetString("37095705934669439343138083508754565189542113879843219016388785533085940283555", 10) + +// 2*d +var d2_fe = edwards25519.FieldElement{ + -21827239, -5839606, -30745221, 13898782, 229458, 15978800, -12551817, -6495438, 29715968, 9444199, +} +var d2_bn = new(big.Int).Mul(big.NewInt(2), d_bn) + +// A is a constant from the Montgomery form of curve25519. +var A_fe = edwards25519.FieldElement{ + 486662, 0, 0, 0, 0, 0, 0, 0, 0, 0, +} +var A_bn, _ = new(big.Int).SetString("486662", 10) + +var feConvTable = []struct { + name string + fe edwards25519.FieldElement + bn *big.Int +}{ + { + name: "d", + fe: d_fe, + bn: d_bn, + }, + { + name: "d2", + fe: d2_fe, + bn: d2_bn, + }, + { + name: "A", + fe: A_fe, + bn: A_bn, + }, +} + +func TestFeFromBig(t *testing.T) { + var _ = Ed25519().(ed25519Curve) + var fe edwards25519.FieldElement + for _, c := range feConvTable { + feFromBig(&fe, c.bn) + for i := 0; i < len(fe); i++ { + if fe[i] != c.fe[i] { + t.Error("bn->fe radix conversion failed") + } + } + } +} + +func TestFeToBig(t *testing.T) { + var curve = Ed25519().(ed25519Curve) + var bn, expectedBN = new(big.Int), new(big.Int) + for _, c := range feConvTable { + expectedBN.Mod(c.bn, curve.P) + feToBig(bn, &c.fe) + if bn.Cmp(expectedBN) != 0 { + t.Errorf("fe->bn radix conversion failed: %s", c.name) + } + } +} + +func TestRadixRoundTrip(t *testing.T) { + if testing.Short() { + t.Skip() + } + + var curve = Ed25519().(ed25519Curve) + var fe edwards25519.FieldElement + var out = new(big.Int) + + for i := 0; i < 1000; i++ { + n, err := rand.Int(rand.Reader, curve.P) + if err != nil { + t.Fatal(err) + } + feFromBig(&fe, n) + feToBig(out, &fe) + if n.Cmp(out) != 0 { + t.Errorf("fe<>bn failed for %x", n.Bytes()) + } + } +} + +func TestIsOnCurve(t *testing.T) { + ed := Ed25519() + if !ed.IsOnCurve(ed.Params().Gx, ed.Params().Gy) { + t.Error("ed25519 base point not on curve") + } + + x, y := new(big.Int), new(big.Int) + if ed.IsOnCurve(x, y) { + t.Error("(0,0) is not on the curve") + } +} + +func TestAdd(t *testing.T) { + c := Ed25519().(ed25519Curve) + Bx, By := c.Params().Gx, c.Params().Gy + B2x, B2y := c.Add(Bx, By, Bx, By) + + if !c.IsOnCurve(B2x, B2y) { + t.Error("B+B is not on the curve") + } +} + +func TestDouble(t *testing.T) { + c := Ed25519().(ed25519Curve) + Gx, Gy := c.Params().Gx, c.Params().Gy + G2x, G2y := c.Double(Gx, Gy) + + Ax, Ay := c.Add(Gx, Gy, Gx, Gy) + + if Ax.Cmp(G2x) != 0 || Ay.Cmp(G2y) != 0 { + t.Errorf("double(B) != B+B") + } +} + +// Test vector generated by instrumenting x/crypto/ed25519 GenerateKey +// seed: c240344fcc6615dda52da98149377ad2b13fdba2bc39a50ba9f3afb2cbd4abaa +// expanded: f04154b9d80963bb4c76214ece8a1049bdd16fbfc5003aff9835a59643ace276 +// public: 65a8343a83ec15e55050f12fc22f2c81a4fe7327c8da1524441f9ce5e5bc27dd + +func TestScalarMultBase(t *testing.T) { + c := Ed25519() + + // endian-swapped because x/crypto assumes this is always little endian from raw bytes + // and scalarFromBytes assumes it's coming from big.Int Bytes() + a, _ := hex.DecodeString("76e2ac4396a53598ff3a00c5bf6fd1bd49108ace4e21764cbb6309d8b95441f0") + if len(a) != 32 { + t.Errorf("failed decoding") + } + + Ax, Ay := c.ScalarBaseMult(a) + + if !c.IsOnCurve(Ax, Ay) { + t.Error("scalarmultbase result was off-curve") + } + + var A edwards25519.ExtendedGroupElement + pub, _ := hex.DecodeString("65a8343a83ec15e55050f12fc22f2c81a4fe7327c8da1524441f9ce5e5bc27dd") + var pubBytes [32]byte + copy(pubBytes[:], pub) + A.FromBytes(&pubBytes) + Bx, By := extendedToAffine(&A) + + if Ax.Cmp(Bx) != 0 || Ay.Cmp(By) != 0 { + t.Error("scalarmultbase disagrees with x/crypto/ed25519") + } +} + +func TestScalarMultBaseIdentity(t *testing.T) { + var c = Ed25519() + var one = new(big.Int).Set(bigOne) + Ax, Ay := c.ScalarBaseMult(one.Bytes()) + + if Ax.Cmp(c.Params().Gx) != 0 || Ay.Cmp(c.Params().Gy) != 0 { + t.Errorf("precomputed 1*B != B") + } + + Ax, Ay = c.ScalarMult(Ax, Ay, one.Bytes()) + + if Ax.Cmp(c.Params().Gx) != 0 || Ay.Cmp(c.Params().Gy) != 0 { + t.Errorf("arbitrary 1*B != B") + } +} + +func TestScalarMultBaseInfinity(t *testing.T) { + c := Ed25519() + a, _ := hex.DecodeString("0000000000000000000000000000000000000000000000000000000000000000") + if len(a) != 32 { + t.Errorf("failed decoding") + } + + Ax, Ay := c.ScalarBaseMult(a) + + if !c.IsOnCurve(Ax, Ay) { + t.Error("scalarmultbase result was off-curve") + } + + if Ax.Cmp(bigZero) != 0 || Ay.Cmp(bigOne) != 0 { + t.Error("scalarmultbase by 0 was not point at infinity") + } +} + +func TestScalarMultsAgreeAtInfinity(t *testing.T) { + c := Ed25519() + a, _ := hex.DecodeString("0000000000000000000000000000000000000000000000000000000000000000") + if len(a) != 32 { + t.Errorf("failed decoding") + } + + Ax, Ay := c.ScalarBaseMult(a) + Bx, By := c.ScalarMult(c.Params().Gx, c.Params().Gy, a) + + if Ax.Cmp(Bx) != 0 || Ay.Cmp(By) != 0 { + t.Error("scalarmultbase disagrees with scalarmult") + } +} + +func TestScalarMultsAgreeElsewhere(t *testing.T) { + c := Ed25519() + // head -c 32 /dev/urandom | sha256sum + a, _ := hex.DecodeString("c07eea55b3322f15099b6cf4d2b7e99d3d0fa6807f6fc7a46b5f7cb78daad4e0") + + Ax, Ay := c.ScalarBaseMult(a) + Bx, By := c.ScalarMult(c.Params().Gx, c.Params().Gy, a) + + if Ax.Cmp(Bx) != 0 || Ay.Cmp(By) != 0 { + t.Error("scalarmultbase disagrees with scalarmult") + } + + if !c.IsOnCurve(Bx, By) { + t.Error("scalarmult is returning off-curve points") + } +} + +// TEST INTERFACE + +func TestMarshalingRoundTrip(t *testing.T) { + ed := Ed25519() + + a, _ := hex.DecodeString("c07eea55b3322f15099b6cf4d2b7e99d3d0fa6807f6fc7a46b5f7cb78daad4e0") + Ax, Ay := ed.ScalarBaseMult(a) + + if !ed.IsOnCurve(Ax, Ay) { + t.Error("scalarBaseMult is returning off-curve points") + } + + sec1A := elliptic.Marshal(ed, Ax, Ay) + Bx, By := elliptic.Unmarshal(ed, sec1A) + + if Ax.Cmp(Bx) != 0 || Ay.Cmp(By) != 0 { + t.Error("point did not survive elliptic.Marshal roundtrip") + } + + if !testing.Short() { + for i := 0; i < 100; i++ { + _, err := io.ReadFull(rand.Reader, a) + if err != nil { + t.Fatal(err) + } + Ax, Ay := ed.ScalarBaseMult(a) + + if !ed.IsOnCurve(Ax, Ay) { + t.Error("scalarBaseMult is returning off-curve points") + } + + sec1A := elliptic.Marshal(ed, Ax, Ay) + Bx, By := elliptic.Unmarshal(ed, sec1A) + + if Ax.Cmp(Bx) != 0 || Ay.Cmp(By) != 0 { + t.Error("point did not survive elliptic.Marshal roundtrip") + } + } + } +} + +// TEST APPLICATION + +func generateKey(r io.Reader) (sk *[32]byte, pk []byte, err error) { + if r == nil { + r = rand.Reader + } + + ed := Ed25519() + + sk = new([32]byte) + _, err = io.ReadFull(r, sk[:]) + if err != nil { + return nil, nil, err + } + + digest := sha512.Sum512(sk[:]) + digest[0] &= 248 + digest[31] &= 127 + digest[31] |= 64 + + // take the first half of expanded bytes & reverse; big.Int expects big-endian + reverseDigest := reverse32(digest[:32]) + scalar := new(big.Int).SetBytes(reverseDigest) + + // A = a*B + Ax, Ay := ed.ScalarBaseMult(scalar.Bytes()) + + // This works with the standard elliptic package marshaling + publicKey := elliptic.Marshal(ed, Ax, Ay) + + // but we can also render the compressed Edwards format + compressedEdwardsY := make([]byte, 32) + + // x, y here will be big-endian byte strings + x, y := publicKey[1:33], publicKey[33:] + + // RFC 8032: To form the encoding of the point, copy the least significant + // bit of the x-coordinate to the most significant bit of the final octet. + copy(compressedEdwardsY[:], y) + compressedEdwardsY[0] |= x[31] << 7 + compressedEdwardsY = reverse32(compressedEdwardsY) + + return sk, compressedEdwardsY, err +} + +func reverse32(b []byte) []byte { + var tmp = make([]byte, 32) + for i := 0; i <= 31; i++ { + tmp[i] = b[31-i] + } + return tmp +} + +// Test vector generated by instrumenting x/crypto/ed25519 GenerateKey(). These +// are raw values. The edwards code interprets them as little-endian, so they +// need to be reversed before use with big.Int. +var genKeyTest = struct { + seed, expanded, public string +}{ + seed: "c240344fcc6615dda52da98149377ad2b13fdba2bc39a50ba9f3afb2cbd4abaa", + expanded: "f04154b9d80963bb4c76214ece8a1049bdd16fbfc5003aff9835a59643ace276", + public: "65a8343a83ec15e55050f12fc22f2c81a4fe7327c8da1524441f9ce5e5bc27dd", +} + +func TestEdDSAGenerateKey(t *testing.T) { + fakeRandom, _ := hex.DecodeString(genKeyTest.seed) + fakeReader := bytes.NewBuffer(fakeRandom) + + sk, pk, err := generateKey(fakeReader) + if err != nil { + t.Fatal(err) + } + + expectedPK, _ := hex.DecodeString(genKeyTest.public) + if !bytes.Equal(sk[:], fakeRandom) || !bytes.Equal(pk, expectedPK) { + t.Error("generateKey output did not match test vector") + } +} + +// BENCHMARKS + +func BenchmarkIsOnCurve(b *testing.B) { + ed := Ed25519() + Gx, Gy := ed.Params().Gx, ed.Params().Gy + for i := 0; i < b.N; i++ { + if !ed.IsOnCurve(Gx, Gy) { + b.Error("ed25519 base point not on curve") + } + } +} + +func BenchmarkAdd(b *testing.B) { + ed := Ed25519() + Gx, Gy := ed.Params().Gx, ed.Params().Gy + for i := 0; i < b.N; i++ { + Gx, Gy = ed.Add(Gx, Gy, Gx, Gy) + } +} + +func BenchmarkDouble(b *testing.B) { + ed := Ed25519() + Gx, Gy := ed.Params().Gx, ed.Params().Gy + for i := 0; i < b.N; i++ { + Gx, Gy = ed.Double(Gx, Gy) + } +} + +func BenchmarkScalarBaseMult(b *testing.B) { + ed := Ed25519() + + var k [32]byte + _, err := io.ReadFull(rand.Reader, k[:]) + if err != nil { + b.Fatal(err) + } + k[0] &= 248 + k[31] &= 127 + k[31] |= 64 + + for i := 0; i < b.N; i++ { + _, _ = ed.ScalarBaseMult(k[:]) + } +} + +func BenchmarkScalarMult(b *testing.B) { + ed := Ed25519() + Bx, By := ed.Params().Gx, ed.Params().Gy + + var k [32]byte + _, err := io.ReadFull(rand.Reader, k[:]) + if err != nil { + b.Fatal(err) + } + k[0] &= 248 + k[31] &= 127 + k[31] |= 64 + + for i := 0; i < b.N; i++ { + _, _ = ed.ScalarMult(Bx, By, k[:]) + } +} diff --git a/internal/LICENSE b/internal/LICENSE new file mode 100644 index 0000000..6a66aea --- /dev/null +++ b/internal/LICENSE @@ -0,0 +1,27 @@ +Copyright (c) 2009 The Go Authors. All rights reserved. + +Redistribution and use in source and binary forms, with or without +modification, are permitted provided that the following conditions are +met: + + * Redistributions of source code must retain the above copyright +notice, this list of conditions and the following disclaimer. + * Redistributions in binary form must reproduce the above +copyright notice, this list of conditions and the following disclaimer +in the documentation and/or other materials provided with the +distribution. + * Neither the name of Google Inc. nor the names of its +contributors may be used to endorse or promote products derived from +this software without specific prior written permission. + +THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS +"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT +LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR +A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT +OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, +SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT +LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, +DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY +THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT +(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE +OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. diff --git a/internal/edwards25519/const.go b/internal/edwards25519/const.go new file mode 100644 index 0000000..e39f086 --- /dev/null +++ b/internal/edwards25519/const.go @@ -0,0 +1,1422 @@ +// Copyright 2016 The Go Authors. All rights reserved. +// Use of this source code is governed by a BSD-style +// license that can be found in the LICENSE file. + +package edwards25519 + +// These values are from the public domain, “ref10” implementation of ed25519 +// from SUPERCOP. + +// d is a constant in the Edwards curve equation. +var d = FieldElement{ + -10913610, 13857413, -15372611, 6949391, 114729, -8787816, -6275908, -3247719, -18696448, -12055116, +} + +// d2 is 2*d. +var d2 = FieldElement{ + -21827239, -5839606, -30745221, 13898782, 229458, 15978800, -12551817, -6495438, 29715968, 9444199, +} + +// SqrtM1 is the square-root of -1 in the field. +var SqrtM1 = FieldElement{ + -32595792, -7943725, 9377950, 3500415, 12389472, -272473, -25146209, -2005654, 326686, 11406482, +} + +// A is a constant in the Montgomery-form of curve25519. +var A = FieldElement{ + 486662, 0, 0, 0, 0, 0, 0, 0, 0, 0, +} + +// bi contains precomputed multiples of the base-point. See the Ed25519 paper +// for a discussion about how these values are used. +var bi = [8]PreComputedGroupElement{ + { + FieldElement{25967493, -14356035, 29566456, 3660896, -12694345, 4014787, 27544626, -11754271, -6079156, 2047605}, + FieldElement{-12545711, 934262, -2722910, 3049990, -727428, 9406986, 12720692, 5043384, 19500929, -15469378}, + FieldElement{-8738181, 4489570, 9688441, -14785194, 10184609, -12363380, 29287919, 11864899, -24514362, -4438546}, + }, + { + FieldElement{15636291, -9688557, 24204773, -7912398, 616977, -16685262, 27787600, -14772189, 28944400, -1550024}, + FieldElement{16568933, 4717097, -11556148, -1102322, 15682896, -11807043, 16354577, -11775962, 7689662, 11199574}, + FieldElement{30464156, -5976125, -11779434, -15670865, 23220365, 15915852, 7512774, 10017326, -17749093, -9920357}, + }, + { + FieldElement{10861363, 11473154, 27284546, 1981175, -30064349, 12577861, 32867885, 14515107, -15438304, 10819380}, + FieldElement{4708026, 6336745, 20377586, 9066809, -11272109, 6594696, -25653668, 12483688, -12668491, 5581306}, + FieldElement{19563160, 16186464, -29386857, 4097519, 10237984, -4348115, 28542350, 13850243, -23678021, -15815942}, + }, + { + FieldElement{5153746, 9909285, 1723747, -2777874, 30523605, 5516873, 19480852, 5230134, -23952439, -15175766}, + FieldElement{-30269007, -3463509, 7665486, 10083793, 28475525, 1649722, 20654025, 16520125, 30598449, 7715701}, + FieldElement{28881845, 14381568, 9657904, 3680757, -20181635, 7843316, -31400660, 1370708, 29794553, -1409300}, + }, + { + FieldElement{-22518993, -6692182, 14201702, -8745502, -23510406, 8844726, 18474211, -1361450, -13062696, 13821877}, + FieldElement{-6455177, -7839871, 3374702, -4740862, -27098617, -10571707, 31655028, -7212327, 18853322, -14220951}, + FieldElement{4566830, -12963868, -28974889, -12240689, -7602672, -2830569, -8514358, -10431137, 2207753, -3209784}, + }, + { + FieldElement{-25154831, -4185821, 29681144, 7868801, -6854661, -9423865, -12437364, -663000, -31111463, -16132436}, + FieldElement{25576264, -2703214, 7349804, -11814844, 16472782, 9300885, 3844789, 15725684, 171356, 6466918}, + FieldElement{23103977, 13316479, 9739013, -16149481, 817875, -15038942, 8965339, -14088058, -30714912, 16193877}, + }, + { + FieldElement{-33521811, 3180713, -2394130, 14003687, -16903474, -16270840, 17238398, 4729455, -18074513, 9256800}, + FieldElement{-25182317, -4174131, 32336398, 5036987, -21236817, 11360617, 22616405, 9761698, -19827198, 630305}, + FieldElement{-13720693, 2639453, -24237460, -7406481, 9494427, -5774029, -6554551, -15960994, -2449256, -14291300}, + }, + { + FieldElement{-3151181, -5046075, 9282714, 6866145, -31907062, -863023, -18940575, 15033784, 25105118, -7894876}, + FieldElement{-24326370, 15950226, -31801215, -14592823, -11662737, -5090925, 1573892, -2625887, 2198790, -15804619}, + FieldElement{-3099351, 10324967, -2241613, 7453183, -5446979, -2735503, -13812022, -16236442, -32461234, -12290683}, + }, +} + +// base contains precomputed multiples of the base-point. See the Ed25519 paper +// for a discussion about how these values are used. +var base = [32][8]PreComputedGroupElement{ + { + { + FieldElement{25967493, -14356035, 29566456, 3660896, -12694345, 4014787, 27544626, -11754271, -6079156, 2047605}, + FieldElement{-12545711, 934262, -2722910, 3049990, -727428, 9406986, 12720692, 5043384, 19500929, -15469378}, + FieldElement{-8738181, 4489570, 9688441, -14785194, 10184609, -12363380, 29287919, 11864899, -24514362, -4438546}, + }, + { + FieldElement{-12815894, -12976347, -21581243, 11784320, -25355658, -2750717, -11717903, -3814571, -358445, -10211303}, + FieldElement{-21703237, 6903825, 27185491, 6451973, -29577724, -9554005, -15616551, 11189268, -26829678, -5319081}, + FieldElement{26966642, 11152617, 32442495, 15396054, 14353839, -12752335, -3128826, -9541118, -15472047, -4166697}, + }, + { + FieldElement{15636291, -9688557, 24204773, -7912398, 616977, -16685262, 27787600, -14772189, 28944400, -1550024}, + FieldElement{16568933, 4717097, -11556148, -1102322, 15682896, -11807043, 16354577, -11775962, 7689662, 11199574}, + FieldElement{30464156, -5976125, -11779434, -15670865, 23220365, 15915852, 7512774, 10017326, -17749093, -9920357}, + }, + { + FieldElement{-17036878, 13921892, 10945806, -6033431, 27105052, -16084379, -28926210, 15006023, 3284568, -6276540}, + FieldElement{23599295, -8306047, -11193664, -7687416, 13236774, 10506355, 7464579, 9656445, 13059162, 10374397}, + FieldElement{7798556, 16710257, 3033922, 2874086, 28997861, 2835604, 32406664, -3839045, -641708, -101325}, + }, + { + FieldElement{10861363, 11473154, 27284546, 1981175, -30064349, 12577861, 32867885, 14515107, -15438304, 10819380}, + FieldElement{4708026, 6336745, 20377586, 9066809, -11272109, 6594696, -25653668, 12483688, -12668491, 5581306}, + FieldElement{19563160, 16186464, -29386857, 4097519, 10237984, -4348115, 28542350, 13850243, -23678021, -15815942}, + }, + { + FieldElement{-15371964, -12862754, 32573250, 4720197, -26436522, 5875511, -19188627, -15224819, -9818940, -12085777}, + FieldElement{-8549212, 109983, 15149363, 2178705, 22900618, 4543417, 3044240, -15689887, 1762328, 14866737}, + FieldElement{-18199695, -15951423, -10473290, 1707278, -17185920, 3916101, -28236412, 3959421, 27914454, 4383652}, + }, + { + FieldElement{5153746, 9909285, 1723747, -2777874, 30523605, 5516873, 19480852, 5230134, -23952439, -15175766}, + FieldElement{-30269007, -3463509, 7665486, 10083793, 28475525, 1649722, 20654025, 16520125, 30598449, 7715701}, + FieldElement{28881845, 14381568, 9657904, 3680757, -20181635, 7843316, -31400660, 1370708, 29794553, -1409300}, + }, + { + FieldElement{14499471, -2729599, -33191113, -4254652, 28494862, 14271267, 30290735, 10876454, -33154098, 2381726}, + FieldElement{-7195431, -2655363, -14730155, 462251, -27724326, 3941372, -6236617, 3696005, -32300832, 15351955}, + FieldElement{27431194, 8222322, 16448760, -3907995, -18707002, 11938355, -32961401, -2970515, 29551813, 10109425}, + }, + }, + { + { + FieldElement{-13657040, -13155431, -31283750, 11777098, 21447386, 6519384, -2378284, -1627556, 10092783, -4764171}, + FieldElement{27939166, 14210322, 4677035, 16277044, -22964462, -12398139, -32508754, 12005538, -17810127, 12803510}, + FieldElement{17228999, -15661624, -1233527, 300140, -1224870, -11714777, 30364213, -9038194, 18016357, 4397660}, + }, + { + FieldElement{-10958843, -7690207, 4776341, -14954238, 27850028, -15602212, -26619106, 14544525, -17477504, 982639}, + FieldElement{29253598, 15796703, -2863982, -9908884, 10057023, 3163536, 7332899, -4120128, -21047696, 9934963}, + FieldElement{5793303, 16271923, -24131614, -10116404, 29188560, 1206517, -14747930, 4559895, -30123922, -10897950}, + }, + { + FieldElement{-27643952, -11493006, 16282657, -11036493, 28414021, -15012264, 24191034, 4541697, -13338309, 5500568}, + FieldElement{12650548, -1497113, 9052871, 11355358, -17680037, -8400164, -17430592, 12264343, 10874051, 13524335}, + FieldElement{25556948, -3045990, 714651, 2510400, 23394682, -10415330, 33119038, 5080568, -22528059, 5376628}, + }, + { + FieldElement{-26088264, -4011052, -17013699, -3537628, -6726793, 1920897, -22321305, -9447443, 4535768, 1569007}, + FieldElement{-2255422, 14606630, -21692440, -8039818, 28430649, 8775819, -30494562, 3044290, 31848280, 12543772}, + FieldElement{-22028579, 2943893, -31857513, 6777306, 13784462, -4292203, -27377195, -2062731, 7718482, 14474653}, + }, + { + FieldElement{2385315, 2454213, -22631320, 46603, -4437935, -15680415, 656965, -7236665, 24316168, -5253567}, + FieldElement{13741529, 10911568, -33233417, -8603737, -20177830, -1033297, 33040651, -13424532, -20729456, 8321686}, + FieldElement{21060490, -2212744, 15712757, -4336099, 1639040, 10656336, 23845965, -11874838, -9984458, 608372}, + }, + { + FieldElement{-13672732, -15087586, -10889693, -7557059, -6036909, 11305547, 1123968, -6780577, 27229399, 23887}, + FieldElement{-23244140, -294205, -11744728, 14712571, -29465699, -2029617, 12797024, -6440308, -1633405, 16678954}, + FieldElement{-29500620, 4770662, -16054387, 14001338, 7830047, 9564805, -1508144, -4795045, -17169265, 4904953}, + }, + { + FieldElement{24059557, 14617003, 19037157, -15039908, 19766093, -14906429, 5169211, 16191880, 2128236, -4326833}, + FieldElement{-16981152, 4124966, -8540610, -10653797, 30336522, -14105247, -29806336, 916033, -6882542, -2986532}, + FieldElement{-22630907, 12419372, -7134229, -7473371, -16478904, 16739175, 285431, 2763829, 15736322, 4143876}, + }, + { + FieldElement{2379352, 11839345, -4110402, -5988665, 11274298, 794957, 212801, -14594663, 23527084, -16458268}, + FieldElement{33431127, -11130478, -17838966, -15626900, 8909499, 8376530, -32625340, 4087881, -15188911, -14416214}, + FieldElement{1767683, 7197987, -13205226, -2022635, -13091350, 448826, 5799055, 4357868, -4774191, -16323038}, + }, + }, + { + { + FieldElement{6721966, 13833823, -23523388, -1551314, 26354293, -11863321, 23365147, -3949732, 7390890, 2759800}, + FieldElement{4409041, 2052381, 23373853, 10530217, 7676779, -12885954, 21302353, -4264057, 1244380, -12919645}, + FieldElement{-4421239, 7169619, 4982368, -2957590, 30256825, -2777540, 14086413, 9208236, 15886429, 16489664}, + }, + { + FieldElement{1996075, 10375649, 14346367, 13311202, -6874135, -16438411, -13693198, 398369, -30606455, -712933}, + FieldElement{-25307465, 9795880, -2777414, 14878809, -33531835, 14780363, 13348553, 12076947, -30836462, 5113182}, + FieldElement{-17770784, 11797796, 31950843, 13929123, -25888302, 12288344, -30341101, -7336386, 13847711, 5387222}, + }, + { + FieldElement{-18582163, -3416217, 17824843, -2340966, 22744343, -10442611, 8763061, 3617786, -19600662, 10370991}, + FieldElement{20246567, -14369378, 22358229, -543712, 18507283, -10413996, 14554437, -8746092, 32232924, 16763880}, + FieldElement{9648505, 10094563, 26416693, 14745928, -30374318, -6472621, 11094161, 15689506, 3140038, -16510092}, + }, + { + FieldElement{-16160072, 5472695, 31895588, 4744994, 8823515, 10365685, -27224800, 9448613, -28774454, 366295}, + FieldElement{19153450, 11523972, -11096490, -6503142, -24647631, 5420647, 28344573, 8041113, 719605, 11671788}, + FieldElement{8678025, 2694440, -6808014, 2517372, 4964326, 11152271, -15432916, -15266516, 27000813, -10195553}, + }, + { + FieldElement{-15157904, 7134312, 8639287, -2814877, -7235688, 10421742, 564065, 5336097, 6750977, -14521026}, + FieldElement{11836410, -3979488, 26297894, 16080799, 23455045, 15735944, 1695823, -8819122, 8169720, 16220347}, + FieldElement{-18115838, 8653647, 17578566, -6092619, -8025777, -16012763, -11144307, -2627664, -5990708, -14166033}, + }, + { + FieldElement{-23308498, -10968312, 15213228, -10081214, -30853605, -11050004, 27884329, 2847284, 2655861, 1738395}, + FieldElement{-27537433, -14253021, -25336301, -8002780, -9370762, 8129821, 21651608, -3239336, -19087449, -11005278}, + FieldElement{1533110, 3437855, 23735889, 459276, 29970501, 11335377, 26030092, 5821408, 10478196, 8544890}, + }, + { + FieldElement{32173121, -16129311, 24896207, 3921497, 22579056, -3410854, 19270449, 12217473, 17789017, -3395995}, + FieldElement{-30552961, -2228401, -15578829, -10147201, 13243889, 517024, 15479401, -3853233, 30460520, 1052596}, + FieldElement{-11614875, 13323618, 32618793, 8175907, -15230173, 12596687, 27491595, -4612359, 3179268, -9478891}, + }, + { + FieldElement{31947069, -14366651, -4640583, -15339921, -15125977, -6039709, -14756777, -16411740, 19072640, -9511060}, + FieldElement{11685058, 11822410, 3158003, -13952594, 33402194, -4165066, 5977896, -5215017, 473099, 5040608}, + FieldElement{-20290863, 8198642, -27410132, 11602123, 1290375, -2799760, 28326862, 1721092, -19558642, -3131606}, + }, + }, + { + { + FieldElement{7881532, 10687937, 7578723, 7738378, -18951012, -2553952, 21820786, 8076149, -27868496, 11538389}, + FieldElement{-19935666, 3899861, 18283497, -6801568, -15728660, -11249211, 8754525, 7446702, -5676054, 5797016}, + FieldElement{-11295600, -3793569, -15782110, -7964573, 12708869, -8456199, 2014099, -9050574, -2369172, -5877341}, + }, + { + FieldElement{-22472376, -11568741, -27682020, 1146375, 18956691, 16640559, 1192730, -3714199, 15123619, 10811505}, + FieldElement{14352098, -3419715, -18942044, 10822655, 32750596, 4699007, -70363, 15776356, -28886779, -11974553}, + FieldElement{-28241164, -8072475, -4978962, -5315317, 29416931, 1847569, -20654173, -16484855, 4714547, -9600655}, + }, + { + FieldElement{15200332, 8368572, 19679101, 15970074, -31872674, 1959451, 24611599, -4543832, -11745876, 12340220}, + FieldElement{12876937, -10480056, 33134381, 6590940, -6307776, 14872440, 9613953, 8241152, 15370987, 9608631}, + FieldElement{-4143277, -12014408, 8446281, -391603, 4407738, 13629032, -7724868, 15866074, -28210621, -8814099}, + }, + { + FieldElement{26660628, -15677655, 8393734, 358047, -7401291, 992988, -23904233, 858697, 20571223, 8420556}, + FieldElement{14620715, 13067227, -15447274, 8264467, 14106269, 15080814, 33531827, 12516406, -21574435, -12476749}, + FieldElement{236881, 10476226, 57258, -14677024, 6472998, 2466984, 17258519, 7256740, 8791136, 15069930}, + }, + { + FieldElement{1276410, -9371918, 22949635, -16322807, -23493039, -5702186, 14711875, 4874229, -30663140, -2331391}, + FieldElement{5855666, 4990204, -13711848, 7294284, -7804282, 1924647, -1423175, -7912378, -33069337, 9234253}, + FieldElement{20590503, -9018988, 31529744, -7352666, -2706834, 10650548, 31559055, -11609587, 18979186, 13396066}, + }, + { + FieldElement{24474287, 4968103, 22267082, 4407354, 24063882, -8325180, -18816887, 13594782, 33514650, 7021958}, + FieldElement{-11566906, -6565505, -21365085, 15928892, -26158305, 4315421, -25948728, -3916677, -21480480, 12868082}, + FieldElement{-28635013, 13504661, 19988037, -2132761, 21078225, 6443208, -21446107, 2244500, -12455797, -8089383}, + }, + { + FieldElement{-30595528, 13793479, -5852820, 319136, -25723172, -6263899, 33086546, 8957937, -15233648, 5540521}, + FieldElement{-11630176, -11503902, -8119500, -7643073, 2620056, 1022908, -23710744, -1568984, -16128528, -14962807}, + FieldElement{23152971, 775386, 27395463, 14006635, -9701118, 4649512, 1689819, 892185, -11513277, -15205948}, + }, + { + FieldElement{9770129, 9586738, 26496094, 4324120, 1556511, -3550024, 27453819, 4763127, -19179614, 5867134}, + FieldElement{-32765025, 1927590, 31726409, -4753295, 23962434, -16019500, 27846559, 5931263, -29749703, -16108455}, + FieldElement{27461885, -2977536, 22380810, 1815854, -23033753, -3031938, 7283490, -15148073, -19526700, 7734629}, + }, + }, + { + { + FieldElement{-8010264, -9590817, -11120403, 6196038, 29344158, -13430885, 7585295, -3176626, 18549497, 15302069}, + FieldElement{-32658337, -6171222, -7672793, -11051681, 6258878, 13504381, 10458790, -6418461, -8872242, 8424746}, + FieldElement{24687205, 8613276, -30667046, -3233545, 1863892, -1830544, 19206234, 7134917, -11284482, -828919}, + }, + { + FieldElement{11334899, -9218022, 8025293, 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FieldElement{-2637277, -13483075, 8488727, -14303896, 12728761, -1622493, 7141596, 11724556, 22761615, -10134141}, + }, + { + FieldElement{16918416, 11729663, -18083579, 3022987, -31015732, -13339659, -28741185, -12227393, 32851222, 11717399}, + FieldElement{11166634, 7338049, -6722523, 4531520, -29468672, -7302055, 31474879, 3483633, -1193175, -4030831}, + FieldElement{-185635, 9921305, 31456609, -13536438, -12013818, 13348923, 33142652, 6546660, -19985279, -3948376}, + }, + { + FieldElement{-32460596, 11266712, -11197107, -7899103, 31703694, 3855903, -8537131, -12833048, -30772034, -15486313}, + FieldElement{-18006477, 12709068, 3991746, -6479188, -21491523, -10550425, -31135347, -16049879, 10928917, 3011958}, + FieldElement{-6957757, -15594337, 31696059, 334240, 29576716, 14796075, -30831056, -12805180, 18008031, 10258577}, + }, + { + FieldElement{-22448644, 15655569, 7018479, -4410003, -30314266, -1201591, -1853465, 1367120, 25127874, 6671743}, + FieldElement{29701166, -14373934, -10878120, 9279288, -17568, 13127210, 21382910, 11042292, 25838796, 4642684}, + FieldElement{-20430234, 14955537, -24126347, 8124619, -5369288, -5990470, 30468147, -13900640, 18423289, 4177476}, + }, + }, +} diff --git a/internal/edwards25519/edwards25519.go b/internal/edwards25519/edwards25519.go new file mode 100644 index 0000000..5f8b994 --- /dev/null +++ b/internal/edwards25519/edwards25519.go @@ -0,0 +1,1771 @@ +// Copyright 2016 The Go Authors. All rights reserved. +// Use of this source code is governed by a BSD-style +// license that can be found in the LICENSE file. + +package edwards25519 + +// This code is a port of the public domain, “ref10” implementation of ed25519 +// from SUPERCOP. + +// FieldElement represents an element of the field GF(2^255 - 19). An element +// t, entries t[0]...t[9], represents the integer t[0]+2^26 t[1]+2^51 t[2]+2^77 +// t[3]+2^102 t[4]+...+2^230 t[9]. Bounds on each t[i] vary depending on +// context. +type FieldElement [10]int32 + +var zero FieldElement + +func FeZero(fe *FieldElement) { + copy(fe[:], zero[:]) +} + +func FeOne(fe *FieldElement) { + FeZero(fe) + fe[0] = 1 +} + +func FeAdd(dst, a, b *FieldElement) { + dst[0] = a[0] + b[0] + dst[1] = a[1] + b[1] + dst[2] = a[2] + b[2] + dst[3] = a[3] + b[3] + dst[4] = a[4] + b[4] + dst[5] = a[5] + b[5] + dst[6] = a[6] + b[6] + dst[7] = a[7] + b[7] + dst[8] = a[8] + b[8] + dst[9] = a[9] + b[9] +} + +func FeSub(dst, a, b *FieldElement) { + dst[0] = a[0] - b[0] + dst[1] = a[1] - b[1] + dst[2] = a[2] - b[2] + dst[3] = a[3] - b[3] + dst[4] = a[4] - b[4] + dst[5] = a[5] - b[5] + dst[6] = a[6] - b[6] + dst[7] = a[7] - b[7] + dst[8] = a[8] - b[8] + dst[9] = a[9] - b[9] +} + +func FeCopy(dst, src *FieldElement) { + copy(dst[:], src[:]) +} + +// Replace (f,g) with (g,g) if b == 1; +// replace (f,g) with (f,g) if b == 0. +// +// Preconditions: b in {0,1}. +func FeCMove(f, g *FieldElement, b int32) { + b = -b + f[0] ^= b & (f[0] ^ g[0]) + f[1] ^= b & (f[1] ^ g[1]) + f[2] ^= b & (f[2] ^ g[2]) + f[3] ^= b & (f[3] ^ g[3]) + f[4] ^= b & (f[4] ^ g[4]) + f[5] ^= b & (f[5] ^ g[5]) + f[6] ^= b & (f[6] ^ g[6]) + f[7] ^= b & (f[7] ^ g[7]) + f[8] ^= b & (f[8] ^ g[8]) + f[9] ^= b & (f[9] ^ g[9]) +} + +func load3(in []byte) int64 { + var r int64 + r = int64(in[0]) + r |= int64(in[1]) << 8 + r |= int64(in[2]) << 16 + return r +} + +func load4(in []byte) int64 { + var r int64 + r = int64(in[0]) + r |= int64(in[1]) << 8 + r |= int64(in[2]) << 16 + r |= int64(in[3]) << 24 + return r +} + +func FeFromBytes(dst *FieldElement, src *[32]byte) { + h0 := load4(src[:]) + h1 := load3(src[4:]) << 6 + h2 := load3(src[7:]) << 5 + h3 := load3(src[10:]) << 3 + h4 := load3(src[13:]) << 2 + h5 := load4(src[16:]) + h6 := load3(src[20:]) << 7 + h7 := load3(src[23:]) << 5 + h8 := load3(src[26:]) << 4 + h9 := (load3(src[29:]) & 8388607) << 2 + + FeCombine(dst, h0, h1, h2, h3, h4, h5, h6, h7, h8, h9) +} + +// FeToBytes marshals h to s. +// Preconditions: +// |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. +// +// Write p=2^255-19; q=floor(h/p). +// Basic claim: q = floor(2^(-255)(h + 19 2^(-25)h9 + 2^(-1))). +// +// Proof: +// Have |h|<=p so |q|<=1 so |19^2 2^(-255) q|<1/4. +// Also have |h-2^230 h9|<2^230 so |19 2^(-255)(h-2^230 h9)|<1/4. +// +// Write y=2^(-1)-19^2 2^(-255)q-19 2^(-255)(h-2^230 h9). +// Then 0> 25 + q = (h[0] + q) >> 26 + q = (h[1] + q) >> 25 + q = (h[2] + q) >> 26 + q = (h[3] + q) >> 25 + q = (h[4] + q) >> 26 + q = (h[5] + q) >> 25 + q = (h[6] + q) >> 26 + q = (h[7] + q) >> 25 + q = (h[8] + q) >> 26 + q = (h[9] + q) >> 25 + + // Goal: Output h-(2^255-19)q, which is between 0 and 2^255-20. + h[0] += 19 * q + // Goal: Output h-2^255 q, which is between 0 and 2^255-20. + + carry[0] = h[0] >> 26 + h[1] += carry[0] + h[0] -= carry[0] << 26 + carry[1] = h[1] >> 25 + h[2] += carry[1] + h[1] -= carry[1] << 25 + carry[2] = h[2] >> 26 + h[3] += carry[2] + h[2] -= carry[2] << 26 + carry[3] = h[3] >> 25 + h[4] += carry[3] + h[3] -= carry[3] << 25 + carry[4] = h[4] >> 26 + h[5] += carry[4] + h[4] -= carry[4] << 26 + carry[5] = h[5] >> 25 + h[6] += carry[5] + h[5] -= carry[5] << 25 + carry[6] = h[6] >> 26 + h[7] += carry[6] + h[6] -= carry[6] << 26 + carry[7] = h[7] >> 25 + h[8] += carry[7] + h[7] -= carry[7] << 25 + carry[8] = h[8] >> 26 + h[9] += carry[8] + h[8] -= carry[8] << 26 + carry[9] = h[9] >> 25 + h[9] -= carry[9] << 25 + // h10 = carry9 + + // Goal: Output h[0]+...+2^255 h10-2^255 q, which is between 0 and 2^255-20. + // Have h[0]+...+2^230 h[9] between 0 and 2^255-1; + // evidently 2^255 h10-2^255 q = 0. + // Goal: Output h[0]+...+2^230 h[9]. + + s[0] = byte(h[0] >> 0) + s[1] = byte(h[0] >> 8) + s[2] = byte(h[0] >> 16) + s[3] = byte((h[0] >> 24) | (h[1] << 2)) + s[4] = byte(h[1] >> 6) + s[5] = byte(h[1] >> 14) + s[6] = byte((h[1] >> 22) | (h[2] << 3)) + s[7] = byte(h[2] >> 5) + s[8] = byte(h[2] >> 13) + s[9] = byte((h[2] >> 21) | (h[3] << 5)) + s[10] = byte(h[3] >> 3) + s[11] = byte(h[3] >> 11) + s[12] = byte((h[3] >> 19) | (h[4] << 6)) + s[13] = byte(h[4] >> 2) + s[14] = byte(h[4] >> 10) + s[15] = byte(h[4] >> 18) + s[16] = byte(h[5] >> 0) + s[17] = byte(h[5] >> 8) + s[18] = byte(h[5] >> 16) + s[19] = byte((h[5] >> 24) | (h[6] << 1)) + s[20] = byte(h[6] >> 7) + s[21] = byte(h[6] >> 15) + s[22] = byte((h[6] >> 23) | (h[7] << 3)) + s[23] = byte(h[7] >> 5) + s[24] = byte(h[7] >> 13) + s[25] = byte((h[7] >> 21) | (h[8] << 4)) + s[26] = byte(h[8] >> 4) + s[27] = byte(h[8] >> 12) + s[28] = byte((h[8] >> 20) | (h[9] << 6)) + s[29] = byte(h[9] >> 2) + s[30] = byte(h[9] >> 10) + s[31] = byte(h[9] >> 18) +} + +func FeIsNegative(f *FieldElement) byte { + var s [32]byte + FeToBytes(&s, f) + return s[0] & 1 +} + +func FeIsNonZero(f *FieldElement) int32 { + var s [32]byte + FeToBytes(&s, f) + var x uint8 + for _, b := range s { + x |= b + } + x |= x >> 4 + x |= x >> 2 + x |= x >> 1 + return int32(x & 1) +} + +// FeNeg sets h = -f +// +// Preconditions: +// |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. +// +// Postconditions: +// |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. +func FeNeg(h, f *FieldElement) { + h[0] = -f[0] + h[1] = -f[1] + h[2] = -f[2] + h[3] = -f[3] + h[4] = -f[4] + h[5] = -f[5] + h[6] = -f[6] + h[7] = -f[7] + h[8] = -f[8] + h[9] = -f[9] +} + +func FeCombine(h *FieldElement, h0, h1, h2, h3, h4, h5, h6, h7, h8, h9 int64) { + var c0, c1, c2, c3, c4, c5, c6, c7, c8, c9 int64 + + /* + |h0| <= (1.1*1.1*2^52*(1+19+19+19+19)+1.1*1.1*2^50*(38+38+38+38+38)) + i.e. |h0| <= 1.2*2^59; narrower ranges for h2, h4, h6, h8 + |h1| <= (1.1*1.1*2^51*(1+1+19+19+19+19+19+19+19+19)) + i.e. |h1| <= 1.5*2^58; narrower ranges for h3, h5, h7, h9 + */ + + c0 = (h0 + (1 << 25)) >> 26 + h1 += c0 + h0 -= c0 << 26 + c4 = (h4 + (1 << 25)) >> 26 + h5 += c4 + h4 -= c4 << 26 + /* |h0| <= 2^25 */ + /* |h4| <= 2^25 */ + /* |h1| <= 1.51*2^58 */ + /* |h5| <= 1.51*2^58 */ + + c1 = (h1 + (1 << 24)) >> 25 + h2 += c1 + h1 -= c1 << 25 + c5 = (h5 + (1 << 24)) >> 25 + h6 += c5 + h5 -= c5 << 25 + /* |h1| <= 2^24; from now on fits into int32 */ + /* |h5| <= 2^24; from now on fits into int32 */ + /* |h2| <= 1.21*2^59 */ + /* |h6| <= 1.21*2^59 */ + + c2 = (h2 + (1 << 25)) >> 26 + h3 += c2 + h2 -= c2 << 26 + c6 = (h6 + (1 << 25)) >> 26 + h7 += c6 + h6 -= c6 << 26 + /* |h2| <= 2^25; from now on fits into int32 unchanged */ + /* |h6| <= 2^25; from now on fits into int32 unchanged */ + /* |h3| <= 1.51*2^58 */ + /* |h7| <= 1.51*2^58 */ + + c3 = (h3 + (1 << 24)) >> 25 + h4 += c3 + h3 -= c3 << 25 + c7 = (h7 + (1 << 24)) >> 25 + h8 += c7 + h7 -= c7 << 25 + /* |h3| <= 2^24; from now on fits into int32 unchanged */ + /* |h7| <= 2^24; from now on fits into int32 unchanged */ + /* |h4| <= 1.52*2^33 */ + /* |h8| <= 1.52*2^33 */ + + c4 = (h4 + (1 << 25)) >> 26 + h5 += c4 + h4 -= c4 << 26 + c8 = (h8 + (1 << 25)) >> 26 + h9 += c8 + h8 -= c8 << 26 + /* |h4| <= 2^25; from now on fits into int32 unchanged */ + /* |h8| <= 2^25; from now on fits into int32 unchanged */ + /* |h5| <= 1.01*2^24 */ + /* |h9| <= 1.51*2^58 */ + + c9 = (h9 + (1 << 24)) >> 25 + h0 += c9 * 19 + h9 -= c9 << 25 + /* |h9| <= 2^24; from now on fits into int32 unchanged */ + /* |h0| <= 1.8*2^37 */ + + c0 = (h0 + (1 << 25)) >> 26 + h1 += c0 + h0 -= c0 << 26 + /* |h0| <= 2^25; from now on fits into int32 unchanged */ + /* |h1| <= 1.01*2^24 */ + + h[0] = int32(h0) + h[1] = int32(h1) + h[2] = int32(h2) + h[3] = int32(h3) + h[4] = int32(h4) + h[5] = int32(h5) + h[6] = int32(h6) + h[7] = int32(h7) + h[8] = int32(h8) + h[9] = int32(h9) +} + +// FeMul calculates h = f * g +// Can overlap h with f or g. +// +// Preconditions: +// |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. +// |g| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. +// +// Postconditions: +// |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. +// +// Notes on implementation strategy: +// +// Using schoolbook multiplication. +// Karatsuba would save a little in some cost models. +// +// Most multiplications by 2 and 19 are 32-bit precomputations; +// cheaper than 64-bit postcomputations. +// +// There is one remaining multiplication by 19 in the carry chain; +// one *19 precomputation can be merged into this, +// but the resulting data flow is considerably less clean. +// +// There are 12 carries below. +// 10 of them are 2-way parallelizable and vectorizable. +// Can get away with 11 carries, but then data flow is much deeper. +// +// With tighter constraints on inputs, can squeeze carries into int32. +func FeMul(h, f, g *FieldElement) { + f0 := int64(f[0]) + f1 := int64(f[1]) + f2 := int64(f[2]) + f3 := int64(f[3]) + f4 := int64(f[4]) + f5 := int64(f[5]) + f6 := int64(f[6]) + f7 := int64(f[7]) + f8 := int64(f[8]) + f9 := int64(f[9]) + + f1_2 := int64(2 * f[1]) + f3_2 := int64(2 * f[3]) + f5_2 := int64(2 * f[5]) + f7_2 := int64(2 * f[7]) + f9_2 := int64(2 * f[9]) + + g0 := int64(g[0]) + g1 := int64(g[1]) + g2 := int64(g[2]) + g3 := int64(g[3]) + g4 := int64(g[4]) + g5 := int64(g[5]) + g6 := int64(g[6]) + g7 := int64(g[7]) + g8 := int64(g[8]) + g9 := int64(g[9]) + + g1_19 := int64(19 * g[1]) /* 1.4*2^29 */ + g2_19 := int64(19 * g[2]) /* 1.4*2^30; still ok */ + g3_19 := int64(19 * g[3]) + g4_19 := int64(19 * g[4]) + g5_19 := int64(19 * g[5]) + g6_19 := int64(19 * g[6]) + g7_19 := int64(19 * g[7]) + g8_19 := int64(19 * g[8]) + g9_19 := int64(19 * g[9]) + + h0 := f0*g0 + f1_2*g9_19 + f2*g8_19 + f3_2*g7_19 + f4*g6_19 + f5_2*g5_19 + f6*g4_19 + f7_2*g3_19 + f8*g2_19 + f9_2*g1_19 + h1 := f0*g1 + f1*g0 + f2*g9_19 + f3*g8_19 + f4*g7_19 + f5*g6_19 + f6*g5_19 + f7*g4_19 + f8*g3_19 + f9*g2_19 + h2 := f0*g2 + f1_2*g1 + f2*g0 + f3_2*g9_19 + f4*g8_19 + f5_2*g7_19 + f6*g6_19 + f7_2*g5_19 + f8*g4_19 + f9_2*g3_19 + h3 := f0*g3 + f1*g2 + f2*g1 + f3*g0 + f4*g9_19 + f5*g8_19 + f6*g7_19 + f7*g6_19 + f8*g5_19 + f9*g4_19 + h4 := f0*g4 + f1_2*g3 + f2*g2 + f3_2*g1 + f4*g0 + f5_2*g9_19 + f6*g8_19 + f7_2*g7_19 + f8*g6_19 + f9_2*g5_19 + h5 := f0*g5 + f1*g4 + f2*g3 + f3*g2 + f4*g1 + f5*g0 + f6*g9_19 + f7*g8_19 + f8*g7_19 + f9*g6_19 + h6 := f0*g6 + f1_2*g5 + f2*g4 + f3_2*g3 + f4*g2 + f5_2*g1 + f6*g0 + f7_2*g9_19 + f8*g8_19 + f9_2*g7_19 + h7 := f0*g7 + f1*g6 + f2*g5 + f3*g4 + f4*g3 + f5*g2 + f6*g1 + f7*g0 + f8*g9_19 + f9*g8_19 + h8 := f0*g8 + f1_2*g7 + f2*g6 + f3_2*g5 + f4*g4 + f5_2*g3 + f6*g2 + f7_2*g1 + f8*g0 + f9_2*g9_19 + h9 := f0*g9 + f1*g8 + f2*g7 + f3*g6 + f4*g5 + f5*g4 + f6*g3 + f7*g2 + f8*g1 + f9*g0 + + FeCombine(h, h0, h1, h2, h3, h4, h5, h6, h7, h8, h9) +} + +func feSquare(f *FieldElement) (h0, h1, h2, h3, h4, h5, h6, h7, h8, h9 int64) { + f0 := int64(f[0]) + f1 := int64(f[1]) + f2 := int64(f[2]) + f3 := int64(f[3]) + f4 := int64(f[4]) + f5 := int64(f[5]) + f6 := int64(f[6]) + f7 := int64(f[7]) + f8 := int64(f[8]) + f9 := int64(f[9]) + f0_2 := int64(2 * f[0]) + f1_2 := int64(2 * f[1]) + f2_2 := int64(2 * f[2]) + f3_2 := int64(2 * f[3]) + f4_2 := int64(2 * f[4]) + f5_2 := int64(2 * f[5]) + f6_2 := int64(2 * f[6]) + f7_2 := int64(2 * f[7]) + f5_38 := 38 * f5 // 1.31*2^30 + f6_19 := 19 * f6 // 1.31*2^30 + f7_38 := 38 * f7 // 1.31*2^30 + f8_19 := 19 * f8 // 1.31*2^30 + f9_38 := 38 * f9 // 1.31*2^30 + + h0 = f0*f0 + f1_2*f9_38 + f2_2*f8_19 + f3_2*f7_38 + f4_2*f6_19 + f5*f5_38 + h1 = f0_2*f1 + f2*f9_38 + f3_2*f8_19 + f4*f7_38 + f5_2*f6_19 + h2 = f0_2*f2 + f1_2*f1 + f3_2*f9_38 + f4_2*f8_19 + f5_2*f7_38 + f6*f6_19 + h3 = f0_2*f3 + f1_2*f2 + f4*f9_38 + f5_2*f8_19 + f6*f7_38 + h4 = f0_2*f4 + f1_2*f3_2 + f2*f2 + f5_2*f9_38 + f6_2*f8_19 + f7*f7_38 + h5 = f0_2*f5 + f1_2*f4 + f2_2*f3 + f6*f9_38 + f7_2*f8_19 + h6 = f0_2*f6 + f1_2*f5_2 + f2_2*f4 + f3_2*f3 + f7_2*f9_38 + f8*f8_19 + h7 = f0_2*f7 + f1_2*f6 + f2_2*f5 + f3_2*f4 + f8*f9_38 + h8 = f0_2*f8 + f1_2*f7_2 + f2_2*f6 + f3_2*f5_2 + f4*f4 + f9*f9_38 + h9 = f0_2*f9 + f1_2*f8 + f2_2*f7 + f3_2*f6 + f4_2*f5 + + return +} + +// FeSquare calculates h = f*f. Can overlap h with f. +// +// Preconditions: +// |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. +// +// Postconditions: +// |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. +func FeSquare(h, f *FieldElement) { + h0, h1, h2, h3, h4, h5, h6, h7, h8, h9 := feSquare(f) + FeCombine(h, h0, h1, h2, h3, h4, h5, h6, h7, h8, h9) +} + +// FeSquare2 sets h = 2 * f * f +// +// Can overlap h with f. +// +// Preconditions: +// |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc. +// +// Postconditions: +// |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc. +// See fe_mul.c for discussion of implementation strategy. +func FeSquare2(h, f *FieldElement) { + h0, h1, h2, h3, h4, h5, h6, h7, h8, h9 := feSquare(f) + + h0 += h0 + h1 += h1 + h2 += h2 + h3 += h3 + h4 += h4 + h5 += h5 + h6 += h6 + h7 += h7 + h8 += h8 + h9 += h9 + + FeCombine(h, h0, h1, h2, h3, h4, h5, h6, h7, h8, h9) +} + +func FeInvert(out, z *FieldElement) { + var t0, t1, t2, t3 FieldElement + var i int + + FeSquare(&t0, z) // 2^1 + FeSquare(&t1, &t0) // 2^2 + for i = 1; i < 2; i++ { // 2^3 + FeSquare(&t1, &t1) + } + FeMul(&t1, z, &t1) // 2^3 + 2^0 + FeMul(&t0, &t0, &t1) // 2^3 + 2^1 + 2^0 + FeSquare(&t2, &t0) // 2^4 + 2^2 + 2^1 + FeMul(&t1, &t1, &t2) // 2^4 + 2^3 + 2^2 + 2^1 + 2^0 + FeSquare(&t2, &t1) // 5,4,3,2,1 + for i = 1; i < 5; i++ { // 9,8,7,6,5 + FeSquare(&t2, &t2) + } + FeMul(&t1, &t2, &t1) // 9,8,7,6,5,4,3,2,1,0 + FeSquare(&t2, &t1) // 10..1 + for i = 1; i < 10; i++ { // 19..10 + FeSquare(&t2, &t2) + } + FeMul(&t2, &t2, &t1) // 19..0 + FeSquare(&t3, &t2) // 20..1 + for i = 1; i < 20; i++ { // 39..20 + FeSquare(&t3, &t3) + } + FeMul(&t2, &t3, &t2) // 39..0 + FeSquare(&t2, &t2) // 40..1 + for i = 1; i < 10; i++ { // 49..10 + FeSquare(&t2, &t2) + } + FeMul(&t1, &t2, &t1) // 49..0 + FeSquare(&t2, &t1) // 50..1 + for i = 1; i < 50; i++ { // 99..50 + FeSquare(&t2, &t2) + } + FeMul(&t2, &t2, &t1) // 99..0 + FeSquare(&t3, &t2) // 100..1 + for i = 1; i < 100; i++ { // 199..100 + FeSquare(&t3, &t3) + } + FeMul(&t2, &t3, &t2) // 199..0 + FeSquare(&t2, &t2) // 200..1 + for i = 1; i < 50; i++ { // 249..50 + FeSquare(&t2, &t2) + } + FeMul(&t1, &t2, &t1) // 249..0 + FeSquare(&t1, &t1) // 250..1 + for i = 1; i < 5; i++ { // 254..5 + FeSquare(&t1, &t1) + } + FeMul(out, &t1, &t0) // 254..5,3,1,0 +} + +func fePow22523(out, z *FieldElement) { + var t0, t1, t2 FieldElement + var i int + + FeSquare(&t0, z) + for i = 1; i < 1; i++ { + FeSquare(&t0, &t0) + } + FeSquare(&t1, &t0) + for i = 1; i < 2; i++ { + FeSquare(&t1, &t1) + } + FeMul(&t1, z, &t1) + FeMul(&t0, &t0, &t1) + FeSquare(&t0, &t0) + for i = 1; i < 1; i++ { + FeSquare(&t0, &t0) + } + FeMul(&t0, &t1, &t0) + FeSquare(&t1, &t0) + for i = 1; i < 5; i++ { + FeSquare(&t1, &t1) + } + FeMul(&t0, &t1, &t0) + FeSquare(&t1, &t0) + for i = 1; i < 10; i++ { + FeSquare(&t1, &t1) + } + FeMul(&t1, &t1, &t0) + FeSquare(&t2, &t1) + for i = 1; i < 20; i++ { + FeSquare(&t2, &t2) + } + FeMul(&t1, &t2, &t1) + FeSquare(&t1, &t1) + for i = 1; i < 10; i++ { + FeSquare(&t1, &t1) + } + FeMul(&t0, &t1, &t0) + FeSquare(&t1, &t0) + for i = 1; i < 50; i++ { + FeSquare(&t1, &t1) + } + FeMul(&t1, &t1, &t0) + FeSquare(&t2, &t1) + for i = 1; i < 100; i++ { + FeSquare(&t2, &t2) + } + FeMul(&t1, &t2, &t1) + FeSquare(&t1, &t1) + for i = 1; i < 50; i++ { + FeSquare(&t1, &t1) + } + FeMul(&t0, &t1, &t0) + FeSquare(&t0, &t0) + for i = 1; i < 2; i++ { + FeSquare(&t0, &t0) + } + FeMul(out, &t0, z) +} + +// Group elements are members of the elliptic curve -x^2 + y^2 = 1 + d * x^2 * +// y^2 where d = -121665/121666. +// +// Several representations are used: +// ProjectiveGroupElement: (X:Y:Z) satisfying x=X/Z, y=Y/Z +// ExtendedGroupElement: (X:Y:Z:T) satisfying x=X/Z, y=Y/Z, XY=ZT +// CompletedGroupElement: ((X:Z),(Y:T)) satisfying x=X/Z, y=Y/T +// PreComputedGroupElement: (y+x,y-x,2dxy) + +type ProjectiveGroupElement struct { + X, Y, Z FieldElement +} + +type ExtendedGroupElement struct { + X, Y, Z, T FieldElement +} + +type CompletedGroupElement struct { + X, Y, Z, T FieldElement +} + +type PreComputedGroupElement struct { + yPlusX, yMinusX, xy2d FieldElement +} + +type CachedGroupElement struct { + yPlusX, yMinusX, Z, T2d FieldElement +} + +func (p *ProjectiveGroupElement) Zero() { + FeZero(&p.X) + FeOne(&p.Y) + FeOne(&p.Z) +} + +func (p *ProjectiveGroupElement) Double(r *CompletedGroupElement) { + var t0 FieldElement + + FeSquare(&r.X, &p.X) + FeSquare(&r.Z, &p.Y) + FeSquare2(&r.T, &p.Z) + FeAdd(&r.Y, &p.X, &p.Y) + FeSquare(&t0, &r.Y) + FeAdd(&r.Y, &r.Z, &r.X) + FeSub(&r.Z, &r.Z, &r.X) + FeSub(&r.X, &t0, &r.Y) + FeSub(&r.T, &r.T, &r.Z) +} + +func (p *ProjectiveGroupElement) ToBytes(s *[32]byte) { + var recip, x, y FieldElement + + FeInvert(&recip, &p.Z) + FeMul(&x, &p.X, &recip) + FeMul(&y, &p.Y, &recip) + FeToBytes(s, &y) + s[31] ^= FeIsNegative(&x) << 7 +} + +func (p *ExtendedGroupElement) Zero() { + FeZero(&p.X) + FeOne(&p.Y) + FeOne(&p.Z) + FeZero(&p.T) +} + +func (p *ExtendedGroupElement) Double(r *CompletedGroupElement) { + var q ProjectiveGroupElement + p.ToProjective(&q) + q.Double(r) +} + +func (p *ExtendedGroupElement) ToCached(r *CachedGroupElement) { + FeAdd(&r.yPlusX, &p.Y, &p.X) + FeSub(&r.yMinusX, &p.Y, &p.X) + FeCopy(&r.Z, &p.Z) + FeMul(&r.T2d, &p.T, &d2) +} + +func (p *ExtendedGroupElement) ToProjective(r *ProjectiveGroupElement) { + FeCopy(&r.X, &p.X) + FeCopy(&r.Y, &p.Y) + FeCopy(&r.Z, &p.Z) +} + +func (p *ExtendedGroupElement) ToBytes(s *[32]byte) { + var recip, x, y FieldElement + + FeInvert(&recip, &p.Z) + FeMul(&x, &p.X, &recip) + FeMul(&y, &p.Y, &recip) + FeToBytes(s, &y) + s[31] ^= FeIsNegative(&x) << 7 +} + +func (p *ExtendedGroupElement) FromBytes(s *[32]byte) bool { + var u, v, v3, vxx, check FieldElement + + FeFromBytes(&p.Y, s) + FeOne(&p.Z) + FeSquare(&u, &p.Y) + FeMul(&v, &u, &d) + FeSub(&u, &u, &p.Z) // y = y^2-1 + FeAdd(&v, &v, &p.Z) // v = dy^2+1 + + FeSquare(&v3, &v) + FeMul(&v3, &v3, &v) // v3 = v^3 + FeSquare(&p.X, &v3) + FeMul(&p.X, &p.X, &v) + FeMul(&p.X, &p.X, &u) // x = uv^7 + + fePow22523(&p.X, &p.X) // x = (uv^7)^((q-5)/8) + FeMul(&p.X, &p.X, &v3) + FeMul(&p.X, &p.X, &u) // x = uv^3(uv^7)^((q-5)/8) + + var tmpX, tmp2 [32]byte + + FeSquare(&vxx, &p.X) + FeMul(&vxx, &vxx, &v) + FeSub(&check, &vxx, &u) // vx^2-u + if FeIsNonZero(&check) == 1 { + FeAdd(&check, &vxx, &u) // vx^2+u + if FeIsNonZero(&check) == 1 { + return false + } + FeMul(&p.X, &p.X, &SqrtM1) + + FeToBytes(&tmpX, &p.X) + for i, v := range tmpX { + tmp2[31-i] = v + } + } + + if FeIsNegative(&p.X) != (s[31] >> 7) { + FeNeg(&p.X, &p.X) + } + + FeMul(&p.T, &p.X, &p.Y) + return true +} + +func (p *CompletedGroupElement) ToProjective(r *ProjectiveGroupElement) { + FeMul(&r.X, &p.X, &p.T) + FeMul(&r.Y, &p.Y, &p.Z) + FeMul(&r.Z, &p.Z, &p.T) +} + +func (p *CompletedGroupElement) ToExtended(r *ExtendedGroupElement) { + FeMul(&r.X, &p.X, &p.T) + FeMul(&r.Y, &p.Y, &p.Z) + FeMul(&r.Z, &p.Z, &p.T) + FeMul(&r.T, &p.X, &p.Y) +} + +func (p *PreComputedGroupElement) Zero() { + FeOne(&p.yPlusX) + FeOne(&p.yMinusX) + FeZero(&p.xy2d) +} + +func geAdd(r *CompletedGroupElement, p *ExtendedGroupElement, q *CachedGroupElement) { + var t0 FieldElement + + FeAdd(&r.X, &p.Y, &p.X) + FeSub(&r.Y, &p.Y, &p.X) + FeMul(&r.Z, &r.X, &q.yPlusX) + FeMul(&r.Y, &r.Y, &q.yMinusX) + FeMul(&r.T, &q.T2d, &p.T) + FeMul(&r.X, &p.Z, &q.Z) + FeAdd(&t0, &r.X, &r.X) + FeSub(&r.X, &r.Z, &r.Y) + FeAdd(&r.Y, &r.Z, &r.Y) + FeAdd(&r.Z, &t0, &r.T) + FeSub(&r.T, &t0, &r.T) +} + +func geSub(r *CompletedGroupElement, p *ExtendedGroupElement, q *CachedGroupElement) { + var t0 FieldElement + + FeAdd(&r.X, &p.Y, &p.X) + FeSub(&r.Y, &p.Y, &p.X) + FeMul(&r.Z, &r.X, &q.yMinusX) + FeMul(&r.Y, &r.Y, &q.yPlusX) + FeMul(&r.T, &q.T2d, &p.T) + FeMul(&r.X, &p.Z, &q.Z) + FeAdd(&t0, &r.X, &r.X) + FeSub(&r.X, &r.Z, &r.Y) + FeAdd(&r.Y, &r.Z, &r.Y) + FeSub(&r.Z, &t0, &r.T) + FeAdd(&r.T, &t0, &r.T) +} + +func geMixedAdd(r *CompletedGroupElement, p *ExtendedGroupElement, q *PreComputedGroupElement) { + var t0 FieldElement + + FeAdd(&r.X, &p.Y, &p.X) + FeSub(&r.Y, &p.Y, &p.X) + FeMul(&r.Z, &r.X, &q.yPlusX) + FeMul(&r.Y, &r.Y, &q.yMinusX) + FeMul(&r.T, &q.xy2d, &p.T) + FeAdd(&t0, &p.Z, &p.Z) + FeSub(&r.X, &r.Z, &r.Y) + FeAdd(&r.Y, &r.Z, &r.Y) + FeAdd(&r.Z, &t0, &r.T) + FeSub(&r.T, &t0, &r.T) +} + +func geMixedSub(r *CompletedGroupElement, p *ExtendedGroupElement, q *PreComputedGroupElement) { + var t0 FieldElement + + FeAdd(&r.X, &p.Y, &p.X) + FeSub(&r.Y, &p.Y, &p.X) + FeMul(&r.Z, &r.X, &q.yMinusX) + FeMul(&r.Y, &r.Y, &q.yPlusX) + FeMul(&r.T, &q.xy2d, &p.T) + FeAdd(&t0, &p.Z, &p.Z) + FeSub(&r.X, &r.Z, &r.Y) + FeAdd(&r.Y, &r.Z, &r.Y) + FeSub(&r.Z, &t0, &r.T) + FeAdd(&r.T, &t0, &r.T) +} + +func slide(r *[256]int8, a *[32]byte) { + for i := range r { + r[i] = int8(1 & (a[i>>3] >> uint(i&7))) + } + + for i := range r { + if r[i] != 0 { + for b := 1; b <= 6 && i+b < 256; b++ { + if r[i+b] != 0 { + if r[i]+(r[i+b]<= -15 { + r[i] -= r[i+b] << uint(b) + for k := i + b; k < 256; k++ { + if r[k] == 0 { + r[k] = 1 + break + } + r[k] = 0 + } + } else { + break + } + } + } + } + } +} + +// GeDoubleScalarMultVartime sets r = a*A + b*B +// where a = a[0]+256*a[1]+...+256^31 a[31]. +// and b = b[0]+256*b[1]+...+256^31 b[31]. +// B is the Ed25519 base point (x,4/5) with x positive. +func GeDoubleScalarMultVartime(r *ProjectiveGroupElement, a *[32]byte, A *ExtendedGroupElement, b *[32]byte) { + var aSlide, bSlide [256]int8 + var Ai [8]CachedGroupElement // A,3A,5A,7A,9A,11A,13A,15A + var t CompletedGroupElement + var u, A2 ExtendedGroupElement + var i int + + slide(&aSlide, a) + slide(&bSlide, b) + + A.ToCached(&Ai[0]) + A.Double(&t) + t.ToExtended(&A2) + + for i := 0; i < 7; i++ { + geAdd(&t, &A2, &Ai[i]) + t.ToExtended(&u) + u.ToCached(&Ai[i+1]) + } + + r.Zero() + + for i = 255; i >= 0; i-- { + if aSlide[i] != 0 || bSlide[i] != 0 { + break + } + } + + for ; i >= 0; i-- { + r.Double(&t) + + if aSlide[i] > 0 { + t.ToExtended(&u) + geAdd(&t, &u, &Ai[aSlide[i]/2]) + } else if aSlide[i] < 0 { + t.ToExtended(&u) + geSub(&t, &u, &Ai[(-aSlide[i])/2]) + } + + if bSlide[i] > 0 { + t.ToExtended(&u) + geMixedAdd(&t, &u, &bi[bSlide[i]/2]) + } else if bSlide[i] < 0 { + t.ToExtended(&u) + geMixedSub(&t, &u, &bi[(-bSlide[i])/2]) + } + + t.ToProjective(r) + } +} + +// equal returns 1 if b == c and 0 otherwise, assuming that b and c are +// non-negative. +func equal(b, c int32) int32 { + x := uint32(b ^ c) + x-- + return int32(x >> 31) +} + +// negative returns 1 if b < 0 and 0 otherwise. +func negative(b int32) int32 { + return (b >> 31) & 1 +} + +func PreComputedGroupElementCMove(t, u *PreComputedGroupElement, b int32) { + FeCMove(&t.yPlusX, &u.yPlusX, b) + FeCMove(&t.yMinusX, &u.yMinusX, b) + FeCMove(&t.xy2d, &u.xy2d, b) +} + +func selectPoint(t *PreComputedGroupElement, pos int32, b int32) { + var minusT PreComputedGroupElement + bNegative := negative(b) + bAbs := b - (((-bNegative) & b) << 1) + + t.Zero() + for i := int32(0); i < 8; i++ { + PreComputedGroupElementCMove(t, &base[pos][i], equal(bAbs, i+1)) + } + FeCopy(&minusT.yPlusX, &t.yMinusX) + FeCopy(&minusT.yMinusX, &t.yPlusX) + FeNeg(&minusT.xy2d, &t.xy2d) + PreComputedGroupElementCMove(t, &minusT, bNegative) +} + +// GeScalarMultBase computes h = a*B, where +// a = a[0]+256*a[1]+...+256^31 a[31] +// B is the Ed25519 base point (x,4/5) with x positive. +// +// Preconditions: +// a[31] <= 127 +func GeScalarMultBase(h *ExtendedGroupElement, a *[32]byte) { + var e [64]int8 + + for i, v := range a { + e[2*i] = int8(v & 15) + e[2*i+1] = int8((v >> 4) & 15) + } + + // each e[i] is between 0 and 15 and e[63] is between 0 and 7. + + carry := int8(0) + for i := 0; i < 63; i++ { + e[i] += carry + carry = (e[i] + 8) >> 4 + e[i] -= carry << 4 + } + e[63] += carry + // each e[i] is between -8 and 8. + + h.Zero() + var t PreComputedGroupElement + var r CompletedGroupElement + for i := int32(1); i < 64; i += 2 { + selectPoint(&t, i/2, int32(e[i])) + geMixedAdd(&r, h, &t) + r.ToExtended(h) + } + + var s ProjectiveGroupElement + + h.Double(&r) + r.ToProjective(&s) + s.Double(&r) + r.ToProjective(&s) + s.Double(&r) + r.ToProjective(&s) + s.Double(&r) + r.ToExtended(h) + + for i := int32(0); i < 64; i += 2 { + selectPoint(&t, i/2, int32(e[i])) + geMixedAdd(&r, h, &t) + r.ToExtended(h) + } +} + +// The scalars are GF(2^252 + 27742317777372353535851937790883648493). + +// Input: +// a[0]+256*a[1]+...+256^31*a[31] = a +// b[0]+256*b[1]+...+256^31*b[31] = b +// c[0]+256*c[1]+...+256^31*c[31] = c +// +// Output: +// s[0]+256*s[1]+...+256^31*s[31] = (ab+c) mod l +// where l = 2^252 + 27742317777372353535851937790883648493. +func ScMulAdd(s, a, b, c *[32]byte) { + a0 := 2097151 & load3(a[:]) + a1 := 2097151 & (load4(a[2:]) >> 5) + a2 := 2097151 & (load3(a[5:]) >> 2) + a3 := 2097151 & (load4(a[7:]) >> 7) + a4 := 2097151 & (load4(a[10:]) >> 4) + a5 := 2097151 & (load3(a[13:]) >> 1) + a6 := 2097151 & (load4(a[15:]) >> 6) + a7 := 2097151 & (load3(a[18:]) >> 3) + a8 := 2097151 & load3(a[21:]) + a9 := 2097151 & (load4(a[23:]) >> 5) + a10 := 2097151 & (load3(a[26:]) >> 2) + a11 := (load4(a[28:]) >> 7) + b0 := 2097151 & load3(b[:]) + b1 := 2097151 & (load4(b[2:]) >> 5) + b2 := 2097151 & (load3(b[5:]) >> 2) + b3 := 2097151 & (load4(b[7:]) >> 7) + b4 := 2097151 & (load4(b[10:]) >> 4) + b5 := 2097151 & (load3(b[13:]) >> 1) + b6 := 2097151 & (load4(b[15:]) >> 6) + b7 := 2097151 & (load3(b[18:]) >> 3) + b8 := 2097151 & load3(b[21:]) + b9 := 2097151 & (load4(b[23:]) >> 5) + b10 := 2097151 & (load3(b[26:]) >> 2) + b11 := (load4(b[28:]) >> 7) + c0 := 2097151 & load3(c[:]) + c1 := 2097151 & (load4(c[2:]) >> 5) + c2 := 2097151 & (load3(c[5:]) >> 2) + c3 := 2097151 & (load4(c[7:]) >> 7) + c4 := 2097151 & (load4(c[10:]) >> 4) + c5 := 2097151 & (load3(c[13:]) >> 1) + c6 := 2097151 & (load4(c[15:]) >> 6) + c7 := 2097151 & (load3(c[18:]) >> 3) + c8 := 2097151 & load3(c[21:]) + c9 := 2097151 & (load4(c[23:]) >> 5) + c10 := 2097151 & (load3(c[26:]) >> 2) + c11 := (load4(c[28:]) >> 7) + var carry [23]int64 + + s0 := c0 + a0*b0 + s1 := c1 + a0*b1 + a1*b0 + s2 := c2 + a0*b2 + a1*b1 + a2*b0 + s3 := c3 + a0*b3 + a1*b2 + a2*b1 + a3*b0 + s4 := c4 + a0*b4 + a1*b3 + a2*b2 + a3*b1 + a4*b0 + s5 := c5 + a0*b5 + a1*b4 + a2*b3 + a3*b2 + a4*b1 + a5*b0 + s6 := c6 + a0*b6 + a1*b5 + a2*b4 + a3*b3 + a4*b2 + a5*b1 + a6*b0 + s7 := c7 + a0*b7 + a1*b6 + a2*b5 + a3*b4 + a4*b3 + a5*b2 + a6*b1 + a7*b0 + s8 := c8 + a0*b8 + a1*b7 + a2*b6 + a3*b5 + a4*b4 + a5*b3 + a6*b2 + a7*b1 + a8*b0 + s9 := c9 + a0*b9 + a1*b8 + a2*b7 + a3*b6 + a4*b5 + a5*b4 + a6*b3 + a7*b2 + a8*b1 + a9*b0 + s10 := c10 + a0*b10 + a1*b9 + a2*b8 + a3*b7 + a4*b6 + a5*b5 + a6*b4 + a7*b3 + a8*b2 + a9*b1 + a10*b0 + s11 := c11 + a0*b11 + a1*b10 + a2*b9 + a3*b8 + a4*b7 + a5*b6 + a6*b5 + a7*b4 + a8*b3 + a9*b2 + a10*b1 + a11*b0 + s12 := a1*b11 + a2*b10 + a3*b9 + a4*b8 + a5*b7 + a6*b6 + a7*b5 + a8*b4 + a9*b3 + a10*b2 + a11*b1 + s13 := a2*b11 + a3*b10 + a4*b9 + a5*b8 + a6*b7 + a7*b6 + a8*b5 + a9*b4 + a10*b3 + a11*b2 + s14 := a3*b11 + a4*b10 + a5*b9 + a6*b8 + a7*b7 + a8*b6 + a9*b5 + a10*b4 + a11*b3 + s15 := a4*b11 + a5*b10 + a6*b9 + a7*b8 + a8*b7 + a9*b6 + a10*b5 + a11*b4 + s16 := a5*b11 + a6*b10 + a7*b9 + a8*b8 + a9*b7 + a10*b6 + a11*b5 + s17 := a6*b11 + a7*b10 + a8*b9 + a9*b8 + a10*b7 + a11*b6 + s18 := a7*b11 + a8*b10 + a9*b9 + a10*b8 + a11*b7 + s19 := a8*b11 + a9*b10 + a10*b9 + a11*b8 + s20 := a9*b11 + a10*b10 + a11*b9 + s21 := a10*b11 + a11*b10 + s22 := a11 * b11 + s23 := int64(0) + + carry[0] = (s0 + (1 << 20)) >> 21 + s1 += carry[0] + s0 -= carry[0] << 21 + carry[2] = (s2 + (1 << 20)) >> 21 + s3 += carry[2] + s2 -= carry[2] << 21 + carry[4] = (s4 + (1 << 20)) >> 21 + s5 += carry[4] + s4 -= carry[4] << 21 + carry[6] = (s6 + (1 << 20)) >> 21 + s7 += carry[6] + s6 -= carry[6] << 21 + carry[8] = (s8 + (1 << 20)) >> 21 + s9 += carry[8] + s8 -= carry[8] << 21 + carry[10] = (s10 + (1 << 20)) >> 21 + s11 += carry[10] + s10 -= carry[10] << 21 + carry[12] = (s12 + (1 << 20)) >> 21 + s13 += carry[12] + s12 -= carry[12] << 21 + carry[14] = (s14 + (1 << 20)) >> 21 + s15 += carry[14] + s14 -= carry[14] << 21 + carry[16] = (s16 + (1 << 20)) >> 21 + s17 += carry[16] + s16 -= carry[16] << 21 + carry[18] = (s18 + (1 << 20)) >> 21 + s19 += carry[18] + s18 -= carry[18] << 21 + carry[20] = (s20 + (1 << 20)) >> 21 + s21 += carry[20] + s20 -= carry[20] << 21 + carry[22] = (s22 + (1 << 20)) >> 21 + s23 += carry[22] + s22 -= carry[22] << 21 + + carry[1] = (s1 + (1 << 20)) >> 21 + s2 += carry[1] + s1 -= carry[1] << 21 + carry[3] = (s3 + (1 << 20)) >> 21 + s4 += carry[3] + s3 -= carry[3] << 21 + carry[5] = (s5 + (1 << 20)) >> 21 + s6 += carry[5] + s5 -= carry[5] << 21 + carry[7] = (s7 + (1 << 20)) >> 21 + s8 += carry[7] + s7 -= carry[7] << 21 + carry[9] = (s9 + (1 << 20)) >> 21 + s10 += carry[9] + s9 -= carry[9] << 21 + carry[11] = (s11 + (1 << 20)) >> 21 + s12 += carry[11] + s11 -= carry[11] << 21 + carry[13] = (s13 + (1 << 20)) >> 21 + s14 += carry[13] + s13 -= carry[13] << 21 + carry[15] = (s15 + (1 << 20)) >> 21 + s16 += carry[15] + s15 -= carry[15] << 21 + carry[17] = (s17 + (1 << 20)) >> 21 + s18 += carry[17] + s17 -= carry[17] << 21 + carry[19] = (s19 + (1 << 20)) >> 21 + s20 += carry[19] + s19 -= carry[19] << 21 + carry[21] = (s21 + (1 << 20)) >> 21 + s22 += carry[21] + s21 -= carry[21] << 21 + + s11 += s23 * 666643 + s12 += s23 * 470296 + s13 += s23 * 654183 + s14 -= s23 * 997805 + s15 += s23 * 136657 + s16 -= s23 * 683901 + s23 = 0 + + s10 += s22 * 666643 + s11 += s22 * 470296 + s12 += s22 * 654183 + s13 -= s22 * 997805 + s14 += s22 * 136657 + s15 -= s22 * 683901 + s22 = 0 + + s9 += s21 * 666643 + s10 += s21 * 470296 + s11 += s21 * 654183 + s12 -= s21 * 997805 + s13 += s21 * 136657 + s14 -= s21 * 683901 + s21 = 0 + + s8 += s20 * 666643 + s9 += s20 * 470296 + s10 += s20 * 654183 + s11 -= s20 * 997805 + s12 += s20 * 136657 + s13 -= s20 * 683901 + s20 = 0 + + s7 += s19 * 666643 + s8 += s19 * 470296 + s9 += s19 * 654183 + s10 -= s19 * 997805 + s11 += s19 * 136657 + s12 -= s19 * 683901 + s19 = 0 + + s6 += s18 * 666643 + s7 += s18 * 470296 + s8 += s18 * 654183 + s9 -= s18 * 997805 + s10 += s18 * 136657 + s11 -= s18 * 683901 + s18 = 0 + + carry[6] = (s6 + (1 << 20)) >> 21 + s7 += carry[6] + s6 -= carry[6] << 21 + carry[8] = (s8 + (1 << 20)) >> 21 + s9 += carry[8] + s8 -= carry[8] << 21 + carry[10] = (s10 + (1 << 20)) >> 21 + s11 += carry[10] + s10 -= carry[10] << 21 + carry[12] = (s12 + (1 << 20)) >> 21 + s13 += carry[12] + s12 -= carry[12] << 21 + carry[14] = (s14 + (1 << 20)) >> 21 + s15 += carry[14] + s14 -= carry[14] << 21 + carry[16] = (s16 + (1 << 20)) >> 21 + s17 += carry[16] + s16 -= carry[16] << 21 + + carry[7] = (s7 + (1 << 20)) >> 21 + s8 += carry[7] + s7 -= carry[7] << 21 + carry[9] = (s9 + (1 << 20)) >> 21 + s10 += carry[9] + s9 -= carry[9] << 21 + carry[11] = (s11 + (1 << 20)) >> 21 + s12 += carry[11] + s11 -= carry[11] << 21 + carry[13] = (s13 + (1 << 20)) >> 21 + s14 += carry[13] + s13 -= carry[13] << 21 + carry[15] = (s15 + (1 << 20)) >> 21 + s16 += carry[15] + s15 -= carry[15] << 21 + + s5 += s17 * 666643 + s6 += s17 * 470296 + s7 += s17 * 654183 + s8 -= s17 * 997805 + s9 += s17 * 136657 + s10 -= s17 * 683901 + s17 = 0 + + s4 += s16 * 666643 + s5 += s16 * 470296 + s6 += s16 * 654183 + s7 -= s16 * 997805 + s8 += s16 * 136657 + s9 -= s16 * 683901 + s16 = 0 + + s3 += s15 * 666643 + s4 += s15 * 470296 + s5 += s15 * 654183 + s6 -= s15 * 997805 + s7 += s15 * 136657 + s8 -= s15 * 683901 + s15 = 0 + + s2 += s14 * 666643 + s3 += s14 * 470296 + s4 += s14 * 654183 + s5 -= s14 * 997805 + s6 += s14 * 136657 + s7 -= s14 * 683901 + s14 = 0 + + s1 += s13 * 666643 + s2 += s13 * 470296 + s3 += s13 * 654183 + s4 -= s13 * 997805 + s5 += s13 * 136657 + s6 -= s13 * 683901 + s13 = 0 + + s0 += s12 * 666643 + s1 += s12 * 470296 + s2 += s12 * 654183 + s3 -= s12 * 997805 + s4 += s12 * 136657 + s5 -= s12 * 683901 + s12 = 0 + + carry[0] = (s0 + (1 << 20)) >> 21 + s1 += carry[0] + s0 -= carry[0] << 21 + carry[2] = (s2 + (1 << 20)) >> 21 + s3 += carry[2] + s2 -= carry[2] << 21 + carry[4] = (s4 + (1 << 20)) >> 21 + s5 += carry[4] + s4 -= carry[4] << 21 + carry[6] = (s6 + (1 << 20)) >> 21 + s7 += carry[6] + s6 -= carry[6] << 21 + carry[8] = (s8 + (1 << 20)) >> 21 + s9 += carry[8] + s8 -= carry[8] << 21 + carry[10] = (s10 + (1 << 20)) >> 21 + s11 += carry[10] + s10 -= carry[10] << 21 + + carry[1] = (s1 + (1 << 20)) >> 21 + s2 += carry[1] + s1 -= carry[1] << 21 + carry[3] = (s3 + (1 << 20)) >> 21 + s4 += carry[3] + s3 -= carry[3] << 21 + carry[5] = (s5 + (1 << 20)) >> 21 + s6 += carry[5] + s5 -= carry[5] << 21 + carry[7] = (s7 + (1 << 20)) >> 21 + s8 += carry[7] + s7 -= carry[7] << 21 + carry[9] = (s9 + (1 << 20)) >> 21 + s10 += carry[9] + s9 -= carry[9] << 21 + carry[11] = (s11 + (1 << 20)) >> 21 + s12 += carry[11] + s11 -= carry[11] << 21 + + s0 += s12 * 666643 + s1 += s12 * 470296 + s2 += s12 * 654183 + s3 -= s12 * 997805 + s4 += s12 * 136657 + s5 -= s12 * 683901 + s12 = 0 + + carry[0] = s0 >> 21 + s1 += carry[0] + s0 -= carry[0] << 21 + carry[1] = s1 >> 21 + s2 += carry[1] + s1 -= carry[1] << 21 + carry[2] = s2 >> 21 + s3 += carry[2] + s2 -= carry[2] << 21 + carry[3] = s3 >> 21 + s4 += carry[3] + s3 -= carry[3] << 21 + carry[4] = s4 >> 21 + s5 += carry[4] + s4 -= carry[4] << 21 + carry[5] = s5 >> 21 + s6 += carry[5] + s5 -= carry[5] << 21 + carry[6] = s6 >> 21 + s7 += carry[6] + s6 -= carry[6] << 21 + carry[7] = s7 >> 21 + s8 += carry[7] + s7 -= carry[7] << 21 + carry[8] = s8 >> 21 + s9 += carry[8] + s8 -= carry[8] << 21 + carry[9] = s9 >> 21 + s10 += carry[9] + s9 -= carry[9] << 21 + carry[10] = s10 >> 21 + s11 += carry[10] + s10 -= carry[10] << 21 + carry[11] = s11 >> 21 + s12 += carry[11] + s11 -= carry[11] << 21 + + s0 += s12 * 666643 + s1 += s12 * 470296 + s2 += s12 * 654183 + s3 -= s12 * 997805 + s4 += s12 * 136657 + s5 -= s12 * 683901 + s12 = 0 + + carry[0] = s0 >> 21 + s1 += carry[0] + s0 -= carry[0] << 21 + carry[1] = s1 >> 21 + s2 += carry[1] + s1 -= carry[1] << 21 + carry[2] = s2 >> 21 + s3 += carry[2] + s2 -= carry[2] << 21 + carry[3] = s3 >> 21 + s4 += carry[3] + s3 -= carry[3] << 21 + carry[4] = s4 >> 21 + s5 += carry[4] + s4 -= carry[4] << 21 + carry[5] = s5 >> 21 + s6 += carry[5] + s5 -= carry[5] << 21 + carry[6] = s6 >> 21 + s7 += carry[6] + s6 -= carry[6] << 21 + carry[7] = s7 >> 21 + s8 += carry[7] + s7 -= carry[7] << 21 + carry[8] = s8 >> 21 + s9 += carry[8] + s8 -= carry[8] << 21 + carry[9] = s9 >> 21 + s10 += carry[9] + s9 -= carry[9] << 21 + carry[10] = s10 >> 21 + s11 += carry[10] + s10 -= carry[10] << 21 + + s[0] = byte(s0 >> 0) + s[1] = byte(s0 >> 8) + s[2] = byte((s0 >> 16) | (s1 << 5)) + s[3] = byte(s1 >> 3) + s[4] = byte(s1 >> 11) + s[5] = byte((s1 >> 19) | (s2 << 2)) + s[6] = byte(s2 >> 6) + s[7] = byte((s2 >> 14) | (s3 << 7)) + s[8] = byte(s3 >> 1) + s[9] = byte(s3 >> 9) + s[10] = byte((s3 >> 17) | (s4 << 4)) + s[11] = byte(s4 >> 4) + s[12] = byte(s4 >> 12) + s[13] = byte((s4 >> 20) | (s5 << 1)) + s[14] = byte(s5 >> 7) + s[15] = byte((s5 >> 15) | (s6 << 6)) + s[16] = byte(s6 >> 2) + s[17] = byte(s6 >> 10) + s[18] = byte((s6 >> 18) | (s7 << 3)) + s[19] = byte(s7 >> 5) + s[20] = byte(s7 >> 13) + s[21] = byte(s8 >> 0) + s[22] = byte(s8 >> 8) + s[23] = byte((s8 >> 16) | (s9 << 5)) + s[24] = byte(s9 >> 3) + s[25] = byte(s9 >> 11) + s[26] = byte((s9 >> 19) | (s10 << 2)) + s[27] = byte(s10 >> 6) + s[28] = byte((s10 >> 14) | (s11 << 7)) + s[29] = byte(s11 >> 1) + s[30] = byte(s11 >> 9) + s[31] = byte(s11 >> 17) +} + +// Input: +// s[0]+256*s[1]+...+256^63*s[63] = s +// +// Output: +// s[0]+256*s[1]+...+256^31*s[31] = s mod l +// where l = 2^252 + 27742317777372353535851937790883648493. +func ScReduce(out *[32]byte, s *[64]byte) { + s0 := 2097151 & load3(s[:]) + s1 := 2097151 & (load4(s[2:]) >> 5) + s2 := 2097151 & (load3(s[5:]) >> 2) + s3 := 2097151 & (load4(s[7:]) >> 7) + s4 := 2097151 & (load4(s[10:]) >> 4) + s5 := 2097151 & (load3(s[13:]) >> 1) + s6 := 2097151 & (load4(s[15:]) >> 6) + s7 := 2097151 & (load3(s[18:]) >> 3) + s8 := 2097151 & load3(s[21:]) + s9 := 2097151 & (load4(s[23:]) >> 5) + s10 := 2097151 & (load3(s[26:]) >> 2) + s11 := 2097151 & (load4(s[28:]) >> 7) + s12 := 2097151 & (load4(s[31:]) >> 4) + s13 := 2097151 & (load3(s[34:]) >> 1) + s14 := 2097151 & (load4(s[36:]) >> 6) + s15 := 2097151 & (load3(s[39:]) >> 3) + s16 := 2097151 & load3(s[42:]) + s17 := 2097151 & (load4(s[44:]) >> 5) + s18 := 2097151 & (load3(s[47:]) >> 2) + s19 := 2097151 & (load4(s[49:]) >> 7) + s20 := 2097151 & (load4(s[52:]) >> 4) + s21 := 2097151 & (load3(s[55:]) >> 1) + s22 := 2097151 & (load4(s[57:]) >> 6) + s23 := (load4(s[60:]) >> 3) + + s11 += s23 * 666643 + s12 += s23 * 470296 + s13 += s23 * 654183 + s14 -= s23 * 997805 + s15 += s23 * 136657 + s16 -= s23 * 683901 + s23 = 0 + + s10 += s22 * 666643 + s11 += s22 * 470296 + s12 += s22 * 654183 + s13 -= s22 * 997805 + s14 += s22 * 136657 + s15 -= s22 * 683901 + s22 = 0 + + s9 += s21 * 666643 + s10 += s21 * 470296 + s11 += s21 * 654183 + s12 -= s21 * 997805 + s13 += s21 * 136657 + s14 -= s21 * 683901 + s21 = 0 + + s8 += s20 * 666643 + s9 += s20 * 470296 + s10 += s20 * 654183 + s11 -= s20 * 997805 + s12 += s20 * 136657 + s13 -= s20 * 683901 + s20 = 0 + + s7 += s19 * 666643 + s8 += s19 * 470296 + s9 += s19 * 654183 + s10 -= s19 * 997805 + s11 += s19 * 136657 + s12 -= s19 * 683901 + s19 = 0 + + s6 += s18 * 666643 + s7 += s18 * 470296 + s8 += s18 * 654183 + s9 -= s18 * 997805 + s10 += s18 * 136657 + s11 -= s18 * 683901 + s18 = 0 + + var carry [17]int64 + + carry[6] = (s6 + (1 << 20)) >> 21 + s7 += carry[6] + s6 -= carry[6] << 21 + carry[8] = (s8 + (1 << 20)) >> 21 + s9 += carry[8] + s8 -= carry[8] << 21 + carry[10] = (s10 + (1 << 20)) >> 21 + s11 += carry[10] + s10 -= carry[10] << 21 + carry[12] = (s12 + (1 << 20)) >> 21 + s13 += carry[12] + s12 -= carry[12] << 21 + carry[14] = (s14 + (1 << 20)) >> 21 + s15 += carry[14] + s14 -= carry[14] << 21 + carry[16] = (s16 + (1 << 20)) >> 21 + s17 += carry[16] + s16 -= carry[16] << 21 + + carry[7] = (s7 + (1 << 20)) >> 21 + s8 += carry[7] + s7 -= carry[7] << 21 + carry[9] = (s9 + (1 << 20)) >> 21 + s10 += carry[9] + s9 -= carry[9] << 21 + carry[11] = (s11 + (1 << 20)) >> 21 + s12 += carry[11] + s11 -= carry[11] << 21 + carry[13] = (s13 + (1 << 20)) >> 21 + s14 += carry[13] + s13 -= carry[13] << 21 + carry[15] = (s15 + (1 << 20)) >> 21 + s16 += carry[15] + s15 -= carry[15] << 21 + + s5 += s17 * 666643 + s6 += s17 * 470296 + s7 += s17 * 654183 + s8 -= s17 * 997805 + s9 += s17 * 136657 + s10 -= s17 * 683901 + s17 = 0 + + s4 += s16 * 666643 + s5 += s16 * 470296 + s6 += s16 * 654183 + s7 -= s16 * 997805 + s8 += s16 * 136657 + s9 -= s16 * 683901 + s16 = 0 + + s3 += s15 * 666643 + s4 += s15 * 470296 + s5 += s15 * 654183 + s6 -= s15 * 997805 + s7 += s15 * 136657 + s8 -= s15 * 683901 + s15 = 0 + + s2 += s14 * 666643 + s3 += s14 * 470296 + s4 += s14 * 654183 + s5 -= s14 * 997805 + s6 += s14 * 136657 + s7 -= s14 * 683901 + s14 = 0 + + s1 += s13 * 666643 + s2 += s13 * 470296 + s3 += s13 * 654183 + s4 -= s13 * 997805 + s5 += s13 * 136657 + s6 -= s13 * 683901 + s13 = 0 + + s0 += s12 * 666643 + s1 += s12 * 470296 + s2 += s12 * 654183 + s3 -= s12 * 997805 + s4 += s12 * 136657 + s5 -= s12 * 683901 + s12 = 0 + + carry[0] = (s0 + (1 << 20)) >> 21 + s1 += carry[0] + s0 -= carry[0] << 21 + carry[2] = (s2 + (1 << 20)) >> 21 + s3 += carry[2] + s2 -= carry[2] << 21 + carry[4] = (s4 + (1 << 20)) >> 21 + s5 += carry[4] + s4 -= carry[4] << 21 + carry[6] = (s6 + (1 << 20)) >> 21 + s7 += carry[6] + s6 -= carry[6] << 21 + carry[8] = (s8 + (1 << 20)) >> 21 + s9 += carry[8] + s8 -= carry[8] << 21 + carry[10] = (s10 + (1 << 20)) >> 21 + s11 += carry[10] + s10 -= carry[10] << 21 + + carry[1] = (s1 + (1 << 20)) >> 21 + s2 += carry[1] + s1 -= carry[1] << 21 + carry[3] = (s3 + (1 << 20)) >> 21 + s4 += carry[3] + s3 -= carry[3] << 21 + carry[5] = (s5 + (1 << 20)) >> 21 + s6 += carry[5] + s5 -= carry[5] << 21 + carry[7] = (s7 + (1 << 20)) >> 21 + s8 += carry[7] + s7 -= carry[7] << 21 + carry[9] = (s9 + (1 << 20)) >> 21 + s10 += carry[9] + s9 -= carry[9] << 21 + carry[11] = (s11 + (1 << 20)) >> 21 + s12 += carry[11] + s11 -= carry[11] << 21 + + s0 += s12 * 666643 + s1 += s12 * 470296 + s2 += s12 * 654183 + s3 -= s12 * 997805 + s4 += s12 * 136657 + s5 -= s12 * 683901 + s12 = 0 + + carry[0] = s0 >> 21 + s1 += carry[0] + s0 -= carry[0] << 21 + carry[1] = s1 >> 21 + s2 += carry[1] + s1 -= carry[1] << 21 + carry[2] = s2 >> 21 + s3 += carry[2] + s2 -= carry[2] << 21 + carry[3] = s3 >> 21 + s4 += carry[3] + s3 -= carry[3] << 21 + carry[4] = s4 >> 21 + s5 += carry[4] + s4 -= carry[4] << 21 + carry[5] = s5 >> 21 + s6 += carry[5] + s5 -= carry[5] << 21 + carry[6] = s6 >> 21 + s7 += carry[6] + s6 -= carry[6] << 21 + carry[7] = s7 >> 21 + s8 += carry[7] + s7 -= carry[7] << 21 + carry[8] = s8 >> 21 + s9 += carry[8] + s8 -= carry[8] << 21 + carry[9] = s9 >> 21 + s10 += carry[9] + s9 -= carry[9] << 21 + carry[10] = s10 >> 21 + s11 += carry[10] + s10 -= carry[10] << 21 + carry[11] = s11 >> 21 + s12 += carry[11] + s11 -= carry[11] << 21 + + s0 += s12 * 666643 + s1 += s12 * 470296 + s2 += s12 * 654183 + s3 -= s12 * 997805 + s4 += s12 * 136657 + s5 -= s12 * 683901 + s12 = 0 + + carry[0] = s0 >> 21 + s1 += carry[0] + s0 -= carry[0] << 21 + carry[1] = s1 >> 21 + s2 += carry[1] + s1 -= carry[1] << 21 + carry[2] = s2 >> 21 + s3 += carry[2] + s2 -= carry[2] << 21 + carry[3] = s3 >> 21 + s4 += carry[3] + s3 -= carry[3] << 21 + carry[4] = s4 >> 21 + s5 += carry[4] + s4 -= carry[4] << 21 + carry[5] = s5 >> 21 + s6 += carry[5] + s5 -= carry[5] << 21 + carry[6] = s6 >> 21 + s7 += carry[6] + s6 -= carry[6] << 21 + carry[7] = s7 >> 21 + s8 += carry[7] + s7 -= carry[7] << 21 + carry[8] = s8 >> 21 + s9 += carry[8] + s8 -= carry[8] << 21 + carry[9] = s9 >> 21 + s10 += carry[9] + s9 -= carry[9] << 21 + carry[10] = s10 >> 21 + s11 += carry[10] + s10 -= carry[10] << 21 + + out[0] = byte(s0 >> 0) + out[1] = byte(s0 >> 8) + out[2] = byte((s0 >> 16) | (s1 << 5)) + out[3] = byte(s1 >> 3) + out[4] = byte(s1 >> 11) + out[5] = byte((s1 >> 19) | (s2 << 2)) + out[6] = byte(s2 >> 6) + out[7] = byte((s2 >> 14) | (s3 << 7)) + out[8] = byte(s3 >> 1) + out[9] = byte(s3 >> 9) + out[10] = byte((s3 >> 17) | (s4 << 4)) + out[11] = byte(s4 >> 4) + out[12] = byte(s4 >> 12) + out[13] = byte((s4 >> 20) | (s5 << 1)) + out[14] = byte(s5 >> 7) + out[15] = byte((s5 >> 15) | (s6 << 6)) + out[16] = byte(s6 >> 2) + out[17] = byte(s6 >> 10) + out[18] = byte((s6 >> 18) | (s7 << 3)) + out[19] = byte(s7 >> 5) + out[20] = byte(s7 >> 13) + out[21] = byte(s8 >> 0) + out[22] = byte(s8 >> 8) + out[23] = byte((s8 >> 16) | (s9 << 5)) + out[24] = byte(s9 >> 3) + out[25] = byte(s9 >> 11) + out[26] = byte((s9 >> 19) | (s10 << 2)) + out[27] = byte(s10 >> 6) + out[28] = byte((s10 >> 14) | (s11 << 7)) + out[29] = byte(s11 >> 1) + out[30] = byte(s11 >> 9) + out[31] = byte(s11 >> 17) +} From 26b25b0ac0a2ce68d46900ae48472dfd30b59f5a Mon Sep 17 00:00:00 2001 From: George Tankersley Date: Mon, 22 May 2017 00:00:00 +0000 Subject: [PATCH 02/46] Add LICENSE --- LICENSE | 27 +++++++++++++++++++++++++++ const.go | 4 ++++ ed25519.go | 4 ++++ ed25519_test.go | 4 ++++ internal/LICENSE | 27 --------------------------- 5 files changed, 39 insertions(+), 27 deletions(-) create mode 100644 LICENSE delete mode 100644 internal/LICENSE diff --git a/LICENSE b/LICENSE new file mode 100644 index 0000000..80bb50e --- /dev/null +++ b/LICENSE @@ -0,0 +1,27 @@ +Copyright 2016 The Go Authors +Copyright 2017 George Tankersley + +Redistribution and use in source and binary forms, with or without +modification, are permitted provided that the following conditions are met: + +1. Redistributions of source code must retain the above copyright notice, this +list of conditions and the following disclaimer. + +2. Redistributions in binary form must reproduce the above copyright notice, +this list of conditions and the following disclaimer in the documentation +and/or other materials provided with the distribution. + +3. Neither the name of the copyright holder nor the names of its contributors +may be used to endorse or promote products derived from this software without +specific prior written permission. + +THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND +ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED +WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE +DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE +FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL +DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR +SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER +CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, +OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE +OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. diff --git a/const.go b/const.go index a92ff0d..ab0e1d0 100644 --- a/const.go +++ b/const.go @@ -1,3 +1,7 @@ +// Copyright (c) 2017 George Tankersley. All rights reserved. +// Use of this source code is governed by a BSD-style +// license that can be found in the LICENSE file. + package ed25519 import ( diff --git a/ed25519.go b/ed25519.go index af14a95..b4c3085 100644 --- a/ed25519.go +++ b/ed25519.go @@ -1,3 +1,7 @@ +// Copyright (c) 2017 George Tankersley. All rights reserved. +// Use of this source code is governed by a BSD-style +// license that can be found in the LICENSE file. + // Implements an elliptic.Curve interface over the ed25519 curve. package ed25519 diff --git a/ed25519_test.go b/ed25519_test.go index 2734089..840eca9 100644 --- a/ed25519_test.go +++ b/ed25519_test.go @@ -1,3 +1,7 @@ +// Copyright (c) 2017 George Tankersley. All rights reserved. +// Use of this source code is governed by a BSD-style +// license that can be found in the LICENSE file. + package ed25519 import ( diff --git a/internal/LICENSE b/internal/LICENSE deleted file mode 100644 index 6a66aea..0000000 --- a/internal/LICENSE +++ /dev/null @@ -1,27 +0,0 @@ -Copyright (c) 2009 The Go Authors. All rights reserved. - -Redistribution and use in source and binary forms, with or without -modification, are permitted provided that the following conditions are -met: - - * Redistributions of source code must retain the above copyright -notice, this list of conditions and the following disclaimer. - * Redistributions in binary form must reproduce the above -copyright notice, this list of conditions and the following disclaimer -in the documentation and/or other materials provided with the -distribution. - * Neither the name of Google Inc. nor the names of its -contributors may be used to endorse or promote products derived from -this software without specific prior written permission. - -THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS -"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT -LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR -A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT -OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, -SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT -LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, -DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY -THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT -(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE -OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. From 259e08bc290bf3e08a00b5734e5741bf042d2f7c Mon Sep 17 00:00:00 2001 From: George Tankersley Date: Sun, 28 May 2017 00:00:00 +0000 Subject: [PATCH 03/46] Use mdbl-2008-bbjlp directly Since we are always converting from affine, we know that Z1=1. This formula is slightly faster and avoids converting through CompletedGroupElement unnecessarily. Assumptions: Z1=1. Cost: 2M + 4S + 1*a + 7add + 1*2. Source: 2008 Bernstein-Birkner-Joye-Lange-Peters, https://eprint.iacr.org/2008/013, plus Z1=1, plus standard simplification. Explicit formulas: B = (X1+Y1)^2 C = X1^2 D = Y1^2 E = a*C F = E+D X3 = (B-C-D)*(F-2) Y3 = F*(E-D) Z3 = F2-2*F https://hyperelliptic.org/EFD/g1p/auto-twisted-projective.html#doubling-mdbl-2008-bbjlp --- ed25519.go | 47 +++++++++++++++++++++++++++++++++++++++-------- ed25519_test.go | 9 ++++++++- 2 files changed, 47 insertions(+), 9 deletions(-) diff --git a/ed25519.go b/ed25519.go index b4c3085..3b4a5d4 100644 --- a/ed25519.go +++ b/ed25519.go @@ -76,7 +76,6 @@ func (curve ed25519Curve) Add(x1, y1, x2, y2 *big.Int) (x, y *big.Int) { // TODO We know Z1=1 and Z2=1 here, so mmadd-2008-hwcd-3 (6M + 1S + 1*k + 9add) could apply func extendedAdd(out, p1, p2 *edwards25519.ExtendedGroupElement) { var tmp1, tmp2, A, B, C, D, E, F, G, H edwards25519.FieldElement - edwards25519.FeSub(&tmp1, &p1.Y, &p1.X) // tmp1 <-- Y1-X1 edwards25519.FeSub(&tmp2, &p2.Y, &p2.X) // tmp2 <-- Y2-X2 edwards25519.FeMul(&A, &tmp1, &tmp2) // A <-- tmp1*tmp2 = (Y1-X1)*(Y2-X2) @@ -97,15 +96,47 @@ func extendedAdd(out, p1, p2 *edwards25519.ExtendedGroupElement) { edwards25519.FeMul(&out.Z, &F, &G) // Z3 <-- F*G } -// Double returns 2*(x,y). -// TODO: cheaper to reimplement? the typed path is aff->proj->completed->proj->aff +// Double returns 2*(x,y). Because we are always converting from affine, we can +// use "mdbl-2008-bbjlp" which assumes Z1=1. Reusing some intermediate +// values, the cost is 3S + 2M + 9*add + 1*a. func (curve ed25519Curve) Double(x1, y1 *big.Int) (x, y *big.Int) { - var p edwards25519.ProjectiveGroupElement - var r edwards25519.CompletedGroupElement + var p, q edwards25519.ProjectiveGroupElement + var t0, t1 edwards25519.FieldElement + affineToProjective(&p, x1, y1) - p.Double(&r) - r.ToProjective(&p) // 3M - return projectiveToAffine(&p) // 1I + 2M + + // C = X1^2, D = Y1^2 + edwards25519.FeSquare(&t0, &p.X) + edwards25519.FeSquare(&t1, &p.Y) + edwards25519.FeMul(&p.Z, &p.X, &p.Y) + + // B = (X1+Y1)^2 = X1^2 + Y1^2 + 2*(X*Y) + edwards25519.FeAdd(&q.X, &t0, &t1) + edwards25519.FeAdd(&q.X, &q.X, &p.Z) + edwards25519.FeAdd(&q.X, &q.X, &p.Z) + + // E = a*C where a = -1 + edwards25519.FeNeg(&q.Z, &t0) + + // F = E + D + edwards25519.FeAdd(&p.X, &q.Z, &t1) + + // X3 = (B-C-D)*(F-2) + edwards25519.FeSub(&p.Y, &q.X, &t0) + edwards25519.FeSub(&p.Y, &p.Y, &t1) + edwards25519.FeSub(&p.Z, &p.X, &feTwo) + edwards25519.FeMul(&q.X, &p.Y, &p.Z) + + // Y3 = F*(E-D) + edwards25519.FeSub(&p.Y, &q.Z, &t1) + edwards25519.FeMul(&q.Y, &p.X, &p.Y) + + // Z3 = F^2 - 2*F + edwards25519.FeSquare(&q.Z, &p.X) + edwards25519.FeSub(&q.Z, &q.Z, &p.X) + edwards25519.FeSub(&q.Z, &q.Z, &p.X) + + return projectiveToAffine(&q) } // ScalarMult returns k*(Bx,By) where k is a number in big-endian form. diff --git a/ed25519_test.go b/ed25519_test.go index 840eca9..217ff75 100644 --- a/ed25519_test.go +++ b/ed25519_test.go @@ -129,7 +129,7 @@ func TestAdd(t *testing.T) { } func TestDouble(t *testing.T) { - c := Ed25519().(ed25519Curve) + c := Ed25519() Gx, Gy := c.Params().Gx, c.Params().Gy G2x, G2y := c.Double(Gx, Gy) @@ -138,6 +138,13 @@ func TestDouble(t *testing.T) { if Ax.Cmp(G2x) != 0 || Ay.Cmp(G2y) != 0 { t.Errorf("double(B) != B+B") } + + G4x, G4y := c.Double(G2x, G2y) + Ax, Ay = c.Add(G2x, G2y, G2x, G2y) + + if Ax.Cmp(G4x) != 0 || Ay.Cmp(G4y) != 0 { + t.Errorf("double(2B) != 2B+2B") + } } // Test vector generated by instrumenting x/crypto/ed25519 GenerateKey From 54cdb2aef0cb378acc8914e3813475fdde1a8748 Mon Sep 17 00:00:00 2001 From: George Tankersley Date: Sun, 28 May 2017 00:00:00 +0000 Subject: [PATCH 04/46] Add documentation --- doc.go | 8 ++++++++ ed25519.go | 2 +- 2 files changed, 9 insertions(+), 1 deletion(-) create mode 100644 doc.go diff --git a/doc.go b/doc.go new file mode 100644 index 0000000..f2a98d5 --- /dev/null +++ b/doc.go @@ -0,0 +1,8 @@ +// Package ed25519 implements an elliptic.Curve interface on top of the twisted +// Edwards curve -x^2 + y^2 = 1 + -(121665/121666)*x^2*y^2. This is better +// known as the Edwards curve equivalent to curve25519, and is the curve used +// by the Ed25519 signature scheme. +// +// Because of the Curve interface, this package takes input in affine (x,y) +// pairs instead of the more standard compressed Edwards y. +package ed25519 diff --git a/ed25519.go b/ed25519.go index 3b4a5d4..ca3d7ec 100644 --- a/ed25519.go +++ b/ed25519.go @@ -2,7 +2,6 @@ // Use of this source code is governed by a BSD-style // license that can be found in the LICENSE file. -// Implements an elliptic.Curve interface over the ed25519 curve. package ed25519 import ( @@ -34,6 +33,7 @@ func initEd25519Params() { ed25519Params.BitSize = 256 } +// Ed25519 returns a Curve that implements Ed25519. func Ed25519() elliptic.Curve { once.Do(initEd25519Params) return ed25519 From ca2029ab22ee9e328cf509d46a0061b7de5500b4 Mon Sep 17 00:00:00 2001 From: George Tankersley Date: Sun, 25 Jun 2017 00:00:00 +0000 Subject: [PATCH 05/46] Add radix51 FieldElement implementation This contains both a pure Go and an amd64 assembly implementation of operations over GF(2^255-19) using radix 2^51. This results in notable speedups when using the assembly, but doesn't help much in pure Go - most of the possible gains are lost to the lack of widening multiply for 64 bit integers. --- ed25519_test.go | 38 ++++ internal/radix51/const.go | 9 + internal/radix51/fe.go | 300 ++++++++++++++++++++++++++++ internal/radix51/fe_mul.go | 122 +++++++++++ internal/radix51/fe_mul_amd64.go | 6 + internal/radix51/fe_mul_amd64.s | 198 ++++++++++++++++++ internal/radix51/fe_square.go | 94 +++++++++ internal/radix51/fe_square_amd64.go | 6 + internal/radix51/fe_square_amd64.s | 146 ++++++++++++++ internal/radix51/fe_test.go | 175 ++++++++++++++++ internal/radix51/mul.go | 14 ++ 11 files changed, 1108 insertions(+) create mode 100644 internal/radix51/const.go create mode 100644 internal/radix51/fe.go create mode 100644 internal/radix51/fe_mul.go create mode 100644 internal/radix51/fe_mul_amd64.go create mode 100644 internal/radix51/fe_mul_amd64.s create mode 100644 internal/radix51/fe_square.go create mode 100644 internal/radix51/fe_square_amd64.go create mode 100644 internal/radix51/fe_square_amd64.s create mode 100644 internal/radix51/fe_test.go create mode 100644 internal/radix51/mul.go diff --git a/ed25519_test.go b/ed25519_test.go index 217ff75..0702e5b 100644 --- a/ed25519_test.go +++ b/ed25519_test.go @@ -15,6 +15,7 @@ import ( "testing" "github.com/gtank/ed25519/internal/edwards25519" + "github.com/gtank/ed25519/internal/radix51" ) // TEST MATH @@ -428,3 +429,40 @@ func BenchmarkScalarMult(b *testing.B) { _, _ = ed.ScalarMult(Bx, By, k[:]) } } + +// A is a constant from the Montgomery form of curve25519. +var radix25A = edwards25519.FieldElement{ + 486662, 0, 0, 0, 0, 0, 0, 0, 0, 0, +} + +var radix51A = radix51.FieldElement{ + 486662, 0, 0, 0, 0, +} + +func BenchmarkFeMul25(b *testing.B) { + var h edwards25519.FieldElement + for i := 0; i < b.N; i++ { + edwards25519.FeMul(&h, &radix25A, &radix25A) + } +} + +func BenchmarkFeMul51(b *testing.B) { + var h radix51.FieldElement + for i := 0; i < b.N; i++ { + radix51.FeMul(&h, &radix51A, &radix51A) + } +} + +func BenchmarkFeSquare25(b *testing.B) { + var h edwards25519.FieldElement + for i := 0; i < b.N; i++ { + edwards25519.FeSquare(&h, &radix25A) + } +} + +func BenchmarkFeSquare51(b *testing.B) { + var h radix51.FieldElement + for i := 0; i < b.N; i++ { + radix51.FeSquare(&h, &radix51A) + } +} diff --git a/internal/radix51/const.go b/internal/radix51/const.go new file mode 100644 index 0000000..9b3f6b2 --- /dev/null +++ b/internal/radix51/const.go @@ -0,0 +1,9 @@ +// Copyright 2017 George Tankersley. All rights reserved. + +// Constants used in the implementation of GF(2^255-19) field arithmetic. +package radix51 + +const ( + // The vaule 2^51-1, used in carry propagation + maskLow51Bits = uint64(1)<<51 - 1 +) diff --git a/internal/radix51/fe.go b/internal/radix51/fe.go new file mode 100644 index 0000000..09ebc86 --- /dev/null +++ b/internal/radix51/fe.go @@ -0,0 +1,300 @@ +// Copyright 2017 George Tankersley. All rights reserved. + +// Field arithmetic in radix 2^51 representation. This code is a port of the +// public domain amd64-51-30k version of ed25519 from SUPERCOP. +package radix51 + +// FieldElement represents an element of the field GF(2^255-19). An element t +// represents the integer t[0] + t[1]*2^51 + t[2]*2^102 + t[3]*2^153 + +// t[4]*2^204. +type FieldElement [5]uint64 + +func (v *FieldElement) FeZero() { + v[0] = 0 + v[1] = 0 + v[2] = 0 + v[3] = 0 + v[4] = 0 +} + +func (v *FieldElement) FeOne() { + v[0] = 1 + v[1] = 0 + v[2] = 0 + v[3] = 0 + v[4] = 0 +} + +// SetInt sets the receiving FieldElement to the specified small integer. +func (v *FieldElement) SetInt(x uint64) { + v[0] = x + v[1] = 0 + v[2] = 0 + v[3] = 0 + v[4] = 0 +} + +func FeReduce(t, v *FieldElement) { + // Copy v + *t = *v + + // Lev v = v[0] + v[1]*2^51 + v[2]*2^102 + v[3]*2^153 + v[4]*2^204 + // Reduce each limb below 2^51, propagating carries. + t[1] += t[0] >> 51 + t[0] = t[0] & maskLow51Bits + t[2] += t[1] >> 51 + t[1] = t[1] & maskLow51Bits + t[3] += t[2] >> 51 + t[2] = t[2] & maskLow51Bits + t[4] += t[3] >> 51 + t[3] = t[3] & maskLow51Bits + t[0] += (t[4] >> 51) * 19 + t[4] = t[4] & maskLow51Bits + + // We now hate a field element t < 2^255, but need t <= 2^255-19 + // TODO Document why this works. It's the elaborate comment about r = h-pq etc etc. + + // Get the carry bit + c := (t[0] + 19) >> 51 + c = (t[1] + c) >> 51 + c = (t[2] + c) >> 51 + c = (t[3] + c) >> 51 + c = (t[4] + c) >> 51 + + t[0] += 19 * c + + t[1] += t[0] >> 51 + t[0] = t[0] & maskLow51Bits + t[2] += t[1] >> 51 + t[1] = t[1] & maskLow51Bits + t[3] += t[2] >> 51 + t[2] = t[2] & maskLow51Bits + t[4] += t[3] >> 51 + t[3] = t[3] & maskLow51Bits + // no additional carry + t[4] = t[4] & maskLow51Bits +} + +// FeAdd sets out = a + b. Long sequences of additions without reduction that +// let coefficients grow larger than 54 bits would be a problem. Paper +// cautions: "do not have such sequences of additions". +func FeAdd(out, a, b *FieldElement) { + out[0] = a[0] + b[0] + out[1] = a[1] + b[1] + out[2] = a[2] + b[2] + out[3] = a[3] + b[3] + out[4] = a[4] + b[4] +} + +// FeSub sets out = a - b +func FeSub(out, a, b *FieldElement) { + var t FieldElement + t = *b + + // Reduce each limb below 2^51, propagating carries. Ensures that results + // fit within the limbs. This would not be required for reduced input. + t[1] += t[0] >> 51 + t[0] = t[0] & maskLow51Bits + t[2] += t[1] >> 51 + t[1] = t[1] & maskLow51Bits + t[3] += t[2] >> 51 + t[2] = t[2] & maskLow51Bits + t[4] += t[3] >> 51 + t[3] = t[3] & maskLow51Bits + t[0] += (t[4] >> 51) * 19 + t[4] = t[4] & maskLow51Bits + + // This is slightly more complicated. Because we use unsigned coefficients, we + // first add a multiple of p and then subtract. + out[0] = (a[0] + 0xFFFFFFFFFFFDA) - t[0] + out[1] = (a[1] + 0xFFFFFFFFFFFFE) - t[1] + out[2] = (a[2] + 0xFFFFFFFFFFFFE) - t[2] + out[3] = (a[3] + 0xFFFFFFFFFFFFE) - t[3] + out[4] = (a[4] + 0xFFFFFFFFFFFFE) - t[4] +} + +// FeNeg sets out = -a +func FeNeg(out, a *FieldElement) { + var t FieldElement + t.SetInt(0) + FeSub(out, &t, a) +} + +// FeInvert sets out = 1/z mod p by calculating z^(p-2), p-2 = 2^255 - 21. +func FeInvert(out, z *FieldElement) { + // Inversion is implemented as exponentiation with exponent p − 2. It uses the + // same sequence of 255 squarings and 11 multiplications as [Curve25519]. + var z2, z9, z11, z2_5_0, z2_10_0, z2_20_0, z2_50_0, z2_100_0, t FieldElement + + FeSquare(&z2, z) // 2 + FeSquare(&t, &z2) // 4 + FeSquare(&t, &t) // 8 + FeMul(&z9, &t, z) // 9 + FeMul(&z11, &z9, &z2) // 11 + FeSquare(&t, &z11) // 22 + FeMul(&z2_5_0, &t, &z9) // 2^5 - 2^0 = 31 + + FeSquare(&t, &z2_5_0) // 2^6 - 2^1 + for i := 0; i < 4; i++ { + FeSquare(&t, &t) // 2^10 - 2^5 + } + FeMul(&z2_10_0, &t, &z2_5_0) // 2^10 - 2^0 + + FeSquare(&t, &z2_10_0) // 2^11 - 2^1 + for i := 0; i < 9; i++ { + FeSquare(&t, &t) // 2^20 - 2^10 + } + FeMul(&z2_20_0, &t, &z2_10_0) // 2^20 - 2^0 + + FeSquare(&t, &z2_20_0) // 2^21 - 2^1 + for i := 0; i < 19; i++ { + FeSquare(&t, &t) // 2^40 - 2^20 + } + FeMul(&t, &t, &z2_20_0) // 2^40 - 2^0 + + FeSquare(&t, &t) // 2^41 - 2^1 + for i := 0; i < 9; i++ { + FeSquare(&t, &t) // 2^50 - 2^10 + } + FeMul(&z2_50_0, &t, &z2_10_0) // 2^50 - 2^0 + + FeSquare(&t, &z2_50_0) // 2^51 - 2^1 + for i := 0; i < 49; i++ { + FeSquare(&t, &t) // 2^100 - 2^50 + } + FeMul(&z2_100_0, &t, &z2_50_0) // 2^100 - 2^0 + + FeSquare(&t, &z2_100_0) // 2^101 - 2^1 + for i := 0; i < 99; i++ { + FeSquare(&t, &t) // 2^200 - 2^100 + } + FeMul(&t, &t, &z2_100_0) // 2^200 - 2^0 + + FeSquare(&t, &t) // 2^201 - 2^1 + for i := 0; i < 49; i++ { + FeSquare(&t, &t) // 2^250 - 2^50 + } + FeMul(&t, &t, &z2_50_0) // 2^250 - 2^0 + + FeSquare(&t, &t) // 2^251 - 2^1 + FeSquare(&t, &t) // 2^252 - 2^2 + FeSquare(&t, &t) // 2^253 - 2^3 + FeSquare(&t, &t) // 2^254 - 2^4 + FeSquare(&t, &t) // 2^255 - 2^5 + + FeMul(out, &t, &z11) // 2^255 - 21 +} + +// FeCSwap implements a constant-time conditional swap. That is: +// (a, b) = (b, a) if c = 1 +// (a, b) = (a, b) if c = 0 +func FeCSwap(a, b *FieldElement, c uint64) { + var t FieldElement + + t[0] ^= c & (a[0] ^ b[0]) + t[1] ^= c & (a[1] ^ b[1]) + t[2] ^= c & (a[2] ^ b[2]) + t[3] ^= c & (a[3] ^ b[3]) + t[4] ^= c & (a[4] ^ b[4]) + + a[0] ^= t[0] + a[1] ^= t[1] + a[2] ^= t[2] + a[3] ^= t[3] + a[4] ^= t[4] + + b[0] ^= t[0] + b[1] ^= t[1] + b[2] ^= t[2] + b[3] ^= t[3] + b[4] ^= t[4] +} + +func FeFromBytes(v *FieldElement, x *[32]byte) { + v[0] = uint64(x[0]) + v[0] |= uint64(x[1]) << 8 + v[0] |= uint64(x[2]) << 16 + v[0] |= uint64(x[3]) << 24 + v[0] |= uint64(x[4]) << 32 + v[0] |= uint64(x[5]) << 40 + v[0] |= uint64(x[6]&7) << 48 + + v[1] = uint64(x[6]) >> 3 + v[1] |= uint64(x[7]) << 5 + v[1] |= uint64(x[8]) << 13 + v[1] |= uint64(x[9]) << 21 + v[1] |= uint64(x[10]) << 29 + v[1] |= uint64(x[11]) << 37 + v[1] |= uint64(x[12]&63) << 45 + + v[2] = uint64(x[12]) >> 6 + v[2] |= uint64(x[13]) << 2 + v[2] |= uint64(x[14]) << 10 + v[2] |= uint64(x[15]) << 18 + v[2] |= uint64(x[16]) << 26 + v[2] |= uint64(x[17]) << 34 + v[2] |= uint64(x[18]) << 42 + v[2] |= uint64(x[19]&1) << 50 + + v[3] = uint64(x[19]) >> 1 + v[3] |= uint64(x[20]) << 7 + v[3] |= uint64(x[21]) << 15 + v[3] |= uint64(x[22]) << 23 + v[3] |= uint64(x[23]) << 31 + v[3] |= uint64(x[24]) << 39 + v[3] |= uint64(x[25]&15) << 47 + + v[4] = uint64(x[25]) >> 4 + v[4] |= uint64(x[26]) << 4 + v[4] |= uint64(x[27]) << 12 + v[4] |= uint64(x[28]) << 20 + v[4] |= uint64(x[29]) << 28 + v[4] |= uint64(x[30]) << 36 + v[4] |= uint64(x[31]&127) << 44 +} + +func FeToBytes(r *[32]byte, v *FieldElement) { + var t FieldElement + FeReduce(&t, v) + + r[0] = byte(t[0] & 0xff) + r[1] = byte((t[0] >> 8) & 0xff) + r[2] = byte((t[0] >> 16) & 0xff) + r[3] = byte((t[0] >> 24) & 0xff) + r[4] = byte((t[0] >> 32) & 0xff) + r[5] = byte((t[0] >> 40) & 0xff) + r[6] = byte((t[0] >> 48)) + + r[6] ^= byte((t[1] << 3) & 0xf8) + r[7] = byte((t[1] >> 5) & 0xff) + r[8] = byte((t[1] >> 13) & 0xff) + r[9] = byte((t[1] >> 21) & 0xff) + r[10] = byte((t[1] >> 29) & 0xff) + r[11] = byte((t[1] >> 37) & 0xff) + r[12] = byte((t[1] >> 45)) + + r[12] ^= byte((t[2] << 6) & 0xc0) + r[13] = byte((t[2] >> 2) & 0xff) + r[14] = byte((t[2] >> 10) & 0xff) + r[15] = byte((t[2] >> 18) & 0xff) + r[16] = byte((t[2] >> 26) & 0xff) + r[17] = byte((t[2] >> 34) & 0xff) + r[18] = byte((t[2] >> 42) & 0xff) + r[19] = byte((t[2] >> 50)) + + r[19] ^= byte((t[3] << 1) & 0xfe) + r[20] = byte((t[3] >> 7) & 0xff) + r[21] = byte((t[3] >> 15) & 0xff) + r[22] = byte((t[3] >> 23) & 0xff) + r[23] = byte((t[3] >> 31) & 0xff) + r[24] = byte((t[3] >> 39) & 0xff) + r[25] = byte((t[3] >> 47)) + + r[25] ^= byte((t[4] << 4) & 0xf0) + r[26] = byte((t[4] >> 4) & 0xff) + r[27] = byte((t[4] >> 12) & 0xff) + r[28] = byte((t[4] >> 20) & 0xff) + r[29] = byte((t[4] >> 28) & 0xff) + r[30] = byte((t[4] >> 36) & 0xff) + r[31] = byte((t[4] >> 44)) +} diff --git a/internal/radix51/fe_mul.go b/internal/radix51/fe_mul.go new file mode 100644 index 0000000..68b6849 --- /dev/null +++ b/internal/radix51/fe_mul.go @@ -0,0 +1,122 @@ +// +build !amd64 noasm + +package radix51 + +// FeMul sets out = a * b +func FeMul(out, x, y *FieldElement) { + var x0, x1, x2, x3, x4 uint64 + var y0, y1, y2, y3, y4 uint64 + + x0 = x[0] + x1 = x[1] + x2 = x[2] + x3 = x[3] + x4 = x[4] + + y0 = y[0] + y1 = y[1] + y2 = y[2] + y3 = y[3] + y4 = y[4] + + // Reduction can be carried out simultaneously to multiplication. For + // example, we do not compute a coefficient r_5 . Whenever the result of a + // mul instruction belongs to r_5 , for example in the multiplication of + // x_3*y_2 , we multiply one of the inputs by 19 and add the result to r_0. + + x1_19 := x1 * 19 + x2_19 := x2 * 19 + x3_19 := x3 * 19 + x4_19 := x4 * 19 + + // calculate r0 = x0*y0 + 19*(x1*y4 + x2*y3 + x3*y2 + x4*y1) + r00, r01 := mul64x64(0, 0, x0, y0) + r00, r01 = mul64x64(r00, r01, x1_19, y4) + r00, r01 = mul64x64(r00, r01, x2_19, y3) + r00, r01 = mul64x64(r00, r01, x3_19, y2) + r00, r01 = mul64x64(r00, r01, x4_19, y1) + + // calculate r1 = x0*y1 + x1*y0 + 19*(x2*y4 + x3*y3 + x4*y2) + r10, r11 := mul64x64(0, 0, x0, y1) + r10, r11 = mul64x64(r10, r11, x1, y0) + r10, r11 = mul64x64(r10, r11, x2_19, y4) + r10, r11 = mul64x64(r10, r11, x3_19, y3) + r10, r11 = mul64x64(r10, r11, x4_19, y2) + + // calculate r2 = x0*y2 + x1*y1 + x2*y0 + 19*(x3*y4 + x4*y3) + r20, r21 := mul64x64(0, 0, x0, y2) + r20, r21 = mul64x64(r20, r21, x1, y1) + r20, r21 = mul64x64(r20, r21, x2, y0) + r20, r21 = mul64x64(r20, r21, x3_19, y4) + r20, r21 = mul64x64(r20, r21, x4_19, y3) + + // calculate r3 = x0*y3 + x1*y2 + x2*y1 + x3*y0 + 19*x4*y4 + r30, r31 := mul64x64(0, 0, x0, y3) + r30, r31 = mul64x64(r30, r31, x1, y2) + r30, r31 = mul64x64(r30, r31, x2, y1) + r30, r31 = mul64x64(r30, r31, x3, y0) + r30, r31 = mul64x64(r30, r31, x4_19, y4) + + // calculate r4 = x0*y4 + x1*y3 + x2*y2 + x3*y1 + x4*y0 + r40, r41 := mul64x64(0, 0, x0, y4) + r40, r41 = mul64x64(r40, r41, x1, y3) + r40, r41 = mul64x64(r40, r41, x2, y2) + r40, r41 = mul64x64(r40, r41, x3, y1) + r40, r41 = mul64x64(r40, r41, x4, y0) + + // After the multiplication we need to reduce (carry) the 5 coefficients to + // obtain a result with coefficients that are at most slightly larger than + // 2^51 . Denote the two registers holding coefficient r_0 as r_00 and r_01 + // with r_0 = 2^64*r_01 + r_00 . Similarly denote the two registers holding + // coefficient r_1 as r_10 and r_11 . We first shift r_01 left by 13, while + // shifting in the most significant bits of r_00 (shld instruction) and + // then compute the logical and of r_00 with 2^51 − 1. We do the same with + // r_10 and r_11 and add r_01 into r_10 after the logical and with 2^51 − + // 1. We proceed this way for coefficients r_2,...,r_4; register r_41 is + // multiplied by 19 before adding it to r_00 . + + r01 = (r01 << 13) | (r00 >> 51) + r00 &= maskLow51Bits + + r11 = (r11 << 13) | (r10 >> 51) + r10 &= maskLow51Bits + r10 += r01 + + r21 = (r21 << 13) | (r20 >> 51) + r20 &= maskLow51Bits + r20 += r11 + + r31 = (r31 << 13) | (r30 >> 51) + r30 &= maskLow51Bits + r30 += r21 + + r41 = (r41 << 13) | (r40 >> 51) + r40 &= maskLow51Bits + r40 += r31 + + r41 *= 19 + r00 += r41 + + // Now all 5 coefficients fit into 64-bit registers but are still too large + // to be used as input to another multiplication. We therefore carry from + // r_0 to r_1 , from r_1 to r_2 , from r_2 to r_3 , from r_3 to r_4 , and + // finally from r_4 to r_0 . Each of these carries is done as one copy, one + // right shift by 51, one logical and with 2^51 − 1, and one addition. + + r10 += r00 >> 51 + r00 &= maskLow51Bits + r20 += r10 >> 51 + r10 &= maskLow51Bits + r30 += r20 >> 51 + r20 &= maskLow51Bits + r40 += r30 >> 51 + r30 &= maskLow51Bits + r00 += (r40 >> 51) * 19 + r40 &= maskLow51Bits + + out[0] = r00 + out[1] = r10 + out[2] = r20 + out[3] = r30 + out[4] = r40 +} diff --git a/internal/radix51/fe_mul_amd64.go b/internal/radix51/fe_mul_amd64.go new file mode 100644 index 0000000..c76b4db --- /dev/null +++ b/internal/radix51/fe_mul_amd64.go @@ -0,0 +1,6 @@ +// +build amd64,!noasm + +package radix51 + +// go:noescape +func FeMul(out, a, b *FieldElement) diff --git a/internal/radix51/fe_mul_amd64.s b/internal/radix51/fe_mul_amd64.s new file mode 100644 index 0000000..07ebb0c --- /dev/null +++ b/internal/radix51/fe_mul_amd64.s @@ -0,0 +1,198 @@ +// Based on assembly generated by PeachPy. Equivalent to the Go in fe_mul.go, +// which was originally based on the amd64-51-30k assembly in SUPERCOP. + +// +build amd64,!noasm + +// func FeMul(outp *uint64, xp *uint64, yp *uint64) +TEXT ·FeMul(SB),$0-24 + MOVQ outp+0(FP), DI + MOVQ xp+8(FP), BX + MOVQ yp+16(FP), CX + + // Calculate r0 + MOVQ 0(BX), AX // rax <-- x0 + MULQ 0(CX) // rdx, rax <-- x0*y0 + MOVQ AX, SI // r00 = rax + MOVQ DX, BP // r01 = rdx + + MOVQ 8(BX), DX // rdx <-- x1 + IMUL3Q $19, DX, AX // rax <-- x1*19 + MULQ 32(CX) // rdx, rax <-- x1_19*y4 + ADDQ AX, SI // r00 += rax + ADCQ DX, BP // r01 += rdx + + MOVQ 16(BX), DX // rdx <-- x2 + IMUL3Q $19, DX, AX // rax <-- x2*19 + MULQ 24(CX) // rdx, rax <-- x2_19*y3 + ADDQ AX, SI // r00 += rax + ADCQ DX, BP // r01 += rdx + + MOVQ 24(BX), DX // rdx <-- x3 + IMUL3Q $19, DX, AX // rax <-- x3*19 + MULQ 16(CX) // rdx, rax <-- x3_19 * y2 + ADDQ AX, SI // r00 += rax + ADCQ DX, BP // r01 += rdx + + MOVQ 32(BX), DX // rdx <-- x4 + IMUL3Q $19, DX, AX // rax <-- x4*19 + MULQ 8(CX) // rdx rax <-- x4_19*y1 + ADDQ AX, SI // r00 += rax + ADCQ DX, BP // r01 += rdx + + // Calculate r1 + MOVQ 0(BX), AX + MULQ 8(CX) + MOVQ AX, R8 // r10 + MOVQ DX, R9 // r11 + + MOVQ 8(BX), AX + MULQ 0(CX) + ADDQ AX, R8 + ADCQ DX, R9 + + MOVQ 16(BX), DX + IMUL3Q $19, DX, AX + MULQ 32(CX) + ADDQ AX, R8 + ADCQ DX, R9 + + MOVQ 24(BX), DX + IMUL3Q $19, DX, AX + MULQ 24(CX) + ADDQ AX, R8 + ADCQ DX, R9 + + MOVQ 32(BX), DX + IMUL3Q $19, DX, AX + MULQ 16(CX) + ADDQ AX, R8 + ADCQ DX, R9 + + // Calculate r2 + MOVQ 0(BX), AX + MULQ 16(CX) + MOVQ AX, R10 // r20 + MOVQ DX, R11 // r21 + + MOVQ 8(BX), AX + MULQ 8(CX) + ADDQ AX, R10 + ADCQ DX, R11 + + MOVQ 16(BX), AX + MULQ 0(CX) + ADDQ AX, R10 + ADCQ DX, R11 + + MOVQ 24(BX), DX + IMUL3Q $19, DX, AX + MULQ 32(CX) + ADDQ AX, R10 + ADCQ DX, R11 + + MOVQ 32(BX), DX + IMUL3Q $19, DX, AX + MULQ 24(CX) + ADDQ AX, R10 + ADCQ DX, R11 + + // Calculate r3 + MOVQ 0(BX), AX + MULQ 24(CX) + MOVQ AX, R12 // r30 + MOVQ DX, R13 // r31 + + MOVQ 8(BX), AX + MULQ 16(CX) + ADDQ AX, R12 + ADCQ DX, R13 + + MOVQ 16(BX), AX + MULQ 8(CX) + ADDQ AX, R12 + ADCQ DX, R13 + + MOVQ 24(BX), AX + MULQ 0(CX) + ADDQ AX, R12 + ADCQ DX, R13 + + MOVQ 32(BX), DX + IMUL3Q $19, DX, AX + MULQ 32(CX) + ADDQ AX, R12 + ADCQ DX, R13 + + // Calculate r4 + MOVQ 0(BX), AX + MULQ 32(CX) + MOVQ AX, R14 // r40 + MOVQ DX, R15 // r41 + + MOVQ 8(BX), AX + MULQ 24(CX) + ADDQ AX, R14 + ADCQ DX, R15 + + MOVQ 16(BX), AX + MULQ 16(CX) + ADDQ AX, R14 + ADCQ DX, R15 + + MOVQ 24(BX), AX + MULQ 8(CX) + ADDQ AX, R14 + ADCQ DX, R15 + + MOVQ 32(BX), AX + MULQ 0(CX) + ADDQ AX, R14 + ADCQ DX, R15 + + + MOVQ $2251799813685247, AX // (1<<51) - 1 + SHLQ $13, SI, BP // r01 = shld with r00 + ANDQ AX, SI // r00 &= mask51 + SHLQ $13, R8, R9 // r11 = shld with r10 + ANDQ AX, R8 // r10 &= mask51 + ADDQ BP, R8 // r10 += r01 + SHLQ $13, R10, R11 // r21 = shld with r20 + ANDQ AX, R10 // r20 &= mask51 + ADDQ R9, R10 // r20 += r11 + SHLQ $13, R12, R13 // r31 = shld with r30 + ANDQ AX, R12 // r30 &= mask51 + ADDQ R11, R12 // r30 += r21 + SHLQ $13, R14, R15 // r41 = shld with r40 + ANDQ AX, R14 // r40 &= mask51 + ADDQ R13, R14 // r40 += r31 + IMUL3Q $19, R15, R15 // r41 = r41*19 + ADDQ R15, SI // r00 += r41 + + MOVQ SI, DX // rdx <-- r00 + SHRQ $51, DX // rdx <-- r00 >> 51 + ADDQ DX, R8 // r10 += r00 >> 51 + MOVQ R8, DX // rdx <-- r10 + SHRQ $51, DX // rdx <-- r10 >> 51 + ANDQ AX, SI // r00 &= mask51 + ADDQ DX, R10 // r20 += r10 >> 51 + MOVQ R10, DX // rdx <-- r20 + SHRQ $51, DX // rdx <-- r20 >> 51 + ANDQ AX, R8 // r10 &= mask51 + ADDQ DX, R12 // r30 += r20 >> 51 + MOVQ R12, DX // rdx <-- r30 + SHRQ $51, DX // rdx <-- r30 >> 51 + ANDQ AX, R10 // r20 &= mask51 + ADDQ DX, R14 // r40 += r30 >> 51 + MOVQ R14, DX // rdx <-- r40 + SHRQ $51, DX // rdx <-- r40 >> 51 + ANDQ AX, R12 // r30 &= mask51 + IMUL3Q $19, DX, DX // rdx <-- (r40 >> 51) * 19 + ADDQ DX, SI // r00 += (r40 >> 51) *19 + ANDQ AX, R14 // r40 &= mask51 + + MOVQ SI, 0(DI) + MOVQ R8, 8(DI) + MOVQ R10, 16(DI) + MOVQ R12, 24(DI) + MOVQ R14, 32(DI) + RET diff --git a/internal/radix51/fe_square.go b/internal/radix51/fe_square.go new file mode 100644 index 0000000..fa738eb --- /dev/null +++ b/internal/radix51/fe_square.go @@ -0,0 +1,94 @@ +// +build !amd64 noasm + +package radix51 + +// FeSquare sets out = x*x +func FeSquare(out, x *FieldElement) { + // Squaring needs only 15 mul instructions. Some inputs are multiplied by 2; + // this is combined with multiplication by 19 where possible. The coefficient + // reduction after squaring is the same as for multiplication. + + var x0, x1, x2, x3, x4 uint64 + + x0 = x[0] + x1 = x[1] + x2 = x[2] + x3 = x[3] + x4 = x[4] + + x0_2 := x0 << 1 + x1_2 := x1 << 1 + + x1_38 := x1 * 38 + x2_38 := x2 * 38 + x3_38 := x3 * 38 + + x3_19 := x3 * 19 + x4_19 := x4 * 19 + + // r0 = x0*x0 + x1*38*x4 + x2*38*x3 + r00, r01 := mul64x64(0, 0, x0, x0) + r00, r01 = mul64x64(r00, r01, x1_38, x4) + r00, r01 = mul64x64(r00, r01, x2_38, x3) + + // r1 = x0*2*x1 + x2*38*x4 + x3*19*x3 + r10, r11 := mul64x64(0, 0, x0_2, x1) + r10, r11 = mul64x64(r10, r11, x2_38, x4) + r10, r11 = mul64x64(r10, r11, x3_19, x3) + + // r2 = x0*2*x2 + x1*x1 + x3*38*x4 + r20, r21 := mul64x64(0, 0, x0_2, x2) + r20, r21 = mul64x64(r20, r21, x1, x1) + r20, r21 = mul64x64(r20, r21, x3_38, x4) + + // r3 = x0*2*x3 + x1*2*x2 + x4*19*x4 + r30, r31 := mul64x64(0, 0, x0_2, x3) + r30, r31 = mul64x64(r30, r31, x1_2, x2) + r30, r31 = mul64x64(r30, r31, x4_19, x4) + + // r4 = x0*2*x4 + x1*2*x3 + x2*x2 + r40, r41 := mul64x64(0, 0, x0_2, x4) + r40, r41 = mul64x64(r40, r41, x1_2, x3) + r40, r41 = mul64x64(r40, r41, x2, x2) + + // Same reduction + + r01 = (r01 << 13) | (r00 >> 51) + r00 &= maskLow51Bits + + r11 = (r11 << 13) | (r10 >> 51) + r10 &= maskLow51Bits + r10 += r01 + + r21 = (r21 << 13) | (r20 >> 51) + r20 &= maskLow51Bits + r20 += r11 + + r31 = (r31 << 13) | (r30 >> 51) + r30 &= maskLow51Bits + r30 += r21 + + r41 = (r41 << 13) | (r40 >> 51) + r40 &= maskLow51Bits + r40 += r31 + + r41 *= 19 + r00 += r41 + + r10 += r00 >> 51 + r00 &= maskLow51Bits + r20 += r10 >> 51 + r10 &= maskLow51Bits + r30 += r20 >> 51 + r20 &= maskLow51Bits + r40 += r30 >> 51 + r30 &= maskLow51Bits + r00 += (r40 >> 51) * 19 + r40 &= maskLow51Bits + + out[0] = r00 + out[1] = r10 + out[2] = r20 + out[3] = r30 + out[4] = r40 +} diff --git a/internal/radix51/fe_square_amd64.go b/internal/radix51/fe_square_amd64.go new file mode 100644 index 0000000..c089146 --- /dev/null +++ b/internal/radix51/fe_square_amd64.go @@ -0,0 +1,6 @@ +// +build amd64,!noasm + +package radix51 + +// go:noescape +func FeSquare(out, x *FieldElement) diff --git a/internal/radix51/fe_square_amd64.s b/internal/radix51/fe_square_amd64.s new file mode 100644 index 0000000..f9af5c8 --- /dev/null +++ b/internal/radix51/fe_square_amd64.s @@ -0,0 +1,146 @@ +// +build amd64,!noasm + +// func FeSquare(outp *uint64, xp *uint64) +TEXT ·FeSquare(SB),4,$0-16 + MOVQ outp+0(FP), DI + MOVQ xp+8(FP), SI + + // r0 = x0*x0 + x1*38*x4 + x2*38*x3 + MOVQ 0(SI), AX + MULQ 0(SI) + MOVQ AX, CX // r00 + MOVQ DX, R8 // r01 + + MOVQ 8(SI), DX + IMUL3Q $38, DX, AX + MULQ 32(SI) + ADDQ AX, CX + ADCQ DX, R8 + + MOVQ 16(SI), DX + IMUL3Q $38, DX, AX + MULQ 24(SI) + ADDQ AX, CX + ADCQ DX, R8 + + // r1 = x0*2*x1 + x2*38*x4 + x3*19*x3 + MOVQ 0(SI), AX + SHLQ $1, AX + MULQ 8(SI) + MOVQ AX, R9 // r10 + MOVQ DX, R10 // r11 + + MOVQ 16(SI), DX + IMUL3Q $38, DX, AX + MULQ 32(SI) + ADDQ AX, R9 + ADCQ DX, R10 + + MOVQ 24(SI), DX + IMUL3Q $19, DX, AX + MULQ 24(SI) + ADDQ AX, R9 + ADCQ DX, R10 + + // r2 = x0*2*x2 + x1*x1 + x3*38*x4 + MOVQ 0(SI), AX + SHLQ $1, AX + MULQ 16(SI) + MOVQ AX, R11 // r20 + MOVQ DX, R12 // r21 + + MOVQ 8(SI), AX + MULQ 8(SI) + ADDQ AX, R11 + ADCQ DX, R12 + + MOVQ 24(SI), DX + IMUL3Q $38, DX, AX + MULQ 32(SI) + ADDQ AX, R11 + ADCQ DX, R12 + + // r3 = x0*2*x3 + x1*2*x2 + x4*19*x4 + MOVQ 0(SI), AX + SHLQ $1, AX + MULQ 24(SI) + MOVQ AX, R13 // r30 + MOVQ DX, R14 // r31 + + MOVQ 8(SI), AX + SHLQ $1, AX + MULQ 16(SI) + ADDQ AX, R13 + ADCQ DX, R14 + + MOVQ 32(SI), DX + IMUL3Q $19, DX, AX + MULQ 32(SI) + ADDQ AX, R13 + ADCQ DX, R14 + + // r4 = x0*2*x4 + x1*2*x3 + x2*x2 + MOVQ 0(SI), AX + SHLQ $1, AX + MULQ 32(SI) + MOVQ AX, R15 // r40 + MOVQ DX, BX // r41 + + MOVQ 8(SI), AX + SHLQ $1, AX + MULQ 24(SI) + ADDQ AX, R15 + ADCQ DX, BX + + MOVQ 16(SI), AX + MULQ 16(SI) + ADDQ AX, R15 + ADCQ DX, BX + + // Reduce + MOVQ $2251799813685247, AX // (1<<51) - 1 + SHLQ $13, CX, R8 // r01 = shld with r00 + ANDQ AX, CX // r00 &= mask51 + SHLQ $13, R9, R10 // r11 = shld with r10 + ANDQ AX, R9 // r10 &= mask51 + ADDQ R8, R9 // r10 += r01 + SHLQ $13, R11, R12 // r21 = shld with r20 + ANDQ AX, R11 // r20 &= mask51 + ADDQ R10, R11 // r20 += r11 + SHLQ $13, R13, R14 // r31 = shld with r30 + ANDQ AX, R13 // r30 &= mask51 + ADDQ R12, R13 // r30 += r21 + SHLQ $13, R15, BX // r41 = shld with r40 + ANDQ AX, R15 // r40 &= mask51 + ADDQ R14, R15 // r40 += r31 + IMUL3Q $19, BX, DX // r41 = r41*19 + ADDQ DX, CX // r00 += r41 + + MOVQ CX, DX // rdx <-- r00 + SHRQ $51, DX // rdx <-- r00 >> 51 + ADDQ DX, R9 // r10 += r00 >> 51 + MOVQ R9, DX // rdx <-- r10 + SHRQ $51, DX // rdx <-- r10 >> 51 + ANDQ AX, CX // r00 &= mask51 + ADDQ DX, R11 // r20 += r10 >> 51 + MOVQ R11, DX // rdx <-- r20 + SHRQ $51, DX // rdx <-- r20 >> 51 + ANDQ AX, R9 // r10 &= mask51 + ADDQ DX, R13 // r30 += r20 >> 51 + MOVQ R13, DX // rdx <-- r30 + SHRQ $51, DX // rdx <-- r30 >> 51 + ANDQ AX, R11 // r20 &= mask51 + ADDQ DX, R15 // r40 += r30 >> 51 + MOVQ R15, DX // rdx <-- r40 + SHRQ $51, DX // rdx <-- r40 >> 51 + ANDQ AX, R13 // r30 &= mask51 + IMUL3Q $19, DX, DX // rdx <-- (r40 >> 51) * 19 + ADDQ DX, CX // r00 += (r40 >> 51) *19 + ANDQ AX, R15 // r40 &= mask51 + + MOVQ CX, 0(DI) + MOVQ R9, 8(DI) + MOVQ R11, 16(DI) + MOVQ R13, 24(DI) + MOVQ R15, 32(DI) + RET diff --git a/internal/radix51/fe_test.go b/internal/radix51/fe_test.go new file mode 100644 index 0000000..217d910 --- /dev/null +++ b/internal/radix51/fe_test.go @@ -0,0 +1,175 @@ +package radix51 + +import ( + "bytes" + "crypto/rand" + "io" + "testing" + "unsafe" +) + +func TestMul64to128(t *testing.T) { + a := uint64(5) + b := uint64(5) + r0, r1 := mul64x64(0, 0, a, b) + if r0 != 0x19 || r1 != 0 { + t.Errorf("lo-range wide mult failed, got %d + %d*(2**64)", r0, r1) + } + + a = uint64(18014398509481983) // 2^54 - 1 + b = uint64(18014398509481983) // 2^54 - 1 + r0, r1 = mul64x64(0, 0, a, b) + if r0 != 0xff80000000000001 || r1 != 0xfffffffffff { + t.Errorf("hi-range wide mult failed, got %d + %d*(2**64)", r0, r1) + } + + a = uint64(1125899906842661) + b = uint64(2097155) + r0, r1 = mul64x64(0, 0, a, b) + r0, r1 = mul64x64(r0, r1, a, b) + r0, r1 = mul64x64(r0, r1, a, b) + r0, r1 = mul64x64(r0, r1, a, b) + r0, r1 = mul64x64(r0, r1, a, b) + if r0 != 16888498990613035 || r1 != 640 { + t.Errorf("wrong answer: %d + %d*(2**64)", r0, r1) + } +} + +func BenchmarkWideMultInline(t *testing.B) { + var r0, r1, ol, oh uint64 + a := uint64(18014398509481983) // 2^54 - 1 + b := uint64(18014398509481983) // 2^54 - 1 + + for i := 0; i < t.N; i++ { + t1 := (a>>32)*(b&0xFFFFFFFF) + ((a & 0xFFFFFFFF) * (b & 0xFFFFFFFF) >> 32) + t2 := (a&0xFFFFFFFF)*(b>>32) + (t1 & 0xFFFFFFFF) + ol = (a * b) + r0 + cmp := ol < r0 + oh = r1 + (a>>32)*(b>>32) + t1>>32 + t2>>32 + uint64(*(*byte)(unsafe.Pointer(&cmp))) + + r1 = oh + r0 = ol + } +} + +func BenchmarkWideMultCall(t *testing.B) { + var r0, r1 uint64 + a := uint64(18014398509481983) + b := uint64(18014398509481983) + + for i := 0; i < t.N; i++ { + r0, r1 = mul64x64(r0, r1, a, b) + } +} + +func TestFeFromBytesRoundTrip(t *testing.T) { + var in, out [32]byte + var fe, r FieldElement + + in = [32]byte{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, + 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32} + + FeFromBytes(&fe, &in) + FeToBytes(&out, &fe) + + if !bytes.Equal(in[:], out[:]) { + t.Error("Bytes<>FE doesn't roundtrip") + } + + // Random field element + fe[0] = 0x4e645be9215a2 + fe[1] = 0x4e9654922df12 + fe[2] = 0x5829e468b0205 + fe[3] = 0x5e8fca9e0881c + fe[4] = 0x5c490f087d796 + + FeToBytes(&out, &fe) + FeFromBytes(&r, &out) + + for i := 0; i < len(fe); i++ { + if r[i] != fe[i] { + t.Error("FE<>Bytes doesn't roundtrip") + } + } +} + +// Tests self-consistency between FeMul and FeSquare. +func TestSanity(t *testing.T) { + var x FieldElement + var x2, x2sq FieldElement + // var x2Go, x2sqGo FieldElement + + x = [5]uint64{1, 1, 1, 1, 1} + FeMul(&x2, &x, &x) + // FeMulGo(&x2Go, &x, &x) + FeSquare(&x2sq, &x) + // FeSquareGo(&x2sqGo, &x) + + // if !vartimeEqual(x2, x2Go) || !vartimeEqual(x2sq, x2sqGo) || !vartimeEqual(x2, x2sq) { + // t.Fatalf("all ones failed\nmul.s: %d\nmul.g: %d\nsqr.s: %d\nsqr.g: %d\n", x2, x2Go, x2sq, x2sqGo) + // } + + if !vartimeEqual(x2, x2sq) { + t.Fatalf("all ones failed\nmul: %x\nsqr: %x\n", x2, x2sq) + } + + var bytes [32]byte + + _, err := io.ReadFull(rand.Reader, bytes[:]) + if err != nil { + t.Fatal(err) + } + FeFromBytes(&x, &bytes) + + FeMul(&x2, &x, &x) + // FeMulGo(&x2Go, &x, &x) + FeSquare(&x2sq, &x) + // FeSquareGo(&x2sqGo, &x) + + // if !vartimeEqual(x2, x2Go) || !vartimeEqual(x2sq, x2sqGo) || !vartimeEqual(x2, x2sq) { + // t.Fatalf("random field element failed\nfe: %x\n\nmul.s: %x\nmul.g: %x\nsqr.s: %x\nsqr.g: %x\n", x, x2, x2Go, x2sq, x2sqGo) + // } + + if !vartimeEqual(x2, x2sq) { + t.Fatalf("all ones failed\nmul: %x\nsqr: %x\n", x2, x2sq) + } +} + +func vartimeEqual(x, y FieldElement) bool { + for i := 0; i < 5; i++ { + if x[i] != y[i] { + return false + } + } + return true +} + +func TestFeInvert(t *testing.T) { + var x FieldElement = [5]uint64{1, 1, 1, 1, 1} + var one FieldElement = [5]uint64{1, 0, 0, 0, 0} + var xinv, r FieldElement + + FeInvert(&xinv, &x) + FeMul(&r, &x, &xinv) + FeReduce(&r, &r) + + if !vartimeEqual(one, r) { + t.Errorf("inversion identity failed, got: %x", r) + } + + var bytes [32]byte + + _, err := io.ReadFull(rand.Reader, bytes[:]) + if err != nil { + t.Fatal(err) + } + FeFromBytes(&x, &bytes) + + FeInvert(&xinv, &x) + FeMul(&r, &x, &xinv) + FeReduce(&r, &r) + + if !vartimeEqual(one, r) { + t.Errorf("random inversion identity failed, got: %x for field element %x", r, x) + } +} diff --git a/internal/radix51/mul.go b/internal/radix51/mul.go new file mode 100644 index 0000000..dbea36c --- /dev/null +++ b/internal/radix51/mul.go @@ -0,0 +1,14 @@ +package radix51 + +import "unsafe" + +// mul64x64 multiples two 64-bit numbers and adds them to two accumulators. +// This function is written to ensure it inlines. I am so sorry. +func mul64x64(lo, hi, a, b uint64) (ol uint64, oh uint64) { + t1 := (a>>32)*(b&0xFFFFFFFF) + ((a & 0xFFFFFFFF) * (b & 0xFFFFFFFF) >> 32) + t2 := (a&0xFFFFFFFF)*(b>>32) + (t1 & 0xFFFFFFFF) + ol = (a * b) + lo + cmp := ol < lo + oh = hi + (a>>32)*(b>>32) + t1>>32 + t2>>32 + uint64(*(*byte)(unsafe.Pointer(&cmp))) + return +} From 7ae6271616dd630bad95bf07e47b665aa0a635a9 Mon Sep 17 00:00:00 2001 From: George Tankersley Date: Sun, 25 Jun 2017 20:20:59 -0400 Subject: [PATCH 06/46] add equality for field elements --- internal/radix51/const.go | 5 +++++ internal/radix51/fe.go | 8 ++++++++ internal/radix51/fe_test.go | 15 +++++++++++++++ 3 files changed, 28 insertions(+) diff --git a/internal/radix51/const.go b/internal/radix51/const.go index 9b3f6b2..ce3f2ff 100644 --- a/internal/radix51/const.go +++ b/internal/radix51/const.go @@ -7,3 +7,8 @@ const ( // The vaule 2^51-1, used in carry propagation maskLow51Bits = uint64(1)<<51 - 1 ) + +var ( + FieldZero FieldElement = [5]uint64{0, 0, 0, 0, 0} + FieldOne FieldElement = [5]uint64{1, 0, 0, 0, 0} +) diff --git a/internal/radix51/fe.go b/internal/radix51/fe.go index 09ebc86..ff7e5e1 100644 --- a/internal/radix51/fe.go +++ b/internal/radix51/fe.go @@ -210,6 +210,14 @@ func FeCSwap(a, b *FieldElement, c uint64) { b[4] ^= t[4] } +func FeEqual(a, b *FieldElement) uint64 { + var result uint64 + for i := 0; i < 5; i++ { + result |= a[i] ^ b[i] + } + return (result ^ 0) +} + func FeFromBytes(v *FieldElement, x *[32]byte) { v[0] = uint64(x[0]) v[0] |= uint64(x[1]) << 8 diff --git a/internal/radix51/fe_test.go b/internal/radix51/fe_test.go index 217d910..ba6f4c7 100644 --- a/internal/radix51/fe_test.go +++ b/internal/radix51/fe_test.go @@ -144,6 +144,21 @@ func vartimeEqual(x, y FieldElement) bool { return true } +func TestFeEqual(t *testing.T) { + var x FieldElement = [5]uint64{1, 1, 1, 1, 1} + var y FieldElement = [5]uint64{5, 4, 3, 2, 1} + + eq := FeEqual(&x, &x) + if eq != 0 { + t.Errorf("wrong about equality") + } + + eq = FeEqual(&x, &y) + if eq == 0 { + t.Errorf("wrong about inequality") + } +} + func TestFeInvert(t *testing.T) { var x FieldElement = [5]uint64{1, 1, 1, 1, 1} var one FieldElement = [5]uint64{1, 0, 0, 0, 0} From 4d11c7b5a69c8a045beeafc24feb90e540542632 Mon Sep 17 00:00:00 2001 From: George Tankersley Date: Mon, 26 Jun 2017 02:12:45 -0400 Subject: [PATCH 07/46] group logic WIP --- const.go | 17 +- ed25519.go | 220 +++--------- ed25519_test.go | 685 +++++++++++++++++--------------------- internal/group/ge.go | 209 ++++++++++++ internal/radix51/const.go | 4 + internal/radix51/fe.go | 40 ++- 6 files changed, 603 insertions(+), 572 deletions(-) create mode 100644 internal/group/ge.go diff --git a/const.go b/const.go index ab0e1d0..48b0745 100644 --- a/const.go +++ b/const.go @@ -6,21 +6,6 @@ package ed25519 import ( "math/big" - - "github.com/gtank/ed25519/internal/edwards25519" ) -var ( - // d2 is 2*d, a curve-specific constant used in our addition formula. - d2 = edwards25519.FieldElement{ - -21827239, -5839606, -30745221, 13898782, 229458, 15978800, -12551817, -6495438, 29715968, 9444199, - } - - // the number 2 as a field element - feTwo = edwards25519.FieldElement{ - 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, - } - - bigZero = big.NewInt(0) - bigOne = big.NewInt(1) -) +var () diff --git a/ed25519.go b/ed25519.go index ca3d7ec..03e1586 100644 --- a/ed25519.go +++ b/ed25519.go @@ -9,9 +9,13 @@ import ( "math/big" "sync" - "github.com/gtank/ed25519/internal/edwards25519" + "github.com/gtank/ed25519/internal/group" + field "github.com/gtank/ed25519/internal/radix51" ) +var bigZero *big.Int +var bigOne *big.Int + type ed25519Curve struct { *elliptic.CurveParams } @@ -31,6 +35,8 @@ func initEd25519Params() { ed25519Params.Gx, _ = new(big.Int).SetString("15112221349535400772501151409588531511454012693041857206046113283949847762202", 10) ed25519Params.Gy, _ = new(big.Int).SetString("46316835694926478169428394003475163141307993866256225615783033603165251855960", 10) ed25519Params.BitSize = 256 + bigZero = big.NewInt(0) + bigOne = big.NewInt(1) } // Ed25519 returns a Curve that implements Ed25519. @@ -45,98 +51,44 @@ func (curve ed25519Curve) Params() *elliptic.CurveParams { } // IsOnCurve reports whether the given (x,y) lies on the curve by checking that -// -x^2 + y^2 - 1 - dx^2y^2 = 0. +// -x^2 + y^2 - 1 - dx^2y^2 = 0 (mod p). func (curve ed25519Curve) IsOnCurve(x, y *big.Int) bool { - lh := new(big.Int).Mul(x, x) // x^2 - y2 := new(big.Int).Mul(y, y) // y^2 - rh := new(big.Int).Mul(lh, y2) // x^2y^2 - rh.Mul(rh, curve.B) // dx^2y^2 with B repurposed as d - rh.Add(rh, bigOne) // 1 + dx^2y^2 - lh.Neg(lh) // -x^2 - lh.Add(lh, y2) // -x^2 + y^2 - lh.Sub(lh, rh) // -x^2 + y^2 - 1 - dx^2y^2 - lh.Mod(lh, curve.P) // -x^2 + y^2 - 1 - dx^2y^2 mod p - return lh.Cmp(bigZero) == 0 + var feX, feY, feB field.FieldElement + field.FeFromBig(&feX, x) + field.FeFromBig(&feY, y) + field.FeFromBig(&feB, curve.B) + + var lh, y2, rh field.FieldElement + field.FeSquare(&lh, &feX) // x^2 + field.FeSquare(&y2, &feY) // y^2 + field.FeMul(&rh, &lh, &y2) // x^2*y^2 + field.FeMul(&rh, &rh, &feB) // d*x^2*y^2 + field.FeAdd(&rh, &rh, &field.FieldOne) // 1 + d*x^2*y^2 + field.FeNeg(&lh, &lh) // -x^2 + field.FeAdd(&lh, &lh, &y2) // -x^2 + y^2 + field.FeSub(&lh, &lh, &rh) // -x^2 + y^2 - 1 - dx^2y^2 + field.FeReduce(&lh, &lh) // mod p + + return field.FeEqual(&lh, &field.FieldZero) } // Add returns the sum of (x1, y1) and (x2, y2). func (curve ed25519Curve) Add(x1, y1, x2, y2 *big.Int) (x, y *big.Int) { - var p1, p2, p3 edwards25519.ExtendedGroupElement + var p1, p2 group.ExtendedGroupElement - affineToExtended(&p1, x1, y1) - affineToExtended(&p2, x2, y2) - extendedAdd(&p3, &p1, &p2) + p1.FromAffine(x1, y1) + p2.FromAffine(x2, y2) - return extendedToAffine(&p3) // 1I + 2M + return p2.Add(&p1, &p2).ToAffine() } -// The internal edwards25519 package optimizes for a particular sequence of -// operations, so we reimplement addition (add-2008-hwcd-3) here to avoid the -// cost of converting between intermediate representations. -// TODO We know Z1=1 and Z2=1 here, so mmadd-2008-hwcd-3 (6M + 1S + 1*k + 9add) could apply -func extendedAdd(out, p1, p2 *edwards25519.ExtendedGroupElement) { - var tmp1, tmp2, A, B, C, D, E, F, G, H edwards25519.FieldElement - edwards25519.FeSub(&tmp1, &p1.Y, &p1.X) // tmp1 <-- Y1-X1 - edwards25519.FeSub(&tmp2, &p2.Y, &p2.X) // tmp2 <-- Y2-X2 - edwards25519.FeMul(&A, &tmp1, &tmp2) // A <-- tmp1*tmp2 = (Y1-X1)*(Y2-X2) - edwards25519.FeAdd(&tmp1, &p1.Y, &p1.X) // tmp1 <-- Y1+X1 - edwards25519.FeAdd(&tmp2, &p2.Y, &p2.X) // tmp2 <-- Y2+X2 - edwards25519.FeMul(&B, &tmp1, &tmp2) // B <-- tmp1*tmp2 = (Y1+X1)*(Y2+X2) - edwards25519.FeMul(&tmp1, &p1.T, &p2.T) // tmp1 <-- T1*T2 - edwards25519.FeMul(&C, &tmp1, &d2) // C <-- tmp1*2d = T1*2d*T2 - edwards25519.FeMul(&tmp1, &p1.Z, &p2.Z) // tmp1 <-- Z1*Z2 - edwards25519.FeAdd(&D, &tmp1, &tmp1) // D <-- tmp1 + tmp1 = 2*Z1*Z2 - edwards25519.FeSub(&E, &B, &A) // E <-- B-A - edwards25519.FeSub(&F, &D, &C) // F <-- D-C - edwards25519.FeAdd(&G, &D, &C) // G <-- D+C - edwards25519.FeAdd(&H, &B, &A) // H <-- B+A - edwards25519.FeMul(&out.X, &E, &F) // X3 <-- E*F - edwards25519.FeMul(&out.Y, &G, &H) // Y3 <-- G*H - edwards25519.FeMul(&out.T, &E, &H) // T3 <-- E*H - edwards25519.FeMul(&out.Z, &F, &G) // Z3 <-- F*G -} - -// Double returns 2*(x,y). Because we are always converting from affine, we can -// use "mdbl-2008-bbjlp" which assumes Z1=1. Reusing some intermediate -// values, the cost is 3S + 2M + 9*add + 1*a. +// Double returns 2*(x,y). func (curve ed25519Curve) Double(x1, y1 *big.Int) (x, y *big.Int) { - var p, q edwards25519.ProjectiveGroupElement - var t0, t1 edwards25519.FieldElement + var p group.ProjectiveGroupElement - affineToProjective(&p, x1, y1) + p.FromAffine(x1, y1) - // C = X1^2, D = Y1^2 - edwards25519.FeSquare(&t0, &p.X) - edwards25519.FeSquare(&t1, &p.Y) - edwards25519.FeMul(&p.Z, &p.X, &p.Y) - - // B = (X1+Y1)^2 = X1^2 + Y1^2 + 2*(X*Y) - edwards25519.FeAdd(&q.X, &t0, &t1) - edwards25519.FeAdd(&q.X, &q.X, &p.Z) - edwards25519.FeAdd(&q.X, &q.X, &p.Z) - - // E = a*C where a = -1 - edwards25519.FeNeg(&q.Z, &t0) - - // F = E + D - edwards25519.FeAdd(&p.X, &q.Z, &t1) - - // X3 = (B-C-D)*(F-2) - edwards25519.FeSub(&p.Y, &q.X, &t0) - edwards25519.FeSub(&p.Y, &p.Y, &t1) - edwards25519.FeSub(&p.Z, &p.X, &feTwo) - edwards25519.FeMul(&q.X, &p.Y, &p.Z) - - // Y3 = F*(E-D) - edwards25519.FeSub(&p.Y, &q.Z, &t1) - edwards25519.FeMul(&q.Y, &p.X, &p.Y) - - // Z3 = F^2 - 2*F - edwards25519.FeSquare(&q.Z, &p.X) - edwards25519.FeSub(&q.Z, &q.Z, &p.X) - edwards25519.FeSub(&q.Z, &q.Z, &p.X) - - return projectiveToAffine(&q) + return p.Double().ToAffine() } // ScalarMult returns k*(Bx,By) where k is a number in big-endian form. @@ -148,8 +100,7 @@ func (curve ed25519Curve) ScalarMult(x1, y1 *big.Int, k []byte) (x, y *big.Int) return } - var r0, r1 edwards25519.ExtendedGroupElement - var h edwards25519.CompletedGroupElement + var r0, r1 group.ExtendedGroupElement var s [32]byte curve.scalarFromBytes(&s, k) @@ -157,7 +108,7 @@ func (curve ed25519Curve) ScalarMult(x1, y1 *big.Int, k []byte) (x, y *big.Int) // Montgomery ladder init: // R_0 = O, R_1 = P r0.Zero() - affineToExtended(&r1, x1, y1) + r1.FromAffine(x1, y1) // Montgomery ladder step: // R_{1-b} = R_{1-b} + R_{b} @@ -165,17 +116,15 @@ func (curve ed25519Curve) ScalarMult(x1, y1 *big.Int, k []byte) (x, y *big.Int) for i := 255; i >= 0; i-- { var b = int32((s[i/8] >> uint(i&7)) & 1) if b == 0 { - extendedAdd(&r1, &r0, &r1) - r0.Double(&h) - h.ToExtended(&r0) + r1.Add(&r0, &r1) + r0.Double() } else { - extendedAdd(&r0, &r0, &r1) - r1.Double(&h) - h.ToExtended(&r1) + r0.Add(&r0, &r1) + r1.Double() } } - return extendedToAffine(&r0) + return r0.ToAffine() } // scalarFromBytes converts a big-endian value to a fixed-size little-endian @@ -194,84 +143,13 @@ func (curve ed25519Curve) scalarFromBytes(out *[32]byte, in []byte) { } } -// ScalarBaseMult returns k*G, where G is the base point of the group and k is -// an integer in big-endian form. -func (curve ed25519Curve) ScalarBaseMult(k []byte) (x, y *big.Int) { - var p edwards25519.ExtendedGroupElement - var scBytes [32]byte +// // ScalarBaseMult returns k*G, where G is the base point of the group and k is +// // an integer in big-endian form. +// func (curve ed25519Curve) ScalarBaseMult(k []byte) (x, y *big.Int) { +// var p edwards25519.ExtendedGroupElement +// var scBytes [32]byte - curve.scalarFromBytes(&scBytes, k) - edwards25519.GeScalarMultBase(&p, &scBytes) - return extendedToAffine(&p) -} - -// Converts (x,y) to (X:Y:T:Z) extended coordinates, or "P3" in ref10. As -// described in "Twisted Edwards Curves Revisited", Hisil-Wong-Carter-Dawson -// 2008, Section 3.1 (https://eprint.iacr.org/2008/522.pdf) -// See also https://hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html#addition-add-2008-hwcd-3 -func affineToExtended(out *edwards25519.ExtendedGroupElement, x, y *big.Int) { - feFromBig(&out.X, x) - feFromBig(&out.Y, y) - edwards25519.FeMul(&out.T, &out.X, &out.Y) - edwards25519.FeOne(&out.Z) -} - -// Extended coordinates are XYZT with x = X/Z, y = Y/Z, or the "P3" -// representation in ref10. Extended->affine is the same operation as moving -// from projective to affine. Per HWCD, it is safe to move from extended to -// projective by simply ignoring T. -func extendedToAffine(in *edwards25519.ExtendedGroupElement) (*big.Int, *big.Int) { - var x, y, zinv edwards25519.FieldElement - var bigX, bigY = new(big.Int), new(big.Int) - - edwards25519.FeInvert(&zinv, &in.Z) - edwards25519.FeMul(&x, &in.X, &zinv) - edwards25519.FeMul(&y, &in.Y, &zinv) - - feToBig(bigX, &x) - feToBig(bigY, &y) - - return bigX, bigY -} - -// Projective coordinates are XYZ with x = X/Z, y = Y/Z, or the "P2" representation in ref10. -func affineToProjective(out *edwards25519.ProjectiveGroupElement, x, y *big.Int) { - feFromBig(&out.X, x) - feFromBig(&out.Y, y) - edwards25519.FeOne(&out.Z) -} - -func projectiveToAffine(in *edwards25519.ProjectiveGroupElement) (*big.Int, *big.Int) { - var x, y, zinv edwards25519.FieldElement - var bigX, bigY = new(big.Int), new(big.Int) - - edwards25519.FeInvert(&zinv, &in.Z) - edwards25519.FeMul(&x, &in.X, &zinv) - edwards25519.FeMul(&y, &in.Y, &zinv) - - feToBig(bigX, &x) - feToBig(bigY, &y) - - return bigX, bigY -} - -func feFromBig(h *edwards25519.FieldElement, in *big.Int) { - tmp := new(big.Int).Mod(in, ed25519.P) - tmpBytes := tmp.Bytes() - var buf, reverse [32]byte - copy(buf[32-len(tmpBytes):], tmpBytes) - for i := 0; i < 32; i++ { - reverse[i] = buf[31-i] - } - edwards25519.FeFromBytes(h, &reverse) -} - -func feToBig(out *big.Int, h *edwards25519.FieldElement) { - var buf, reverse [32]byte - edwards25519.FeToBytes(&buf, h) - for i := 0; i < 32; i++ { - reverse[i] = buf[31-i] - } - out.SetBytes(reverse[:]) - out.Mod(out, ed25519.P) -} +// curve.scalarFromBytes(&scBytes, k) +// edwards25519.GeScalarMultBase(&p, &scBytes) +// return extendedToAffine(&p) +// } diff --git a/ed25519_test.go b/ed25519_test.go index 0702e5b..dd3f590 100644 --- a/ed25519_test.go +++ b/ed25519_test.go @@ -5,93 +5,20 @@ package ed25519 import ( - "bytes" - "crypto/elliptic" "crypto/rand" - "crypto/sha512" - "encoding/hex" - "io" "math/big" "testing" - "github.com/gtank/ed25519/internal/edwards25519" - "github.com/gtank/ed25519/internal/radix51" + field "github.com/gtank/ed25519/internal/radix51" ) -// TEST MATH - -// d is a constant from the curve equation. it's easy to find the value in multiple formats. -var d_fe = edwards25519.FieldElement{ - -10913610, 13857413, -15372611, 6949391, 114729, -8787816, -6275908, -3247719, -18696448, -12055116, -} -var d_bn, _ = new(big.Int).SetString("37095705934669439343138083508754565189542113879843219016388785533085940283555", 10) - -// 2*d -var d2_fe = edwards25519.FieldElement{ - -21827239, -5839606, -30745221, 13898782, 229458, 15978800, -12551817, -6495438, 29715968, 9444199, -} -var d2_bn = new(big.Int).Mul(big.NewInt(2), d_bn) - -// A is a constant from the Montgomery form of curve25519. -var A_fe = edwards25519.FieldElement{ - 486662, 0, 0, 0, 0, 0, 0, 0, 0, 0, -} -var A_bn, _ = new(big.Int).SetString("486662", 10) - -var feConvTable = []struct { - name string - fe edwards25519.FieldElement - bn *big.Int -}{ - { - name: "d", - fe: d_fe, - bn: d_bn, - }, - { - name: "d2", - fe: d2_fe, - bn: d2_bn, - }, - { - name: "A", - fe: A_fe, - bn: A_bn, - }, -} - -func TestFeFromBig(t *testing.T) { - var _ = Ed25519().(ed25519Curve) - var fe edwards25519.FieldElement - for _, c := range feConvTable { - feFromBig(&fe, c.bn) - for i := 0; i < len(fe); i++ { - if fe[i] != c.fe[i] { - t.Error("bn->fe radix conversion failed") - } - } - } -} - -func TestFeToBig(t *testing.T) { - var curve = Ed25519().(ed25519Curve) - var bn, expectedBN = new(big.Int), new(big.Int) - for _, c := range feConvTable { - expectedBN.Mod(c.bn, curve.P) - feToBig(bn, &c.fe) - if bn.Cmp(expectedBN) != 0 { - t.Errorf("fe->bn radix conversion failed: %s", c.name) - } - } -} - func TestRadixRoundTrip(t *testing.T) { if testing.Short() { t.Skip() } var curve = Ed25519().(ed25519Curve) - var fe edwards25519.FieldElement + var fe field.FieldElement var out = new(big.Int) for i := 0; i < 1000; i++ { @@ -99,8 +26,8 @@ func TestRadixRoundTrip(t *testing.T) { if err != nil { t.Fatal(err) } - feFromBig(&fe, n) - feToBig(out, &fe) + field.FeFromBig(&fe, n) + out = field.FeToBig(&fe) if n.Cmp(out) != 0 { t.Errorf("fe<>bn failed for %x", n.Bytes()) } @@ -119,6 +46,16 @@ func TestIsOnCurve(t *testing.T) { } } +func BenchmarkIsOnCurve(b *testing.B) { + ed := Ed25519() + Gx, Gy := ed.Params().Gx, ed.Params().Gy + for i := 0; i < b.N; i++ { + if !ed.IsOnCurve(Gx, Gy) { + b.Error("ed25519 base point not on curve") + } + } +} + func TestAdd(t *testing.T) { c := Ed25519().(ed25519Curve) Bx, By := c.Params().Gx, c.Params().Gy @@ -129,6 +66,14 @@ func TestAdd(t *testing.T) { } } +func BenchmarkAdd(b *testing.B) { + c := Ed25519() + Gx, Gy := c.Params().Gx, c.Params().Gy + for i := 0; i < b.N; i++ { + Gx, Gy = c.Add(Gx, Gy, Gx, Gy) + } +} + func TestDouble(t *testing.T) { c := Ed25519() Gx, Gy := c.Params().Gx, c.Params().Gy @@ -148,321 +93,303 @@ func TestDouble(t *testing.T) { } } -// Test vector generated by instrumenting x/crypto/ed25519 GenerateKey -// seed: c240344fcc6615dda52da98149377ad2b13fdba2bc39a50ba9f3afb2cbd4abaa -// expanded: f04154b9d80963bb4c76214ece8a1049bdd16fbfc5003aff9835a59643ace276 -// public: 65a8343a83ec15e55050f12fc22f2c81a4fe7327c8da1524441f9ce5e5bc27dd - -func TestScalarMultBase(t *testing.T) { - c := Ed25519() - - // endian-swapped because x/crypto assumes this is always little endian from raw bytes - // and scalarFromBytes assumes it's coming from big.Int Bytes() - a, _ := hex.DecodeString("76e2ac4396a53598ff3a00c5bf6fd1bd49108ace4e21764cbb6309d8b95441f0") - if len(a) != 32 { - t.Errorf("failed decoding") - } - - Ax, Ay := c.ScalarBaseMult(a) - - if !c.IsOnCurve(Ax, Ay) { - t.Error("scalarmultbase result was off-curve") - } - - var A edwards25519.ExtendedGroupElement - pub, _ := hex.DecodeString("65a8343a83ec15e55050f12fc22f2c81a4fe7327c8da1524441f9ce5e5bc27dd") - var pubBytes [32]byte - copy(pubBytes[:], pub) - A.FromBytes(&pubBytes) - Bx, By := extendedToAffine(&A) - - if Ax.Cmp(Bx) != 0 || Ay.Cmp(By) != 0 { - t.Error("scalarmultbase disagrees with x/crypto/ed25519") - } -} - -func TestScalarMultBaseIdentity(t *testing.T) { - var c = Ed25519() - var one = new(big.Int).Set(bigOne) - Ax, Ay := c.ScalarBaseMult(one.Bytes()) - - if Ax.Cmp(c.Params().Gx) != 0 || Ay.Cmp(c.Params().Gy) != 0 { - t.Errorf("precomputed 1*B != B") - } - - Ax, Ay = c.ScalarMult(Ax, Ay, one.Bytes()) - - if Ax.Cmp(c.Params().Gx) != 0 || Ay.Cmp(c.Params().Gy) != 0 { - t.Errorf("arbitrary 1*B != B") - } -} - -func TestScalarMultBaseInfinity(t *testing.T) { - c := Ed25519() - a, _ := hex.DecodeString("0000000000000000000000000000000000000000000000000000000000000000") - if len(a) != 32 { - t.Errorf("failed decoding") - } - - Ax, Ay := c.ScalarBaseMult(a) - - if !c.IsOnCurve(Ax, Ay) { - t.Error("scalarmultbase result was off-curve") - } - - if Ax.Cmp(bigZero) != 0 || Ay.Cmp(bigOne) != 0 { - t.Error("scalarmultbase by 0 was not point at infinity") - } -} - -func TestScalarMultsAgreeAtInfinity(t *testing.T) { - c := Ed25519() - a, _ := hex.DecodeString("0000000000000000000000000000000000000000000000000000000000000000") - if len(a) != 32 { - t.Errorf("failed decoding") - } - - Ax, Ay := c.ScalarBaseMult(a) - Bx, By := c.ScalarMult(c.Params().Gx, c.Params().Gy, a) - - if Ax.Cmp(Bx) != 0 || Ay.Cmp(By) != 0 { - t.Error("scalarmultbase disagrees with scalarmult") - } -} - -func TestScalarMultsAgreeElsewhere(t *testing.T) { - c := Ed25519() - // head -c 32 /dev/urandom | sha256sum - a, _ := hex.DecodeString("c07eea55b3322f15099b6cf4d2b7e99d3d0fa6807f6fc7a46b5f7cb78daad4e0") - - Ax, Ay := c.ScalarBaseMult(a) - Bx, By := c.ScalarMult(c.Params().Gx, c.Params().Gy, a) - - if Ax.Cmp(Bx) != 0 || Ay.Cmp(By) != 0 { - t.Error("scalarmultbase disagrees with scalarmult") - } - - if !c.IsOnCurve(Bx, By) { - t.Error("scalarmult is returning off-curve points") - } -} - -// TEST INTERFACE - -func TestMarshalingRoundTrip(t *testing.T) { - ed := Ed25519() - - a, _ := hex.DecodeString("c07eea55b3322f15099b6cf4d2b7e99d3d0fa6807f6fc7a46b5f7cb78daad4e0") - Ax, Ay := ed.ScalarBaseMult(a) - - if !ed.IsOnCurve(Ax, Ay) { - t.Error("scalarBaseMult is returning off-curve points") - } - - sec1A := elliptic.Marshal(ed, Ax, Ay) - Bx, By := elliptic.Unmarshal(ed, sec1A) - - if Ax.Cmp(Bx) != 0 || Ay.Cmp(By) != 0 { - t.Error("point did not survive elliptic.Marshal roundtrip") - } - - if !testing.Short() { - for i := 0; i < 100; i++ { - _, err := io.ReadFull(rand.Reader, a) - if err != nil { - t.Fatal(err) - } - Ax, Ay := ed.ScalarBaseMult(a) - - if !ed.IsOnCurve(Ax, Ay) { - t.Error("scalarBaseMult is returning off-curve points") - } - - sec1A := elliptic.Marshal(ed, Ax, Ay) - Bx, By := elliptic.Unmarshal(ed, sec1A) - - if Ax.Cmp(Bx) != 0 || Ay.Cmp(By) != 0 { - t.Error("point did not survive elliptic.Marshal roundtrip") - } - } - } -} - -// TEST APPLICATION - -func generateKey(r io.Reader) (sk *[32]byte, pk []byte, err error) { - if r == nil { - r = rand.Reader - } - - ed := Ed25519() - - sk = new([32]byte) - _, err = io.ReadFull(r, sk[:]) - if err != nil { - return nil, nil, err - } - - digest := sha512.Sum512(sk[:]) - digest[0] &= 248 - digest[31] &= 127 - digest[31] |= 64 - - // take the first half of expanded bytes & reverse; big.Int expects big-endian - reverseDigest := reverse32(digest[:32]) - scalar := new(big.Int).SetBytes(reverseDigest) - - // A = a*B - Ax, Ay := ed.ScalarBaseMult(scalar.Bytes()) - - // This works with the standard elliptic package marshaling - publicKey := elliptic.Marshal(ed, Ax, Ay) - - // but we can also render the compressed Edwards format - compressedEdwardsY := make([]byte, 32) - - // x, y here will be big-endian byte strings - x, y := publicKey[1:33], publicKey[33:] - - // RFC 8032: To form the encoding of the point, copy the least significant - // bit of the x-coordinate to the most significant bit of the final octet. - copy(compressedEdwardsY[:], y) - compressedEdwardsY[0] |= x[31] << 7 - compressedEdwardsY = reverse32(compressedEdwardsY) - - return sk, compressedEdwardsY, err -} - -func reverse32(b []byte) []byte { - var tmp = make([]byte, 32) - for i := 0; i <= 31; i++ { - tmp[i] = b[31-i] - } - return tmp -} - -// Test vector generated by instrumenting x/crypto/ed25519 GenerateKey(). These -// are raw values. The edwards code interprets them as little-endian, so they -// need to be reversed before use with big.Int. -var genKeyTest = struct { - seed, expanded, public string -}{ - seed: "c240344fcc6615dda52da98149377ad2b13fdba2bc39a50ba9f3afb2cbd4abaa", - expanded: "f04154b9d80963bb4c76214ece8a1049bdd16fbfc5003aff9835a59643ace276", - public: "65a8343a83ec15e55050f12fc22f2c81a4fe7327c8da1524441f9ce5e5bc27dd", -} - -func TestEdDSAGenerateKey(t *testing.T) { - fakeRandom, _ := hex.DecodeString(genKeyTest.seed) - fakeReader := bytes.NewBuffer(fakeRandom) - - sk, pk, err := generateKey(fakeReader) - if err != nil { - t.Fatal(err) - } - - expectedPK, _ := hex.DecodeString(genKeyTest.public) - if !bytes.Equal(sk[:], fakeRandom) || !bytes.Equal(pk, expectedPK) { - t.Error("generateKey output did not match test vector") - } -} - -// BENCHMARKS - -func BenchmarkIsOnCurve(b *testing.B) { - ed := Ed25519() - Gx, Gy := ed.Params().Gx, ed.Params().Gy - for i := 0; i < b.N; i++ { - if !ed.IsOnCurve(Gx, Gy) { - b.Error("ed25519 base point not on curve") - } - } -} - -func BenchmarkAdd(b *testing.B) { - ed := Ed25519() - Gx, Gy := ed.Params().Gx, ed.Params().Gy - for i := 0; i < b.N; i++ { - Gx, Gy = ed.Add(Gx, Gy, Gx, Gy) - } -} - func BenchmarkDouble(b *testing.B) { - ed := Ed25519() - Gx, Gy := ed.Params().Gx, ed.Params().Gy + c := Ed25519() + Gx, Gy := c.Params().Gx, c.Params().Gy for i := 0; i < b.N; i++ { - Gx, Gy = ed.Double(Gx, Gy) + Gx, Gy = c.Double(Gx, Gy) } } -func BenchmarkScalarBaseMult(b *testing.B) { - ed := Ed25519() +// // Test vector generated by instrumenting x/crypto/ed25519 GenerateKey +// // seed: c240344fcc6615dda52da98149377ad2b13fdba2bc39a50ba9f3afb2cbd4abaa +// // expanded: f04154b9d80963bb4c76214ece8a1049bdd16fbfc5003aff9835a59643ace276 +// // public: 65a8343a83ec15e55050f12fc22f2c81a4fe7327c8da1524441f9ce5e5bc27dd - var k [32]byte - _, err := io.ReadFull(rand.Reader, k[:]) - if err != nil { - b.Fatal(err) - } - k[0] &= 248 - k[31] &= 127 - k[31] |= 64 +// func TestScalarMultBase(t *testing.T) { +// c := Ed25519() - for i := 0; i < b.N; i++ { - _, _ = ed.ScalarBaseMult(k[:]) - } -} +// // endian-swapped because x/crypto assumes this is always little endian from raw bytes +// // and scalarFromBytes assumes it's coming from big.Int Bytes() +// a, _ := hex.DecodeString("76e2ac4396a53598ff3a00c5bf6fd1bd49108ace4e21764cbb6309d8b95441f0") +// if len(a) != 32 { +// t.Errorf("failed decoding") +// } -func BenchmarkScalarMult(b *testing.B) { - ed := Ed25519() - Bx, By := ed.Params().Gx, ed.Params().Gy +// Ax, Ay := c.ScalarBaseMult(a) - var k [32]byte - _, err := io.ReadFull(rand.Reader, k[:]) - if err != nil { - b.Fatal(err) - } - k[0] &= 248 - k[31] &= 127 - k[31] |= 64 +// if !c.IsOnCurve(Ax, Ay) { +// t.Error("scalarmultbase result was off-curve") +// } - for i := 0; i < b.N; i++ { - _, _ = ed.ScalarMult(Bx, By, k[:]) - } -} +// var A edwards25519.ExtendedGroupElement +// pub, _ := hex.DecodeString("65a8343a83ec15e55050f12fc22f2c81a4fe7327c8da1524441f9ce5e5bc27dd") +// var pubBytes [32]byte +// copy(pubBytes[:], pub) +// A.FromBytes(&pubBytes) +// Bx, By := extendedToAffine(&A) -// A is a constant from the Montgomery form of curve25519. -var radix25A = edwards25519.FieldElement{ - 486662, 0, 0, 0, 0, 0, 0, 0, 0, 0, -} +// if Ax.Cmp(Bx) != 0 || Ay.Cmp(By) != 0 { +// t.Error("scalarmultbase disagrees with x/crypto/ed25519") +// } +// } -var radix51A = radix51.FieldElement{ - 486662, 0, 0, 0, 0, -} +// func TestScalarMultBaseIdentity(t *testing.T) { +// var c = Ed25519() +// var one = new(big.Int).Set(bigOne) +// Ax, Ay := c.ScalarBaseMult(one.Bytes()) -func BenchmarkFeMul25(b *testing.B) { - var h edwards25519.FieldElement - for i := 0; i < b.N; i++ { - edwards25519.FeMul(&h, &radix25A, &radix25A) - } -} +// if Ax.Cmp(c.Params().Gx) != 0 || Ay.Cmp(c.Params().Gy) != 0 { +// t.Errorf("precomputed 1*B != B") +// } -func BenchmarkFeMul51(b *testing.B) { - var h radix51.FieldElement - for i := 0; i < b.N; i++ { - radix51.FeMul(&h, &radix51A, &radix51A) - } -} +// Ax, Ay = c.ScalarMult(Ax, Ay, one.Bytes()) -func BenchmarkFeSquare25(b *testing.B) { - var h edwards25519.FieldElement - for i := 0; i < b.N; i++ { - edwards25519.FeSquare(&h, &radix25A) - } -} +// if Ax.Cmp(c.Params().Gx) != 0 || Ay.Cmp(c.Params().Gy) != 0 { +// t.Errorf("arbitrary 1*B != B") +// } +// } -func BenchmarkFeSquare51(b *testing.B) { - var h radix51.FieldElement - for i := 0; i < b.N; i++ { - radix51.FeSquare(&h, &radix51A) - } -} +// func TestScalarMultBaseInfinity(t *testing.T) { +// c := Ed25519() +// a, _ := hex.DecodeString("0000000000000000000000000000000000000000000000000000000000000000") +// if len(a) != 32 { +// t.Errorf("failed decoding") +// } + +// Ax, Ay := c.ScalarBaseMult(a) + +// if !c.IsOnCurve(Ax, Ay) { +// t.Error("scalarmultbase result was off-curve") +// } + +// if Ax.Cmp(bigZero) != 0 || Ay.Cmp(bigOne) != 0 { +// t.Error("scalarmultbase by 0 was not point at infinity") +// } +// } + +// func TestScalarMultsAgreeAtInfinity(t *testing.T) { +// c := Ed25519() +// a, _ := hex.DecodeString("0000000000000000000000000000000000000000000000000000000000000000") +// if len(a) != 32 { +// t.Errorf("failed decoding") +// } + +// Ax, Ay := c.ScalarBaseMult(a) +// Bx, By := c.ScalarMult(c.Params().Gx, c.Params().Gy, a) + +// if Ax.Cmp(Bx) != 0 || Ay.Cmp(By) != 0 { +// t.Error("scalarmultbase disagrees with scalarmult") +// } +// } + +// func TestScalarMultsAgreeElsewhere(t *testing.T) { +// c := Ed25519() +// // head -c 32 /dev/urandom | sha256sum +// a, _ := hex.DecodeString("c07eea55b3322f15099b6cf4d2b7e99d3d0fa6807f6fc7a46b5f7cb78daad4e0") + +// Ax, Ay := c.ScalarBaseMult(a) +// Bx, By := c.ScalarMult(c.Params().Gx, c.Params().Gy, a) + +// if Ax.Cmp(Bx) != 0 || Ay.Cmp(By) != 0 { +// t.Error("scalarmultbase disagrees with scalarmult") +// } + +// if !c.IsOnCurve(Bx, By) { +// t.Error("scalarmult is returning off-curve points") +// } +// } + +// // TEST INTERFACE + +// func TestMarshalingRoundTrip(t *testing.T) { +// ed := Ed25519() + +// a, _ := hex.DecodeString("c07eea55b3322f15099b6cf4d2b7e99d3d0fa6807f6fc7a46b5f7cb78daad4e0") +// Ax, Ay := ed.ScalarBaseMult(a) + +// if !ed.IsOnCurve(Ax, Ay) { +// t.Error("scalarBaseMult is returning off-curve points") +// } + +// sec1A := elliptic.Marshal(ed, Ax, Ay) +// Bx, By := elliptic.Unmarshal(ed, sec1A) + +// if Ax.Cmp(Bx) != 0 || Ay.Cmp(By) != 0 { +// t.Error("point did not survive elliptic.Marshal roundtrip") +// } + +// if !testing.Short() { +// for i := 0; i < 100; i++ { +// _, err := io.ReadFull(rand.Reader, a) +// if err != nil { +// t.Fatal(err) +// } +// Ax, Ay := ed.ScalarBaseMult(a) + +// if !ed.IsOnCurve(Ax, Ay) { +// t.Error("scalarBaseMult is returning off-curve points") +// } + +// sec1A := elliptic.Marshal(ed, Ax, Ay) +// Bx, By := elliptic.Unmarshal(ed, sec1A) + +// if Ax.Cmp(Bx) != 0 || Ay.Cmp(By) != 0 { +// t.Error("point did not survive elliptic.Marshal roundtrip") +// } +// } +// } +// } + +// // TEST APPLICATION + +// func generateKey(r io.Reader) (sk *[32]byte, pk []byte, err error) { +// if r == nil { +// r = rand.Reader +// } + +// ed := Ed25519() + +// sk = new([32]byte) +// _, err = io.ReadFull(r, sk[:]) +// if err != nil { +// return nil, nil, err +// } + +// digest := sha512.Sum512(sk[:]) +// digest[0] &= 248 +// digest[31] &= 127 +// digest[31] |= 64 + +// // take the first half of expanded bytes & reverse; big.Int expects big-endian +// reverseDigest := reverse32(digest[:32]) +// scalar := new(big.Int).SetBytes(reverseDigest) + +// // A = a*B +// Ax, Ay := ed.ScalarBaseMult(scalar.Bytes()) + +// // This works with the standard elliptic package marshaling +// publicKey := elliptic.Marshal(ed, Ax, Ay) + +// // but we can also render the compressed Edwards format +// compressedEdwardsY := make([]byte, 32) + +// // x, y here will be big-endian byte strings +// x, y := publicKey[1:33], publicKey[33:] + +// // RFC 8032: To form the encoding of the point, copy the least significant +// // bit of the x-coordinate to the most significant bit of the final octet. +// copy(compressedEdwardsY[:], y) +// compressedEdwardsY[0] |= x[31] << 7 +// compressedEdwardsY = reverse32(compressedEdwardsY) + +// return sk, compressedEdwardsY, err +// } + +// func reverse32(b []byte) []byte { +// var tmp = make([]byte, 32) +// for i := 0; i <= 31; i++ { +// tmp[i] = b[31-i] +// } +// return tmp +// } + +// // Test vector generated by instrumenting x/crypto/ed25519 GenerateKey(). These +// // are raw values. The edwards code interprets them as little-endian, so they +// // need to be reversed before use with big.Int. +// var genKeyTest = struct { +// seed, expanded, public string +// }{ +// seed: "c240344fcc6615dda52da98149377ad2b13fdba2bc39a50ba9f3afb2cbd4abaa", +// expanded: "f04154b9d80963bb4c76214ece8a1049bdd16fbfc5003aff9835a59643ace276", +// public: "65a8343a83ec15e55050f12fc22f2c81a4fe7327c8da1524441f9ce5e5bc27dd", +// } + +// func TestEdDSAGenerateKey(t *testing.T) { +// fakeRandom, _ := hex.DecodeString(genKeyTest.seed) +// fakeReader := bytes.NewBuffer(fakeRandom) + +// sk, pk, err := generateKey(fakeReader) +// if err != nil { +// t.Fatal(err) +// } + +// expectedPK, _ := hex.DecodeString(genKeyTest.public) +// if !bytes.Equal(sk[:], fakeRandom) || !bytes.Equal(pk, expectedPK) { +// t.Error("generateKey output did not match test vector") +// } +// } + +// // BENCHMARKS + +// func BenchmarkScalarBaseMult(b *testing.B) { +// ed := Ed25519() + +// var k [32]byte +// _, err := io.ReadFull(rand.Reader, k[:]) +// if err != nil { +// b.Fatal(err) +// } +// k[0] &= 248 +// k[31] &= 127 +// k[31] |= 64 + +// for i := 0; i < b.N; i++ { +// _, _ = ed.ScalarBaseMult(k[:]) +// } +// } + +// func BenchmarkScalarMult(b *testing.B) { +// ed := Ed25519() +// Bx, By := ed.Params().Gx, ed.Params().Gy + +// var k [32]byte +// _, err := io.ReadFull(rand.Reader, k[:]) +// if err != nil { +// b.Fatal(err) +// } +// k[0] &= 248 +// k[31] &= 127 +// k[31] |= 64 + +// for i := 0; i < b.N; i++ { +// _, _ = ed.ScalarMult(Bx, By, k[:]) +// } +// } + +// // A is a constant from the Montgomery form of curve25519. +// var radix25A = edwards25519.FieldElement{ +// 486662, 0, 0, 0, 0, 0, 0, 0, 0, 0, +// } + +// var radix51A = radix51.FieldElement{ +// 486662, 0, 0, 0, 0, +// } + +// func BenchmarkFeMul25(b *testing.B) { +// var h edwards25519.FieldElement +// for i := 0; i < b.N; i++ { +// edwards25519.FeMul(&h, &radix25A, &radix25A) +// } +// } + +// func BenchmarkFeMul51(b *testing.B) { +// var h radix51.FieldElement +// for i := 0; i < b.N; i++ { +// radix51.FeMul(&h, &radix51A, &radix51A) +// } +// } + +// func BenchmarkFeSquare25(b *testing.B) { +// var h edwards25519.FieldElement +// for i := 0; i < b.N; i++ { +// edwards25519.FeSquare(&h, &radix25A) +// } +// } + +// func BenchmarkFeSquare51(b *testing.B) { +// var h radix51.FieldElement +// for i := 0; i < b.N; i++ { +// radix51.FeSquare(&h, &radix51A) +// } +// } diff --git a/internal/group/ge.go b/internal/group/ge.go new file mode 100644 index 0000000..6d2aa92 --- /dev/null +++ b/internal/group/ge.go @@ -0,0 +1,209 @@ +// Implements group logic for the Ed25519 curve. + +package group + +import ( + "math/big" + + field "github.com/gtank/ed25519/internal/radix51" +) + +// From EFD https://hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html +// An elliptic curve in twisted Edwards form has parameters a, d and coordinates +// x, y satisfying the following equations: +// +// a * x^2 + y^2 = 1 + d * x^2 * y^2 +// +// Extended coordinates assume a = -1 and represent x, y as (X, Y, Z, T) +// satisfying the following equations: +// +// x = X / Z +// y = Y / Z +// x * y = T / Z +// +// This representation was introduced in the HisilWongCarterDawson paper "Twisted +// Edwards curves revisited" (Asiacrypt 2008). +type ExtendedGroupElement struct { + X, Y, Z, T field.FieldElement +} + +// Converts (x,y) to (X:Y:T:Z) extended coordinates, or "P3" in ref10. As +// described in "Twisted Edwards Curves Revisited", Hisil-Wong-Carter-Dawson +// 2008, Section 3.1 (https://eprint.iacr.org/2008/522.pdf) +// See also https://hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html#addition-add-2008-hwcd-3 +func (v *ExtendedGroupElement) FromAffine(x, y *big.Int) { + field.FeFromBig(&v.X, x) + field.FeFromBig(&v.Y, y) + field.FeMul(&v.T, &v.X, &v.Y) + field.FeOne(&v.Z) +} + +// Extended coordinates are XYZT with x = X/Z, y = Y/Z, or the "P3" +// representation in ref10. Extended->affine is the same operation as moving +// from projective to affine. Per HWCD, it is safe to move from extended to +// projective by simply ignoring T. +func (v *ExtendedGroupElement) ToAffine() (*big.Int, *big.Int) { + var x, y, zinv field.FieldElement + + field.FeInvert(&zinv, &v.Z) + field.FeMul(&x, &v.X, &zinv) + field.FeMul(&y, &v.Y, &zinv) + + return field.FeToBig(&x), field.FeToBig(&y) +} + +// Per HWCD, it is safe to move from extended to projective by simply ignoring T. +func (v *ExtendedGroupElement) ToProjective() *ProjectiveGroupElement { + var p ProjectiveGroupElement + + field.FeCopy(&p.X, &v.X) + field.FeCopy(&p.Y, &v.Y) + field.FeCopy(&p.Z, &v.Z) + + return &p +} + +func (v *ExtendedGroupElement) Zero() *ExtendedGroupElement { + field.FeZero(&v.X) + field.FeOne(&v.Y) + field.FeOne(&v.Z) + field.FeZero(&v.T) + return v +} + +// This is the same addition formula everyone uses, "add-2008-hwcd-3". +// https://hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html#addition-add-2008-hwcd-3 +// TODO We know Z1=1 and Z2=1 here, so mmadd-2008-hwcd-3 (6M + 1S + 1*k + 9add) could apply +func (v *ExtendedGroupElement) Add(p1, p2 *ExtendedGroupElement) *ExtendedGroupElement { + var tmp1, tmp2, A, B, C, D, E, F, G, H field.FieldElement + field.FeSub(&tmp1, &p1.Y, &p1.X) // tmp1 <-- Y1-X1 + field.FeSub(&tmp2, &p2.Y, &p2.X) // tmp2 <-- Y2-X2 + field.FeMul(&A, &tmp1, &tmp2) // A <-- tmp1*tmp2 = (Y1-X1)*(Y2-X2) + field.FeAdd(&tmp1, &p1.Y, &p1.X) // tmp1 <-- Y1+X1 + field.FeAdd(&tmp2, &p2.Y, &p2.X) // tmp2 <-- Y2+X2 + field.FeMul(&B, &tmp1, &tmp2) // B <-- tmp1*tmp2 = (Y1+X1)*(Y2+X2) + field.FeMul(&tmp1, &p1.T, &p2.T) // tmp1 <-- T1*T2 + field.FeMul(&C, &tmp1, &field.D2) // C <-- tmp1*2d = T1*2d*T2 + field.FeMul(&tmp1, &p1.Z, &p2.Z) // tmp1 <-- Z1*Z2 + field.FeAdd(&D, &tmp1, &tmp1) // D <-- tmp1 + tmp1 = 2*Z1*Z2 + field.FeSub(&E, &B, &A) // E <-- B-A + field.FeSub(&F, &D, &C) // F <-- D-C + field.FeAdd(&G, &D, &C) // G <-- D+C + field.FeAdd(&H, &B, &A) // H <-- B+A + field.FeMul(&v.X, &E, &F) // X3 <-- E*F + field.FeMul(&v.Y, &G, &H) // Y3 <-- G*H + field.FeMul(&v.T, &E, &H) // T3 <-- E*H + field.FeMul(&v.Z, &F, &G) // Z3 <-- F*G + return v +} + +func (v *ExtendedGroupElement) Double() *ExtendedGroupElement { + var p ProjectiveGroupElement = v.ToProjective() + return p.Double().ToExtended() +} + +// Projective coordinates are XYZ with x = X/Z, y = Y/Z, or the "P2" +// representation in ref10. This representation has a cheaper doubling formula +// than extended coordinates. +type ProjectiveGroupElement struct { + X, Y, Z field.FieldElement +} + +func (v *ProjectiveGroupElement) FromAffine(x, y *big.Int) { + field.FeFromBig(&v.X, x) + field.FeFromBig(&v.Y, y) + field.FeOne(&v.Z) +} + +func (v *ProjectiveGroupElement) ToAffine() (*big.Int, *big.Int) { + var x, y, zinv field.FieldElement + + field.FeInvert(&zinv, &v.Z) + field.FeMul(&x, &v.X, &zinv) + field.FeMul(&y, &v.Y, &zinv) + + return field.FeToBig(&x), field.FeToBig(&y) +} + +func (v *ProjectiveGroupElement) ToExtended() *ExtendedGroupElement { + var r ExtendedGroupElement + var zinv field.FieldElement + + field.FeCopy(&r.X, &v.X) + field.FeCopy(&r.Y, &v.Y) + field.FeCopy(&r.Z, &v.Z) + + field.FeInvert(&zinv, &v.Z) + field.FeMul(&r.T, &v.X, &v.Y) // XY = ZT + field.FeMul(&r.T, &r.T, &zinv) // T = ZT/Z + + return &r +} + +func (v *ProjectiveGroupElement) Zero() *ProjectiveGroupElement { + field.FeZero(&v.X) + field.FeOne(&v.Y) + field.FeOne(&v.Z) + return v +} + +// Because we are often converting from affine, we can use "mdbl-2008-bbjlp" +// which assumes Z1=1. We also assume a = -1. +// +// Assumptions: Z1 = 1. +// Cost: 2M + 4S + 1*a + 7add + 1*2. +// Source: 2008 BernsteinBirknerJoyeLangePeters +// http://eprint.iacr.org/2008/013, plus Z1=1, plus standard simplification. +// Explicit formulas: +// +// B = (X1+Y1)^2 +// C = X1^2 +// D = Y1^2 +// E = a*C +// F = E+D +// X3 = (B-C-D)*(F-2) +// Y3 = F*(E-D) +// Z3 = F^2-2*F +// +// This assumption is one reason why this package is internal. For instance, it +// will not hold during a Montgomery ladder using extended coordinates. +// TODO: Hand off or switch entirely to dbl-2008-bbjlp when package is public. +func (v *ProjectiveGroupElement) DoubleZ1() *ProjectiveGroupElement { + var p, q ProjectiveGroupElement + var t0, t1 field.FieldElement + + p = *v + + // C = X1^2, D = Y1^2 + field.FeSquare(&t0, &p.X) + field.FeSquare(&t1, &p.Y) + field.FeMul(&p.Z, &p.X, &p.Y) + + // B = (X1+Y1)^2 = X1^2 + Y1^2 + 2*(X*Y) + field.FeAdd(&q.X, &t0, &t1) + field.FeAdd(&q.X, &q.X, &p.Z) + field.FeAdd(&q.X, &q.X, &p.Z) + + // E = a*C where a = -1 + field.FeNeg(&q.Z, &t0) + + // F = E + D + field.FeAdd(&p.X, &q.Z, &t1) + + // X3 = (B-C-D)*(F-2) + field.FeSub(&p.Y, &q.X, &t0) + field.FeSub(&p.Y, &p.Y, &t1) + field.FeSub(&p.Z, &p.X, &field.FieldTwo) + field.FeMul(&q.X, &p.Y, &p.Z) + + // Y3 = F*(E-D) + field.FeSub(&p.Y, &q.Z, &t1) + field.FeMul(&q.Y, &p.X, &p.Y) + + // Z3 = F^2 - 2*F + field.FeSquare(&q.Z, &p.X) + field.FeSub(&q.Z, &q.Z, &p.X) + field.FeSub(&q.Z, &q.Z, &p.X) + + return &q +} diff --git a/internal/radix51/const.go b/internal/radix51/const.go index ce3f2ff..fa6e7b1 100644 --- a/internal/radix51/const.go +++ b/internal/radix51/const.go @@ -11,4 +11,8 @@ const ( var ( FieldZero FieldElement = [5]uint64{0, 0, 0, 0, 0} FieldOne FieldElement = [5]uint64{1, 0, 0, 0, 0} + FieldTwo FieldElement = [5]uint64{2, 0, 0, 0, 0} + + // 2*d, used in addition formula + D2 FieldElement = [5]uint64{1859910466990425, 932731440258426, 1072319116312658, 1815898335770999, 633789495995903} ) diff --git a/internal/radix51/fe.go b/internal/radix51/fe.go index ff7e5e1..68719df 100644 --- a/internal/radix51/fe.go +++ b/internal/radix51/fe.go @@ -4,12 +4,14 @@ // public domain amd64-51-30k version of ed25519 from SUPERCOP. package radix51 +import "math/big" + // FieldElement represents an element of the field GF(2^255-19). An element t // represents the integer t[0] + t[1]*2^51 + t[2]*2^102 + t[3]*2^153 + // t[4]*2^204. type FieldElement [5]uint64 -func (v *FieldElement) FeZero() { +func FeZero(v *FieldElement) { v[0] = 0 v[1] = 0 v[2] = 0 @@ -17,7 +19,7 @@ func (v *FieldElement) FeZero() { v[4] = 0 } -func (v *FieldElement) FeOne() { +func FeOne(v *FieldElement) { v[0] = 1 v[1] = 0 v[2] = 0 @@ -26,7 +28,7 @@ func (v *FieldElement) FeOne() { } // SetInt sets the receiving FieldElement to the specified small integer. -func (v *FieldElement) SetInt(x uint64) { +func SetInt(v *FieldElement, x uint64) { v[0] = x v[1] = 0 v[2] = 0 @@ -116,7 +118,7 @@ func FeSub(out, a, b *FieldElement) { // FeNeg sets out = -a func FeNeg(out, a *FieldElement) { var t FieldElement - t.SetInt(0) + FeZero(&t) FeSub(out, &t, a) } @@ -210,12 +212,17 @@ func FeCSwap(a, b *FieldElement, c uint64) { b[4] ^= t[4] } -func FeEqual(a, b *FieldElement) uint64 { +// Returns true if two field elements are equal. +func FeEqual(a, b *FieldElement) bool { var result uint64 for i := 0; i < 5; i++ { result |= a[i] ^ b[i] } - return (result ^ 0) + return result == 0 +} + +func FeCopy(out, in *FieldElement) { + copy(out[:], in[:]) } func FeFromBytes(v *FieldElement, x *[32]byte) { @@ -306,3 +313,24 @@ func FeToBytes(r *[32]byte, v *FieldElement) { r[30] = byte((t[4] >> 36) & 0xff) r[31] = byte((t[4] >> 44)) } + +func FeFromBig(h *FieldElement, in *big.Int) { + tmpBytes := in.Bytes() + var buf, reverse [32]byte + copy(buf[32-len(tmpBytes):], tmpBytes) + for i := 0; i < 32; i++ { + reverse[i] = buf[31-i] + } + FeFromBytes(h, &reverse) +} + +func FeToBig(h *FieldElement) *big.Int { + var buf, reverse [32]byte + FeToBytes(&buf, h) // does inline reduction + for i := 0; i < 32; i++ { + reverse[i] = buf[31-i] + } + out := new(big.Int) + out.SetBytes(reverse[:]) + return out +} From c9f6a1c18390e12de9fdc9efd811f8cd81c01834 Mon Sep 17 00:00:00 2001 From: George Tankersley Date: Thu, 29 Jun 2017 00:00:00 +0000 Subject: [PATCH 08/46] use faster FeFromBig & a horrible assortment of other random changes --- const.go | 11 -------- ed25519.go | 2 +- ed25519_test.go | 55 ++++++++++++++++--------------------- internal/group/ge.go | 23 ++++++++-------- internal/radix51/fe.go | 30 ++++++++++++++------ internal/radix51/fe_test.go | 4 +-- 6 files changed, 59 insertions(+), 66 deletions(-) delete mode 100644 const.go diff --git a/const.go b/const.go deleted file mode 100644 index 48b0745..0000000 --- a/const.go +++ /dev/null @@ -1,11 +0,0 @@ -// Copyright (c) 2017 George Tankersley. All rights reserved. -// Use of this source code is governed by a BSD-style -// license that can be found in the LICENSE file. - -package ed25519 - -import ( - "math/big" -) - -var () diff --git a/ed25519.go b/ed25519.go index 03e1586..3114253 100644 --- a/ed25519.go +++ b/ed25519.go @@ -88,7 +88,7 @@ func (curve ed25519Curve) Double(x1, y1 *big.Int) (x, y *big.Int) { p.FromAffine(x1, y1) - return p.Double().ToAffine() + return p.DoubleZ1().ToAffine() } // ScalarMult returns k*(Bx,By) where k is a number in big-endian form. diff --git a/ed25519_test.go b/ed25519_test.go index dd3f590..1438990 100644 --- a/ed25519_test.go +++ b/ed25519_test.go @@ -357,39 +357,30 @@ func BenchmarkDouble(b *testing.B) { // } // } -// // A is a constant from the Montgomery form of curve25519. -// var radix25A = edwards25519.FieldElement{ -// 486662, 0, 0, 0, 0, 0, 0, 0, 0, 0, -// } +var radix51A = field.FieldElement{ + 486662, 0, 0, 0, 0, +} -// var radix51A = radix51.FieldElement{ -// 486662, 0, 0, 0, 0, -// } +func BenchmarkFeMul51(b *testing.B) { + var h field.FieldElement + for i := 0; i < b.N; i++ { + field.FeMul(&h, &radix51A, &radix51A) + } +} -// func BenchmarkFeMul25(b *testing.B) { -// var h edwards25519.FieldElement -// for i := 0; i < b.N; i++ { -// edwards25519.FeMul(&h, &radix25A, &radix25A) -// } -// } +func BenchmarkFeSquare51(b *testing.B) { + var h field.FieldElement + for i := 0; i < b.N; i++ { + field.FeSquare(&h, &radix51A) + } +} -// func BenchmarkFeMul51(b *testing.B) { -// var h radix51.FieldElement -// for i := 0; i < b.N; i++ { -// radix51.FeMul(&h, &radix51A, &radix51A) -// } -// } +var concreteCurve = Ed25519().(ed25519Curve) +var randFieldInt, _ = rand.Int(rand.Reader, concreteCurve.P) -// func BenchmarkFeSquare25(b *testing.B) { -// var h edwards25519.FieldElement -// for i := 0; i < b.N; i++ { -// edwards25519.FeSquare(&h, &radix25A) -// } -// } - -// func BenchmarkFeSquare51(b *testing.B) { -// var h radix51.FieldElement -// for i := 0; i < b.N; i++ { -// radix51.FeSquare(&h, &radix51A) -// } -// } +func BenchmarkFeFromBig(b *testing.B) { + var fe field.FieldElement + for i := 0; i < b.N; i++ { + field.FeFromBig(&fe, randFieldInt) + } +} diff --git a/internal/group/ge.go b/internal/group/ge.go index 6d2aa92..9072fdb 100644 --- a/internal/group/ge.go +++ b/internal/group/ge.go @@ -98,8 +98,8 @@ func (v *ExtendedGroupElement) Add(p1, p2 *ExtendedGroupElement) *ExtendedGroupE } func (v *ExtendedGroupElement) Double() *ExtendedGroupElement { - var p ProjectiveGroupElement = v.ToProjective() - return p.Double().ToExtended() + // return v.ToProjective().Double().ToExtended() + panic("not yet implemented") } // Projective coordinates are XYZ with x = X/Z, y = Y/Z, or the "P2" @@ -125,17 +125,16 @@ func (v *ProjectiveGroupElement) ToAffine() (*big.Int, *big.Int) { return field.FeToBig(&x), field.FeToBig(&y) } +// HWCD Section 3: "Given (X : Y : Z) in [projective coordinates] passing to +// [extended coordinates, (X : Y : T : Z)] can be performed in 3M+1S by computing +// (XZ, YZ, XY, Z^2)" func (v *ProjectiveGroupElement) ToExtended() *ExtendedGroupElement { var r ExtendedGroupElement - var zinv field.FieldElement - field.FeCopy(&r.X, &v.X) - field.FeCopy(&r.Y, &v.Y) - field.FeCopy(&r.Z, &v.Z) - - field.FeInvert(&zinv, &v.Z) - field.FeMul(&r.T, &v.X, &v.Y) // XY = ZT - field.FeMul(&r.T, &r.T, &zinv) // T = ZT/Z + field.FeMul(&r.X, &v.X, &v.Z) + field.FeMul(&r.Y, &v.Y, &v.Z) + field.FeMul(&r.T, &v.X, &v.Y) + field.FeSquare(&r.Z, &v.Z) return &r } @@ -166,8 +165,8 @@ func (v *ProjectiveGroupElement) Zero() *ProjectiveGroupElement { // Z3 = F^2-2*F // // This assumption is one reason why this package is internal. For instance, it -// will not hold during a Montgomery ladder using extended coordinates. -// TODO: Hand off or switch entirely to dbl-2008-bbjlp when package is public. +// will not hold throughout a Montgomery ladder using extended coordinates. +// TODO: Check or switch entirely to dbl-2008-bbjlp like everyone else. func (v *ProjectiveGroupElement) DoubleZ1() *ProjectiveGroupElement { var p, q ProjectiveGroupElement var t0, t1 field.FieldElement diff --git a/internal/radix51/fe.go b/internal/radix51/fe.go index 68719df..1af0a28 100644 --- a/internal/radix51/fe.go +++ b/internal/radix51/fe.go @@ -4,7 +4,10 @@ // public domain amd64-51-30k version of ed25519 from SUPERCOP. package radix51 -import "math/big" +import ( + "math/big" + "math/bits" +) // FieldElement represents an element of the field GF(2^255-19). An element t // represents the integer t[0] + t[1]*2^51 + t[2]*2^102 + t[3]*2^153 + @@ -314,14 +317,25 @@ func FeToBytes(r *[32]byte, v *FieldElement) { r[31] = byte((t[4] >> 44)) } -func FeFromBig(h *FieldElement, in *big.Int) { - tmpBytes := in.Bytes() - var buf, reverse [32]byte - copy(buf[32-len(tmpBytes):], tmpBytes) - for i := 0; i < 32; i++ { - reverse[i] = buf[31-i] +func FeFromBig(h *FieldElement, num *big.Int) { + var buf [32]byte + + offset := 0 + words := num.Bits() + numWords := len(words) + + for n := 0; n < numWords; n++ { + word := words[n] + for i := 0; i < bits.UintSize/8; i++ { + if offset >= len(buf) { + break + } + buf[offset] = byte(word >> uint((i << 3))) + offset++ + } } - FeFromBytes(h, &reverse) + + FeFromBytes(h, &buf) } func FeToBig(h *FieldElement) *big.Int { diff --git a/internal/radix51/fe_test.go b/internal/radix51/fe_test.go index ba6f4c7..15a5a06 100644 --- a/internal/radix51/fe_test.go +++ b/internal/radix51/fe_test.go @@ -149,12 +149,12 @@ func TestFeEqual(t *testing.T) { var y FieldElement = [5]uint64{5, 4, 3, 2, 1} eq := FeEqual(&x, &x) - if eq != 0 { + if !eq { t.Errorf("wrong about equality") } eq = FeEqual(&x, &y) - if eq == 0 { + if eq { t.Errorf("wrong about inequality") } } From ce27eaf07c8d7cf1ce82b737209f8b6ab0b0710d Mon Sep 17 00:00:00 2001 From: George Tankersley Date: Fri, 7 Jul 2017 00:00:00 +0000 Subject: [PATCH 09/46] Explain Z1=1 again --- ed25519.go | 1 + 1 file changed, 1 insertion(+) diff --git a/ed25519.go b/ed25519.go index 3114253..05a23d6 100644 --- a/ed25519.go +++ b/ed25519.go @@ -88,6 +88,7 @@ func (curve ed25519Curve) Double(x1, y1 *big.Int) (x, y *big.Int) { p.FromAffine(x1, y1) + // Use the special-case DoubleZ1 here because we know Z will be 1. return p.DoubleZ1().ToAffine() } From 0ba575b405d13b9c4ff6a77497d47dcb80f9cb45 Mon Sep 17 00:00:00 2001 From: George Tankersley Date: Fri, 7 Jul 2017 00:00:00 +0000 Subject: [PATCH 10/46] Implement ScalarMult using Montgomery pattern and dedicated extended-coordinates doubling. This will be slow. --- ed25519_test.go | 50 +++++++++++++++++---------- internal/group/ge.go | 82 +++++++++++++++++++++++++++++++++++++++----- 2 files changed, 105 insertions(+), 27 deletions(-) diff --git a/ed25519_test.go b/ed25519_test.go index 1438990..c979861 100644 --- a/ed25519_test.go +++ b/ed25519_test.go @@ -6,6 +6,7 @@ package ed25519 import ( "crypto/rand" + "io" "math/big" "testing" @@ -101,6 +102,37 @@ func BenchmarkDouble(b *testing.B) { } } +func TestScalarMult(t *testing.T) { + ed := Ed25519() + x, y := ed.Params().Gx, ed.Params().Gy + + twoX, twoY := ed.ScalarMult(x, y, big.NewInt(2).Bytes()) + xPlusX, yPlusY := ed.Add(x, y, x, y) + + if !ed.IsOnCurve(twoX, twoY) { + t.Error("2*B is not on the curve") + } + + if twoX.Cmp(xPlusX) != 0 || twoY.Cmp(yPlusY) != 0 { + t.Errorf("2*B != B+B") + } +} + +func BenchmarkScalarMult(b *testing.B) { + ed := Ed25519() + Bx, By := ed.Params().Gx, ed.Params().Gy + + var k [32]byte + _, err := io.ReadFull(rand.Reader, k[:]) + if err != nil { + b.Fatal(err) + } + + for i := 0; i < b.N; i++ { + _, _ = ed.ScalarMult(Bx, By, k[:]) + } +} + // // Test vector generated by instrumenting x/crypto/ed25519 GenerateKey // // seed: c240344fcc6615dda52da98149377ad2b13fdba2bc39a50ba9f3afb2cbd4abaa // // expanded: f04154b9d80963bb4c76214ece8a1049bdd16fbfc5003aff9835a59643ace276 @@ -339,24 +371,6 @@ func BenchmarkDouble(b *testing.B) { // } // } -// func BenchmarkScalarMult(b *testing.B) { -// ed := Ed25519() -// Bx, By := ed.Params().Gx, ed.Params().Gy - -// var k [32]byte -// _, err := io.ReadFull(rand.Reader, k[:]) -// if err != nil { -// b.Fatal(err) -// } -// k[0] &= 248 -// k[31] &= 127 -// k[31] |= 64 - -// for i := 0; i < b.N; i++ { -// _, _ = ed.ScalarMult(Bx, By, k[:]) -// } -// } - var radix51A = field.FieldElement{ 486662, 0, 0, 0, 0, } diff --git a/internal/group/ge.go b/internal/group/ge.go index 9072fdb..99cb31f 100644 --- a/internal/group/ge.go +++ b/internal/group/ge.go @@ -97,9 +97,71 @@ func (v *ExtendedGroupElement) Add(p1, p2 *ExtendedGroupElement) *ExtendedGroupE return v } +// This implements the explicit formulas from HWCD Section 3.3, "Dedicated +// Doubling in [extended coordinates]". +// +// Explicit formula is as follows. Cost is 4M + 4S + 1D. For Ed25519, a = -1: +// +// A ← X1^2 +// B ← Y1^2 +// C ← 2*Z1^2 +// D ← a*A +// E ← (X1+Y1)^2 − A − B +// G ← D+B +// F ← G−C +// H ← D−B +// X3 ← E*F +// Y3 ← G*H +// T3 ← E*H +// Z3 ← F*G +// +// In ref10/donna/dalek etc, this is instead handled by a faster +// mixed-coordinate doubling that results in a "Completed" group element +// instead of another point in extended coordinates. I have implemented it +// this way to see if more straightforward code is worth the (hopefully small) +// performance tradeoff. func (v *ExtendedGroupElement) Double() *ExtendedGroupElement { - // return v.ToProjective().Double().ToExtended() - panic("not yet implemented") + // TODO: Convert to projective coordinates? Section 4.3 mixed doubling? + // TODO: make a decision about how these APIs work wrt chaining/smashing + // *v = *(v.ToProjective().Double().ToExtended()) + // return v + + var A, B, C, D, E, F, G, H field.FieldElement + + // A ← X1^2, B ← Y1^2 + field.FeSquare(&A, &v.X) + field.FeSquare(&B, &v.Y) + + // C ← 2*Z1^2 + field.FeSquare(&C, &v.Z) + field.FeAdd(&C, &C, &C) // TODO should probably implement FeSquare2 + + // D ← -1*A + field.FeNeg(&D, &A) // implemented as substraction + + // E ← (X1+Y1)^2 − A − B + var t0 field.FieldElement + field.FeAdd(&t0, &v.X, &v.Y) + field.FeSquare(&t0, &t0) + field.FeSub(&E, &t0, &A) + field.FeSub(&E, &E, &B) + + // G ← D+B + field.FeAdd(&G, &D, &B) + // F ← G−C + field.FeSub(&F, &G, &C) + // H ← D−B + field.FeSub(&H, &D, &B) + // X3 ← E*F + field.FeMul(&v.X, &E, &F) + // Y3 ← G*H + field.FeMul(&v.Y, &G, &H) + // T3 ← E*H + field.FeMul(&v.T, &E, &H) + // Z3 ← F*G + field.FeMul(&v.Z, &F, &G) + + return v } // Projective coordinates are XYZ with x = X/Z, y = Y/Z, or the "P2" @@ -165,9 +227,13 @@ func (v *ProjectiveGroupElement) Zero() *ProjectiveGroupElement { // Z3 = F^2-2*F // // This assumption is one reason why this package is internal. For instance, it -// will not hold throughout a Montgomery ladder using extended coordinates. -// TODO: Check or switch entirely to dbl-2008-bbjlp like everyone else. +// will not hold throughout a Montgomery ladder, when we convert to projective +// from possibly arbitrary extended coordinates. func (v *ProjectiveGroupElement) DoubleZ1() *ProjectiveGroupElement { + // TODO This function is inconsistent with the other ones in that it + // returns a copy rather than smashing the receiver. It doesn't matter + // because it is always called on ephemeral intermediate values, but should + // fix. var p, q ProjectiveGroupElement var t0, t1 field.FieldElement @@ -176,12 +242,10 @@ func (v *ProjectiveGroupElement) DoubleZ1() *ProjectiveGroupElement { // C = X1^2, D = Y1^2 field.FeSquare(&t0, &p.X) field.FeSquare(&t1, &p.Y) - field.FeMul(&p.Z, &p.X, &p.Y) - // B = (X1+Y1)^2 = X1^2 + Y1^2 + 2*(X*Y) - field.FeAdd(&q.X, &t0, &t1) - field.FeAdd(&q.X, &q.X, &p.Z) - field.FeAdd(&q.X, &q.X, &p.Z) + // B = (X1+Y1)^2 + field.FeAdd(&p.Z, &p.X, &p.Y) // Z is irrelevant but already allocated + field.FeSquare(&q.X, &p.Z) // E = a*C where a = -1 field.FeNeg(&q.Z, &t0) From f8bd1037cf8a5263daceb06f1dc563f04c54d0bb Mon Sep 17 00:00:00 2001 From: George Tankersley Date: Fri, 7 Jul 2017 00:00:00 +0000 Subject: [PATCH 11/46] Use Bits() for FeToBig. This doesn't help quite as much as it does with FeFromBig, because FeToBig runtime is dominated by the reduction step. Still helps though. --- ed25519_test.go | 10 ++++++++++ internal/radix51/fe.go | 26 ++++++++++++++++++++------ 2 files changed, 30 insertions(+), 6 deletions(-) diff --git a/ed25519_test.go b/ed25519_test.go index c979861..8d0dc0f 100644 --- a/ed25519_test.go +++ b/ed25519_test.go @@ -398,3 +398,13 @@ func BenchmarkFeFromBig(b *testing.B) { field.FeFromBig(&fe, randFieldInt) } } + +var feOnes field.FieldElement = [5]uint64{1, 1, 1, 1, 1} + +//func FeToBig(h *FieldElement) *big.Int { +func BenchmarkFeToBig(b *testing.B) { + for i := 0; i < b.N; i++ { + _ = field.FeToBig(&feOnes) + } + +} diff --git a/internal/radix51/fe.go b/internal/radix51/fe.go index 1af0a28..d78922f 100644 --- a/internal/radix51/fe.go +++ b/internal/radix51/fe.go @@ -339,12 +339,26 @@ func FeFromBig(h *FieldElement, num *big.Int) { } func FeToBig(h *FieldElement) *big.Int { - var buf, reverse [32]byte - FeToBytes(&buf, h) // does inline reduction - for i := 0; i < 32; i++ { - reverse[i] = buf[31-i] + var buf [32]byte + FeToBytes(&buf, h) // does a reduction + + numWords := 256 / bits.UintSize + words := make([]big.Word, numWords) + + offset := 0 + byteSize := uint(bits.UintSize >> 3) + for n := 0; n < numWords; n++ { + word := uint(0) + for i := uint(0); i < byteSize; i++ { + if offset >= len(buf) { + break + } + word |= uint(buf[offset]) << (i << 3) + offset++ + } + words[n] = big.Word(word) } + out := new(big.Int) - out.SetBytes(reverse[:]) - return out + return out.SetBits(words) } From 86c2863f5b2f915dc15dbc8e1a6371bd3b316d30 Mon Sep 17 00:00:00 2001 From: George Tankersley Date: Fri, 7 Jul 2017 00:00:00 +0000 Subject: [PATCH 12/46] Add short randomized segment to TestScalarMult --- ed25519_test.go | 18 +++++++++++++++++- 1 file changed, 17 insertions(+), 1 deletion(-) diff --git a/ed25519_test.go b/ed25519_test.go index 8d0dc0f..3808847 100644 --- a/ed25519_test.go +++ b/ed25519_test.go @@ -116,6 +116,23 @@ func TestScalarMult(t *testing.T) { if twoX.Cmp(xPlusX) != 0 || twoY.Cmp(yPlusY) != 0 { t.Errorf("2*B != B+B") } + + // TODO: fuzz like it's going out of style + if !testing.Short() { + buf := make([]byte, 32) + for i := 0; i < 1000; i++ { + _, err := io.ReadFull(rand.Reader, buf) + if err != nil { + t.Fatal(err) + } + randX, randY := ed.ScalarMult(x, y, buf) + + if !ed.IsOnCurve(randX, randY) { + t.Errorf("scalarMult return off-curve point for scalar %x", buf) + } + } + } + } func BenchmarkScalarMult(b *testing.B) { @@ -406,5 +423,4 @@ func BenchmarkFeToBig(b *testing.B) { for i := 0; i < b.N; i++ { _ = field.FeToBig(&feOnes) } - } From ad98326f8c5113597010b26d2b3e291fd2f63975 Mon Sep 17 00:00:00 2001 From: George Tankersley Date: Fri, 7 Jul 2017 00:00:00 +0000 Subject: [PATCH 13/46] Remove 32-bit code and update license. The assembly is faster on target amd64 platforms. --- LICENSE | 3 +- internal/edwards25519/const.go | 1422 -------------------- internal/edwards25519/edwards25519.go | 1771 ------------------------- 3 files changed, 1 insertion(+), 3195 deletions(-) delete mode 100644 internal/edwards25519/const.go delete mode 100644 internal/edwards25519/edwards25519.go diff --git a/LICENSE b/LICENSE index 80bb50e..10f0ec1 100644 --- a/LICENSE +++ b/LICENSE @@ -1,5 +1,4 @@ -Copyright 2016 The Go Authors -Copyright 2017 George Tankersley +Copyright 2017 George Tankersley. All rights reserved. Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: diff --git a/internal/edwards25519/const.go b/internal/edwards25519/const.go deleted file mode 100644 index e39f086..0000000 --- a/internal/edwards25519/const.go +++ /dev/null @@ -1,1422 +0,0 @@ -// Copyright 2016 The Go Authors. All rights reserved. -// Use of this source code is governed by a BSD-style -// license that can be found in the LICENSE file. - -package edwards25519 - -// These values are from the public domain, “ref10” implementation of ed25519 -// from SUPERCOP. - -// d is a constant in the Edwards curve equation. -var d = FieldElement{ - -10913610, 13857413, -15372611, 6949391, 114729, -8787816, -6275908, -3247719, -18696448, -12055116, -} - -// d2 is 2*d. -var d2 = FieldElement{ - -21827239, -5839606, -30745221, 13898782, 229458, 15978800, -12551817, -6495438, 29715968, 9444199, -} - -// SqrtM1 is the square-root of -1 in the field. -var SqrtM1 = FieldElement{ - -32595792, -7943725, 9377950, 3500415, 12389472, -272473, -25146209, -2005654, 326686, 11406482, -} - -// A is a constant in the Montgomery-form of curve25519. -var A = FieldElement{ - 486662, 0, 0, 0, 0, 0, 0, 0, 0, 0, -} - -// bi contains precomputed multiples of the base-point. See the Ed25519 paper -// for a discussion about how these values are used. -var bi = [8]PreComputedGroupElement{ - { - FieldElement{25967493, -14356035, 29566456, 3660896, -12694345, 4014787, 27544626, -11754271, -6079156, 2047605}, - FieldElement{-12545711, 934262, -2722910, 3049990, -727428, 9406986, 12720692, 5043384, 19500929, -15469378}, - FieldElement{-8738181, 4489570, 9688441, -14785194, 10184609, -12363380, 29287919, 11864899, -24514362, -4438546}, - }, - { - FieldElement{15636291, -9688557, 24204773, -7912398, 616977, -16685262, 27787600, -14772189, 28944400, -1550024}, - FieldElement{16568933, 4717097, -11556148, -1102322, 15682896, -11807043, 16354577, -11775962, 7689662, 11199574}, - FieldElement{30464156, -5976125, -11779434, -15670865, 23220365, 15915852, 7512774, 10017326, -17749093, -9920357}, - }, - { - FieldElement{10861363, 11473154, 27284546, 1981175, -30064349, 12577861, 32867885, 14515107, -15438304, 10819380}, - FieldElement{4708026, 6336745, 20377586, 9066809, -11272109, 6594696, -25653668, 12483688, -12668491, 5581306}, - FieldElement{19563160, 16186464, -29386857, 4097519, 10237984, -4348115, 28542350, 13850243, -23678021, -15815942}, - }, - { - FieldElement{5153746, 9909285, 1723747, -2777874, 30523605, 5516873, 19480852, 5230134, -23952439, -15175766}, - FieldElement{-30269007, -3463509, 7665486, 10083793, 28475525, 1649722, 20654025, 16520125, 30598449, 7715701}, - FieldElement{28881845, 14381568, 9657904, 3680757, -20181635, 7843316, -31400660, 1370708, 29794553, -1409300}, - }, - { - FieldElement{-22518993, -6692182, 14201702, -8745502, -23510406, 8844726, 18474211, -1361450, -13062696, 13821877}, - FieldElement{-6455177, -7839871, 3374702, -4740862, -27098617, -10571707, 31655028, -7212327, 18853322, -14220951}, - FieldElement{4566830, -12963868, -28974889, -12240689, -7602672, -2830569, -8514358, -10431137, 2207753, -3209784}, - }, - { - FieldElement{-25154831, -4185821, 29681144, 7868801, -6854661, -9423865, -12437364, -663000, -31111463, -16132436}, - FieldElement{25576264, -2703214, 7349804, -11814844, 16472782, 9300885, 3844789, 15725684, 171356, 6466918}, - FieldElement{23103977, 13316479, 9739013, -16149481, 817875, -15038942, 8965339, -14088058, -30714912, 16193877}, - }, - { - FieldElement{-33521811, 3180713, -2394130, 14003687, -16903474, -16270840, 17238398, 4729455, -18074513, 9256800}, - FieldElement{-25182317, -4174131, 32336398, 5036987, -21236817, 11360617, 22616405, 9761698, -19827198, 630305}, - FieldElement{-13720693, 2639453, -24237460, -7406481, 9494427, -5774029, -6554551, -15960994, -2449256, -14291300}, - }, - { - FieldElement{-3151181, -5046075, 9282714, 6866145, -31907062, -863023, -18940575, 15033784, 25105118, -7894876}, - FieldElement{-24326370, 15950226, -31801215, -14592823, -11662737, -5090925, 1573892, -2625887, 2198790, -15804619}, - FieldElement{-3099351, 10324967, -2241613, 7453183, -5446979, -2735503, -13812022, -16236442, -32461234, -12290683}, - }, -} - -// base contains precomputed multiples of the base-point. See the Ed25519 paper -// for a discussion about how these values are used. -var base = [32][8]PreComputedGroupElement{ - { - { - FieldElement{25967493, -14356035, 29566456, 3660896, -12694345, 4014787, 27544626, -11754271, -6079156, 2047605}, - FieldElement{-12545711, 934262, -2722910, 3049990, -727428, 9406986, 12720692, 5043384, 19500929, -15469378}, - FieldElement{-8738181, 4489570, 9688441, -14785194, 10184609, -12363380, 29287919, 11864899, -24514362, -4438546}, - }, - { - FieldElement{-12815894, -12976347, -21581243, 11784320, -25355658, -2750717, -11717903, -3814571, -358445, -10211303}, - FieldElement{-21703237, 6903825, 27185491, 6451973, -29577724, -9554005, -15616551, 11189268, -26829678, -5319081}, - FieldElement{26966642, 11152617, 32442495, 15396054, 14353839, -12752335, -3128826, -9541118, -15472047, -4166697}, - }, - { - FieldElement{15636291, -9688557, 24204773, -7912398, 616977, -16685262, 27787600, -14772189, 28944400, -1550024}, - FieldElement{16568933, 4717097, -11556148, -1102322, 15682896, -11807043, 16354577, -11775962, 7689662, 11199574}, - FieldElement{30464156, -5976125, -11779434, -15670865, 23220365, 15915852, 7512774, 10017326, -17749093, -9920357}, - }, - { - FieldElement{-17036878, 13921892, 10945806, -6033431, 27105052, -16084379, -28926210, 15006023, 3284568, -6276540}, - FieldElement{23599295, -8306047, -11193664, -7687416, 13236774, 10506355, 7464579, 9656445, 13059162, 10374397}, - FieldElement{7798556, 16710257, 3033922, 2874086, 28997861, 2835604, 32406664, -3839045, -641708, -101325}, - }, - { - FieldElement{10861363, 11473154, 27284546, 1981175, -30064349, 12577861, 32867885, 14515107, -15438304, 10819380}, - FieldElement{4708026, 6336745, 20377586, 9066809, -11272109, 6594696, -25653668, 12483688, -12668491, 5581306}, - FieldElement{19563160, 16186464, -29386857, 4097519, 10237984, -4348115, 28542350, 13850243, -23678021, -15815942}, - }, - { - FieldElement{-15371964, -12862754, 32573250, 4720197, -26436522, 5875511, -19188627, -15224819, -9818940, -12085777}, - FieldElement{-8549212, 109983, 15149363, 2178705, 22900618, 4543417, 3044240, -15689887, 1762328, 14866737}, - FieldElement{-18199695, -15951423, -10473290, 1707278, -17185920, 3916101, -28236412, 3959421, 27914454, 4383652}, - }, - { - FieldElement{5153746, 9909285, 1723747, -2777874, 30523605, 5516873, 19480852, 5230134, -23952439, -15175766}, - FieldElement{-30269007, -3463509, 7665486, 10083793, 28475525, 1649722, 20654025, 16520125, 30598449, 7715701}, - FieldElement{28881845, 14381568, 9657904, 3680757, -20181635, 7843316, -31400660, 1370708, 29794553, -1409300}, - }, - { - FieldElement{14499471, -2729599, -33191113, -4254652, 28494862, 14271267, 30290735, 10876454, -33154098, 2381726}, - FieldElement{-7195431, -2655363, -14730155, 462251, -27724326, 3941372, -6236617, 3696005, -32300832, 15351955}, - FieldElement{27431194, 8222322, 16448760, -3907995, -18707002, 11938355, -32961401, -2970515, 29551813, 10109425}, - }, - }, - { - { - FieldElement{-13657040, -13155431, -31283750, 11777098, 21447386, 6519384, -2378284, -1627556, 10092783, -4764171}, - FieldElement{27939166, 14210322, 4677035, 16277044, -22964462, -12398139, -32508754, 12005538, -17810127, 12803510}, - FieldElement{17228999, -15661624, -1233527, 300140, -1224870, -11714777, 30364213, -9038194, 18016357, 4397660}, - }, - { - FieldElement{-10958843, -7690207, 4776341, -14954238, 27850028, -15602212, -26619106, 14544525, -17477504, 982639}, - FieldElement{29253598, 15796703, -2863982, -9908884, 10057023, 3163536, 7332899, -4120128, -21047696, 9934963}, - FieldElement{5793303, 16271923, -24131614, -10116404, 29188560, 1206517, -14747930, 4559895, -30123922, -10897950}, - }, - { - FieldElement{-27643952, -11493006, 16282657, -11036493, 28414021, -15012264, 24191034, 4541697, -13338309, 5500568}, - FieldElement{12650548, -1497113, 9052871, 11355358, -17680037, -8400164, -17430592, 12264343, 10874051, 13524335}, - FieldElement{25556948, -3045990, 714651, 2510400, 23394682, -10415330, 33119038, 5080568, -22528059, 5376628}, - }, - { - FieldElement{-26088264, -4011052, -17013699, -3537628, -6726793, 1920897, -22321305, -9447443, 4535768, 1569007}, - FieldElement{-2255422, 14606630, -21692440, -8039818, 28430649, 8775819, -30494562, 3044290, 31848280, 12543772}, - 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FieldElement{-10312768, 3936952, 9156313, -8897683, 16498692, -994647, -27481051, -666732, 3424691, 7540221}, - }, - { - FieldElement{30322361, -6964110, 11361005, -4143317, 7433304, 4989748, -7071422, -16317219, -9244265, 15258046}, - FieldElement{13054562, -2779497, 19155474, 469045, -12482797, 4566042, 5631406, 2711395, 1062915, -5136345}, - FieldElement{-19240248, -11254599, -29509029, -7499965, -5835763, 13005411, -6066489, 12194497, 32960380, 1459310}, - }, - }, - { - { - FieldElement{19852034, 7027924, 23669353, 10020366, 8586503, -6657907, 394197, -6101885, 18638003, -11174937}, - FieldElement{31395534, 15098109, 26581030, 8030562, -16527914, -5007134, 9012486, -7584354, -6643087, -5442636}, - FieldElement{-9192165, -2347377, -1997099, 4529534, 25766844, 607986, -13222, 9677543, -32294889, -6456008}, - }, - { - FieldElement{-2444496, -149937, 29348902, 8186665, 1873760, 12489863, -30934579, -7839692, -7852844, -8138429}, - FieldElement{-15236356, -15433509, 7766470, 746860, 26346930, -10221762, -27333451, 10754588, -9431476, 5203576}, - FieldElement{31834314, 14135496, -770007, 5159118, 20917671, -16768096, -7467973, -7337524, 31809243, 7347066}, - }, - { - FieldElement{-9606723, -11874240, 20414459, 13033986, 13716524, -11691881, 19797970, -12211255, 15192876, -2087490}, - FieldElement{-12663563, -2181719, 1168162, -3804809, 26747877, -14138091, 10609330, 12694420, 33473243, -13382104}, - FieldElement{33184999, 11180355, 15832085, -11385430, -1633671, 225884, 15089336, -11023903, -6135662, 14480053}, - }, - { - FieldElement{31308717, -5619998, 31030840, -1897099, 15674547, -6582883, 5496208, 13685227, 27595050, 8737275}, - FieldElement{-20318852, -15150239, 10933843, -16178022, 8335352, -7546022, -31008351, -12610604, 26498114, 66511}, - FieldElement{22644454, -8761729, -16671776, 4884562, -3105614, -13559366, 30540766, -4286747, -13327787, -7515095}, - }, - { - FieldElement{-28017847, 9834845, 18617207, -2681312, -3401956, -13307506, 8205540, 13585437, -17127465, 15115439}, - FieldElement{23711543, -672915, 31206561, -8362711, 6164647, -9709987, -33535882, -1426096, 8236921, 16492939}, - FieldElement{-23910559, -13515526, -26299483, -4503841, 25005590, -7687270, 19574902, 10071562, 6708380, -6222424}, - }, - { - FieldElement{2101391, -4930054, 19702731, 2367575, -15427167, 1047675, 5301017, 9328700, 29955601, -11678310}, - FieldElement{3096359, 9271816, -21620864, -15521844, -14847996, -7592937, -25892142, -12635595, -9917575, 6216608}, - FieldElement{-32615849, 338663, -25195611, 2510422, -29213566, -13820213, 24822830, -6146567, -26767480, 7525079}, - }, - { - FieldElement{-23066649, -13985623, 16133487, -7896178, -3389565, 778788, -910336, -2782495, -19386633, 11994101}, - FieldElement{21691500, -13624626, -641331, -14367021, 3285881, -3483596, -25064666, 9718258, -7477437, 13381418}, - FieldElement{18445390, -4202236, 14979846, 11622458, -1727110, -3582980, 23111648, -6375247, 28535282, 15779576}, - }, - { - FieldElement{30098053, 3089662, -9234387, 16662135, -21306940, 11308411, -14068454, 12021730, 9955285, -16303356}, - FieldElement{9734894, -14576830, -7473633, -9138735, 2060392, 11313496, -18426029, 9924399, 20194861, 13380996}, - FieldElement{-26378102, -7965207, -22167821, 15789297, -18055342, -6168792, -1984914, 15707771, 26342023, 10146099}, - }, - }, - { - { - FieldElement{-26016874, -219943, 21339191, -41388, 19745256, -2878700, -29637280, 2227040, 21612326, -545728}, - FieldElement{-13077387, 1184228, 23562814, -5970442, -20351244, -6348714, 25764461, 12243797, -20856566, 11649658}, - FieldElement{-10031494, 11262626, 27384172, 2271902, 26947504, -15997771, 39944, 6114064, 33514190, 2333242}, - }, - { - FieldElement{-21433588, -12421821, 8119782, 7219913, -21830522, -9016134, -6679750, -12670638, 24350578, -13450001}, - FieldElement{-4116307, -11271533, -23886186, 4843615, -30088339, 690623, -31536088, -10406836, 8317860, 12352766}, - FieldElement{18200138, -14475911, -33087759, -2696619, -23702521, -9102511, -23552096, -2287550, 20712163, 6719373}, - }, - { - FieldElement{26656208, 6075253, -7858556, 1886072, -28344043, 4262326, 11117530, -3763210, 26224235, -3297458}, - FieldElement{-17168938, -14854097, -3395676, -16369877, -19954045, 14050420, 21728352, 9493610, 18620611, -16428628}, - FieldElement{-13323321, 13325349, 11432106, 5964811, 18609221, 6062965, -5269471, -9725556, -30701573, -16479657}, - }, - { - FieldElement{-23860538, -11233159, 26961357, 1640861, -32413112, -16737940, 12248509, -5240639, 13735342, 1934062}, - FieldElement{25089769, 6742589, 17081145, -13406266, 21909293, -16067981, -15136294, -3765346, -21277997, 5473616}, - FieldElement{31883677, -7961101, 1083432, -11572403, 22828471, 13290673, -7125085, 12469656, 29111212, -5451014}, - }, - { - FieldElement{24244947, -15050407, -26262976, 2791540, -14997599, 16666678, 24367466, 6388839, -10295587, 452383}, - FieldElement{-25640782, -3417841, 5217916, 16224624, 19987036, -4082269, -24236251, -5915248, 15766062, 8407814}, - FieldElement{-20406999, 13990231, 15495425, 16395525, 5377168, 15166495, -8917023, -4388953, -8067909, 2276718}, - }, - { - FieldElement{30157918, 12924066, -17712050, 9245753, 19895028, 3368142, -23827587, 5096219, 22740376, -7303417}, - FieldElement{2041139, -14256350, 7783687, 13876377, -25946985, -13352459, 24051124, 13742383, -15637599, 13295222}, - FieldElement{33338237, -8505733, 12532113, 7977527, 9106186, -1715251, -17720195, -4612972, -4451357, -14669444}, - }, - { - FieldElement{-20045281, 5454097, -14346548, 6447146, 28862071, 1883651, -2469266, -4141880, 7770569, 9620597}, - FieldElement{23208068, 7979712, 33071466, 8149229, 1758231, -10834995, 30945528, -1694323, -33502340, -14767970}, - FieldElement{1439958, -16270480, -1079989, -793782, 4625402, 10647766, -5043801, 1220118, 30494170, -11440799}, - }, - { - FieldElement{-5037580, -13028295, -2970559, -3061767, 15640974, -6701666, -26739026, 926050, -1684339, -13333647}, - FieldElement{13908495, -3549272, 30919928, -6273825, -21521863, 7989039, 9021034, 9078865, 3353509, 4033511}, - FieldElement{-29663431, -15113610, 32259991, -344482, 24295849, -12912123, 23161163, 8839127, 27485041, 7356032}, - }, - }, - { - { - FieldElement{9661027, 705443, 11980065, -5370154, -1628543, 14661173, -6346142, 2625015, 28431036, -16771834}, - FieldElement{-23839233, -8311415, -25945511, 7480958, -17681669, -8354183, -22545972, 14150565, 15970762, 4099461}, - FieldElement{29262576, 16756590, 26350592, -8793563, 8529671, -11208050, 13617293, -9937143, 11465739, 8317062}, - }, - { - FieldElement{-25493081, -6962928, 32500200, -9419051, -23038724, -2302222, 14898637, 3848455, 20969334, -5157516}, - FieldElement{-20384450, -14347713, -18336405, 13884722, -33039454, 2842114, -21610826, -3649888, 11177095, 14989547}, - FieldElement{-24496721, -11716016, 16959896, 2278463, 12066309, 10137771, 13515641, 2581286, -28487508, 9930240}, - }, - { - FieldElement{-17751622, -2097826, 16544300, -13009300, -15914807, -14949081, 18345767, -13403753, 16291481, -5314038}, - FieldElement{-33229194, 2553288, 32678213, 9875984, 8534129, 6889387, -9676774, 6957617, 4368891, 9788741}, - FieldElement{16660756, 7281060, -10830758, 12911820, 20108584, -8101676, -21722536, -8613148, 16250552, -11111103}, - }, - { - FieldElement{-19765507, 2390526, -16551031, 14161980, 1905286, 6414907, 4689584, 10604807, -30190403, 4782747}, - FieldElement{-1354539, 14736941, -7367442, -13292886, 7710542, -14155590, -9981571, 4383045, 22546403, 437323}, - FieldElement{31665577, -12180464, -16186830, 1491339, -18368625, 3294682, 27343084, 2786261, -30633590, -14097016}, - }, - { - FieldElement{-14467279, -683715, -33374107, 7448552, 19294360, 14334329, -19690631, 2355319, -19284671, -6114373}, - FieldElement{15121312, -15796162, 6377020, -6031361, -10798111, -12957845, 18952177, 15496498, -29380133, 11754228}, - FieldElement{-2637277, -13483075, 8488727, -14303896, 12728761, -1622493, 7141596, 11724556, 22761615, -10134141}, - }, - { - FieldElement{16918416, 11729663, -18083579, 3022987, -31015732, -13339659, -28741185, -12227393, 32851222, 11717399}, - FieldElement{11166634, 7338049, -6722523, 4531520, -29468672, -7302055, 31474879, 3483633, -1193175, -4030831}, - FieldElement{-185635, 9921305, 31456609, -13536438, -12013818, 13348923, 33142652, 6546660, -19985279, -3948376}, - }, - { - FieldElement{-32460596, 11266712, -11197107, -7899103, 31703694, 3855903, -8537131, -12833048, -30772034, -15486313}, - FieldElement{-18006477, 12709068, 3991746, -6479188, -21491523, -10550425, -31135347, -16049879, 10928917, 3011958}, - FieldElement{-6957757, -15594337, 31696059, 334240, 29576716, 14796075, -30831056, -12805180, 18008031, 10258577}, - }, - { - FieldElement{-22448644, 15655569, 7018479, -4410003, -30314266, -1201591, -1853465, 1367120, 25127874, 6671743}, - FieldElement{29701166, -14373934, -10878120, 9279288, -17568, 13127210, 21382910, 11042292, 25838796, 4642684}, - FieldElement{-20430234, 14955537, -24126347, 8124619, -5369288, -5990470, 30468147, -13900640, 18423289, 4177476}, - }, - }, -} diff --git a/internal/edwards25519/edwards25519.go b/internal/edwards25519/edwards25519.go deleted file mode 100644 index 5f8b994..0000000 --- a/internal/edwards25519/edwards25519.go +++ /dev/null @@ -1,1771 +0,0 @@ -// Copyright 2016 The Go Authors. All rights reserved. -// Use of this source code is governed by a BSD-style -// license that can be found in the LICENSE file. - -package edwards25519 - -// This code is a port of the public domain, “ref10” implementation of ed25519 -// from SUPERCOP. - -// FieldElement represents an element of the field GF(2^255 - 19). An element -// t, entries t[0]...t[9], represents the integer t[0]+2^26 t[1]+2^51 t[2]+2^77 -// t[3]+2^102 t[4]+...+2^230 t[9]. Bounds on each t[i] vary depending on -// context. -type FieldElement [10]int32 - -var zero FieldElement - -func FeZero(fe *FieldElement) { - copy(fe[:], zero[:]) -} - -func FeOne(fe *FieldElement) { - FeZero(fe) - fe[0] = 1 -} - -func FeAdd(dst, a, b *FieldElement) { - dst[0] = a[0] + b[0] - dst[1] = a[1] + b[1] - dst[2] = a[2] + b[2] - dst[3] = a[3] + b[3] - dst[4] = a[4] + b[4] - dst[5] = a[5] + b[5] - dst[6] = a[6] + b[6] - dst[7] = a[7] + b[7] - dst[8] = a[8] + b[8] - dst[9] = a[9] + b[9] -} - -func FeSub(dst, a, b *FieldElement) { - dst[0] = a[0] - b[0] - dst[1] = a[1] - b[1] - dst[2] = a[2] - b[2] - dst[3] = a[3] - b[3] - dst[4] = a[4] - b[4] - dst[5] = a[5] - b[5] - dst[6] = a[6] - b[6] - dst[7] = a[7] - b[7] - dst[8] = a[8] - b[8] - dst[9] = a[9] - b[9] -} - -func FeCopy(dst, src *FieldElement) { - copy(dst[:], src[:]) -} - -// Replace (f,g) with (g,g) if b == 1; -// replace (f,g) with (f,g) if b == 0. -// -// Preconditions: b in {0,1}. -func FeCMove(f, g *FieldElement, b int32) { - b = -b - f[0] ^= b & (f[0] ^ g[0]) - f[1] ^= b & (f[1] ^ g[1]) - f[2] ^= b & (f[2] ^ g[2]) - f[3] ^= b & (f[3] ^ g[3]) - f[4] ^= b & (f[4] ^ g[4]) - f[5] ^= b & (f[5] ^ g[5]) - f[6] ^= b & (f[6] ^ g[6]) - f[7] ^= b & (f[7] ^ g[7]) - f[8] ^= b & (f[8] ^ g[8]) - f[9] ^= b & (f[9] ^ g[9]) -} - -func load3(in []byte) int64 { - var r int64 - r = int64(in[0]) - r |= int64(in[1]) << 8 - r |= int64(in[2]) << 16 - return r -} - -func load4(in []byte) int64 { - var r int64 - r = int64(in[0]) - r |= int64(in[1]) << 8 - r |= int64(in[2]) << 16 - r |= int64(in[3]) << 24 - return r -} - -func FeFromBytes(dst *FieldElement, src *[32]byte) { - h0 := load4(src[:]) - h1 := load3(src[4:]) << 6 - h2 := load3(src[7:]) << 5 - h3 := load3(src[10:]) << 3 - h4 := load3(src[13:]) << 2 - h5 := load4(src[16:]) - h6 := load3(src[20:]) << 7 - h7 := load3(src[23:]) << 5 - h8 := load3(src[26:]) << 4 - h9 := (load3(src[29:]) & 8388607) << 2 - - FeCombine(dst, h0, h1, h2, h3, h4, h5, h6, h7, h8, h9) -} - -// FeToBytes marshals h to s. -// Preconditions: -// |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. -// -// Write p=2^255-19; q=floor(h/p). -// Basic claim: q = floor(2^(-255)(h + 19 2^(-25)h9 + 2^(-1))). -// -// Proof: -// Have |h|<=p so |q|<=1 so |19^2 2^(-255) q|<1/4. -// Also have |h-2^230 h9|<2^230 so |19 2^(-255)(h-2^230 h9)|<1/4. -// -// Write y=2^(-1)-19^2 2^(-255)q-19 2^(-255)(h-2^230 h9). -// Then 0> 25 - q = (h[0] + q) >> 26 - q = (h[1] + q) >> 25 - q = (h[2] + q) >> 26 - q = (h[3] + q) >> 25 - q = (h[4] + q) >> 26 - q = (h[5] + q) >> 25 - q = (h[6] + q) >> 26 - q = (h[7] + q) >> 25 - q = (h[8] + q) >> 26 - q = (h[9] + q) >> 25 - - // Goal: Output h-(2^255-19)q, which is between 0 and 2^255-20. - h[0] += 19 * q - // Goal: Output h-2^255 q, which is between 0 and 2^255-20. - - carry[0] = h[0] >> 26 - h[1] += carry[0] - h[0] -= carry[0] << 26 - carry[1] = h[1] >> 25 - h[2] += carry[1] - h[1] -= carry[1] << 25 - carry[2] = h[2] >> 26 - h[3] += carry[2] - h[2] -= carry[2] << 26 - carry[3] = h[3] >> 25 - h[4] += carry[3] - h[3] -= carry[3] << 25 - carry[4] = h[4] >> 26 - h[5] += carry[4] - h[4] -= carry[4] << 26 - carry[5] = h[5] >> 25 - h[6] += carry[5] - h[5] -= carry[5] << 25 - carry[6] = h[6] >> 26 - h[7] += carry[6] - h[6] -= carry[6] << 26 - carry[7] = h[7] >> 25 - h[8] += carry[7] - h[7] -= carry[7] << 25 - carry[8] = h[8] >> 26 - h[9] += carry[8] - h[8] -= carry[8] << 26 - carry[9] = h[9] >> 25 - h[9] -= carry[9] << 25 - // h10 = carry9 - - // Goal: Output h[0]+...+2^255 h10-2^255 q, which is between 0 and 2^255-20. - // Have h[0]+...+2^230 h[9] between 0 and 2^255-1; - // evidently 2^255 h10-2^255 q = 0. - // Goal: Output h[0]+...+2^230 h[9]. - - s[0] = byte(h[0] >> 0) - s[1] = byte(h[0] >> 8) - s[2] = byte(h[0] >> 16) - s[3] = byte((h[0] >> 24) | (h[1] << 2)) - s[4] = byte(h[1] >> 6) - s[5] = byte(h[1] >> 14) - s[6] = byte((h[1] >> 22) | (h[2] << 3)) - s[7] = byte(h[2] >> 5) - s[8] = byte(h[2] >> 13) - s[9] = byte((h[2] >> 21) | (h[3] << 5)) - s[10] = byte(h[3] >> 3) - s[11] = byte(h[3] >> 11) - s[12] = byte((h[3] >> 19) | (h[4] << 6)) - s[13] = byte(h[4] >> 2) - s[14] = byte(h[4] >> 10) - s[15] = byte(h[4] >> 18) - s[16] = byte(h[5] >> 0) - s[17] = byte(h[5] >> 8) - s[18] = byte(h[5] >> 16) - s[19] = byte((h[5] >> 24) | (h[6] << 1)) - s[20] = byte(h[6] >> 7) - s[21] = byte(h[6] >> 15) - s[22] = byte((h[6] >> 23) | (h[7] << 3)) - s[23] = byte(h[7] >> 5) - s[24] = byte(h[7] >> 13) - s[25] = byte((h[7] >> 21) | (h[8] << 4)) - s[26] = byte(h[8] >> 4) - s[27] = byte(h[8] >> 12) - s[28] = byte((h[8] >> 20) | (h[9] << 6)) - s[29] = byte(h[9] >> 2) - s[30] = byte(h[9] >> 10) - s[31] = byte(h[9] >> 18) -} - -func FeIsNegative(f *FieldElement) byte { - var s [32]byte - FeToBytes(&s, f) - return s[0] & 1 -} - -func FeIsNonZero(f *FieldElement) int32 { - var s [32]byte - FeToBytes(&s, f) - var x uint8 - for _, b := range s { - x |= b - } - x |= x >> 4 - x |= x >> 2 - x |= x >> 1 - return int32(x & 1) -} - -// FeNeg sets h = -f -// -// Preconditions: -// |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. -// -// Postconditions: -// |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. -func FeNeg(h, f *FieldElement) { - h[0] = -f[0] - h[1] = -f[1] - h[2] = -f[2] - h[3] = -f[3] - h[4] = -f[4] - h[5] = -f[5] - h[6] = -f[6] - h[7] = -f[7] - h[8] = -f[8] - h[9] = -f[9] -} - -func FeCombine(h *FieldElement, h0, h1, h2, h3, h4, h5, h6, h7, h8, h9 int64) { - var c0, c1, c2, c3, c4, c5, c6, c7, c8, c9 int64 - - /* - |h0| <= (1.1*1.1*2^52*(1+19+19+19+19)+1.1*1.1*2^50*(38+38+38+38+38)) - i.e. |h0| <= 1.2*2^59; narrower ranges for h2, h4, h6, h8 - |h1| <= (1.1*1.1*2^51*(1+1+19+19+19+19+19+19+19+19)) - i.e. |h1| <= 1.5*2^58; narrower ranges for h3, h5, h7, h9 - */ - - c0 = (h0 + (1 << 25)) >> 26 - h1 += c0 - h0 -= c0 << 26 - c4 = (h4 + (1 << 25)) >> 26 - h5 += c4 - h4 -= c4 << 26 - /* |h0| <= 2^25 */ - /* |h4| <= 2^25 */ - /* |h1| <= 1.51*2^58 */ - /* |h5| <= 1.51*2^58 */ - - c1 = (h1 + (1 << 24)) >> 25 - h2 += c1 - h1 -= c1 << 25 - c5 = (h5 + (1 << 24)) >> 25 - h6 += c5 - h5 -= c5 << 25 - /* |h1| <= 2^24; from now on fits into int32 */ - /* |h5| <= 2^24; from now on fits into int32 */ - /* |h2| <= 1.21*2^59 */ - /* |h6| <= 1.21*2^59 */ - - c2 = (h2 + (1 << 25)) >> 26 - h3 += c2 - h2 -= c2 << 26 - c6 = (h6 + (1 << 25)) >> 26 - h7 += c6 - h6 -= c6 << 26 - /* |h2| <= 2^25; from now on fits into int32 unchanged */ - /* |h6| <= 2^25; from now on fits into int32 unchanged */ - /* |h3| <= 1.51*2^58 */ - /* |h7| <= 1.51*2^58 */ - - c3 = (h3 + (1 << 24)) >> 25 - h4 += c3 - h3 -= c3 << 25 - c7 = (h7 + (1 << 24)) >> 25 - h8 += c7 - h7 -= c7 << 25 - /* |h3| <= 2^24; from now on fits into int32 unchanged */ - /* |h7| <= 2^24; from now on fits into int32 unchanged */ - /* |h4| <= 1.52*2^33 */ - /* |h8| <= 1.52*2^33 */ - - c4 = (h4 + (1 << 25)) >> 26 - h5 += c4 - h4 -= c4 << 26 - c8 = (h8 + (1 << 25)) >> 26 - h9 += c8 - h8 -= c8 << 26 - /* |h4| <= 2^25; from now on fits into int32 unchanged */ - /* |h8| <= 2^25; from now on fits into int32 unchanged */ - /* |h5| <= 1.01*2^24 */ - /* |h9| <= 1.51*2^58 */ - - c9 = (h9 + (1 << 24)) >> 25 - h0 += c9 * 19 - h9 -= c9 << 25 - /* |h9| <= 2^24; from now on fits into int32 unchanged */ - /* |h0| <= 1.8*2^37 */ - - c0 = (h0 + (1 << 25)) >> 26 - h1 += c0 - h0 -= c0 << 26 - /* |h0| <= 2^25; from now on fits into int32 unchanged */ - /* |h1| <= 1.01*2^24 */ - - h[0] = int32(h0) - h[1] = int32(h1) - h[2] = int32(h2) - h[3] = int32(h3) - h[4] = int32(h4) - h[5] = int32(h5) - h[6] = int32(h6) - h[7] = int32(h7) - h[8] = int32(h8) - h[9] = int32(h9) -} - -// FeMul calculates h = f * g -// Can overlap h with f or g. -// -// Preconditions: -// |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. -// |g| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. -// -// Postconditions: -// |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. -// -// Notes on implementation strategy: -// -// Using schoolbook multiplication. -// Karatsuba would save a little in some cost models. -// -// Most multiplications by 2 and 19 are 32-bit precomputations; -// cheaper than 64-bit postcomputations. -// -// There is one remaining multiplication by 19 in the carry chain; -// one *19 precomputation can be merged into this, -// but the resulting data flow is considerably less clean. -// -// There are 12 carries below. -// 10 of them are 2-way parallelizable and vectorizable. -// Can get away with 11 carries, but then data flow is much deeper. -// -// With tighter constraints on inputs, can squeeze carries into int32. -func FeMul(h, f, g *FieldElement) { - f0 := int64(f[0]) - f1 := int64(f[1]) - f2 := int64(f[2]) - f3 := int64(f[3]) - f4 := int64(f[4]) - f5 := int64(f[5]) - f6 := int64(f[6]) - f7 := int64(f[7]) - f8 := int64(f[8]) - f9 := int64(f[9]) - - f1_2 := int64(2 * f[1]) - f3_2 := int64(2 * f[3]) - f5_2 := int64(2 * f[5]) - f7_2 := int64(2 * f[7]) - f9_2 := int64(2 * f[9]) - - g0 := int64(g[0]) - g1 := int64(g[1]) - g2 := int64(g[2]) - g3 := int64(g[3]) - g4 := int64(g[4]) - g5 := int64(g[5]) - g6 := int64(g[6]) - g7 := int64(g[7]) - g8 := int64(g[8]) - g9 := int64(g[9]) - - g1_19 := int64(19 * g[1]) /* 1.4*2^29 */ - g2_19 := int64(19 * g[2]) /* 1.4*2^30; still ok */ - g3_19 := int64(19 * g[3]) - g4_19 := int64(19 * g[4]) - g5_19 := int64(19 * g[5]) - g6_19 := int64(19 * g[6]) - g7_19 := int64(19 * g[7]) - g8_19 := int64(19 * g[8]) - g9_19 := int64(19 * g[9]) - - h0 := f0*g0 + f1_2*g9_19 + f2*g8_19 + f3_2*g7_19 + f4*g6_19 + f5_2*g5_19 + f6*g4_19 + f7_2*g3_19 + f8*g2_19 + f9_2*g1_19 - h1 := f0*g1 + f1*g0 + f2*g9_19 + f3*g8_19 + f4*g7_19 + f5*g6_19 + f6*g5_19 + f7*g4_19 + f8*g3_19 + f9*g2_19 - h2 := f0*g2 + f1_2*g1 + f2*g0 + f3_2*g9_19 + f4*g8_19 + f5_2*g7_19 + f6*g6_19 + f7_2*g5_19 + f8*g4_19 + f9_2*g3_19 - h3 := f0*g3 + f1*g2 + f2*g1 + f3*g0 + f4*g9_19 + f5*g8_19 + f6*g7_19 + f7*g6_19 + f8*g5_19 + f9*g4_19 - h4 := f0*g4 + f1_2*g3 + f2*g2 + f3_2*g1 + f4*g0 + f5_2*g9_19 + f6*g8_19 + f7_2*g7_19 + f8*g6_19 + f9_2*g5_19 - h5 := f0*g5 + f1*g4 + f2*g3 + f3*g2 + f4*g1 + f5*g0 + f6*g9_19 + f7*g8_19 + f8*g7_19 + f9*g6_19 - h6 := f0*g6 + f1_2*g5 + f2*g4 + f3_2*g3 + f4*g2 + f5_2*g1 + f6*g0 + f7_2*g9_19 + f8*g8_19 + f9_2*g7_19 - h7 := f0*g7 + f1*g6 + f2*g5 + f3*g4 + f4*g3 + f5*g2 + f6*g1 + f7*g0 + f8*g9_19 + f9*g8_19 - h8 := f0*g8 + f1_2*g7 + f2*g6 + f3_2*g5 + f4*g4 + f5_2*g3 + f6*g2 + f7_2*g1 + f8*g0 + f9_2*g9_19 - h9 := f0*g9 + f1*g8 + f2*g7 + f3*g6 + f4*g5 + f5*g4 + f6*g3 + f7*g2 + f8*g1 + f9*g0 - - FeCombine(h, h0, h1, h2, h3, h4, h5, h6, h7, h8, h9) -} - -func feSquare(f *FieldElement) (h0, h1, h2, h3, h4, h5, h6, h7, h8, h9 int64) { - f0 := int64(f[0]) - f1 := int64(f[1]) - f2 := int64(f[2]) - f3 := int64(f[3]) - f4 := int64(f[4]) - f5 := int64(f[5]) - f6 := int64(f[6]) - f7 := int64(f[7]) - f8 := int64(f[8]) - f9 := int64(f[9]) - f0_2 := int64(2 * f[0]) - f1_2 := int64(2 * f[1]) - f2_2 := int64(2 * f[2]) - f3_2 := int64(2 * f[3]) - f4_2 := int64(2 * f[4]) - f5_2 := int64(2 * f[5]) - f6_2 := int64(2 * f[6]) - f7_2 := int64(2 * f[7]) - f5_38 := 38 * f5 // 1.31*2^30 - f6_19 := 19 * f6 // 1.31*2^30 - f7_38 := 38 * f7 // 1.31*2^30 - f8_19 := 19 * f8 // 1.31*2^30 - f9_38 := 38 * f9 // 1.31*2^30 - - h0 = f0*f0 + f1_2*f9_38 + f2_2*f8_19 + f3_2*f7_38 + f4_2*f6_19 + f5*f5_38 - h1 = f0_2*f1 + f2*f9_38 + f3_2*f8_19 + f4*f7_38 + f5_2*f6_19 - h2 = f0_2*f2 + f1_2*f1 + f3_2*f9_38 + f4_2*f8_19 + f5_2*f7_38 + f6*f6_19 - h3 = f0_2*f3 + f1_2*f2 + f4*f9_38 + f5_2*f8_19 + f6*f7_38 - h4 = f0_2*f4 + f1_2*f3_2 + f2*f2 + f5_2*f9_38 + f6_2*f8_19 + f7*f7_38 - h5 = f0_2*f5 + f1_2*f4 + f2_2*f3 + f6*f9_38 + f7_2*f8_19 - h6 = f0_2*f6 + f1_2*f5_2 + f2_2*f4 + f3_2*f3 + f7_2*f9_38 + f8*f8_19 - h7 = f0_2*f7 + f1_2*f6 + f2_2*f5 + f3_2*f4 + f8*f9_38 - h8 = f0_2*f8 + f1_2*f7_2 + f2_2*f6 + f3_2*f5_2 + f4*f4 + f9*f9_38 - h9 = f0_2*f9 + f1_2*f8 + f2_2*f7 + f3_2*f6 + f4_2*f5 - - return -} - -// FeSquare calculates h = f*f. Can overlap h with f. -// -// Preconditions: -// |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc. -// -// Postconditions: -// |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc. -func FeSquare(h, f *FieldElement) { - h0, h1, h2, h3, h4, h5, h6, h7, h8, h9 := feSquare(f) - FeCombine(h, h0, h1, h2, h3, h4, h5, h6, h7, h8, h9) -} - -// FeSquare2 sets h = 2 * f * f -// -// Can overlap h with f. -// -// Preconditions: -// |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc. -// -// Postconditions: -// |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc. -// See fe_mul.c for discussion of implementation strategy. -func FeSquare2(h, f *FieldElement) { - h0, h1, h2, h3, h4, h5, h6, h7, h8, h9 := feSquare(f) - - h0 += h0 - h1 += h1 - h2 += h2 - h3 += h3 - h4 += h4 - h5 += h5 - h6 += h6 - h7 += h7 - h8 += h8 - h9 += h9 - - FeCombine(h, h0, h1, h2, h3, h4, h5, h6, h7, h8, h9) -} - -func FeInvert(out, z *FieldElement) { - var t0, t1, t2, t3 FieldElement - var i int - - FeSquare(&t0, z) // 2^1 - FeSquare(&t1, &t0) // 2^2 - for i = 1; i < 2; i++ { // 2^3 - FeSquare(&t1, &t1) - } - FeMul(&t1, z, &t1) // 2^3 + 2^0 - FeMul(&t0, &t0, &t1) // 2^3 + 2^1 + 2^0 - FeSquare(&t2, &t0) // 2^4 + 2^2 + 2^1 - FeMul(&t1, &t1, &t2) // 2^4 + 2^3 + 2^2 + 2^1 + 2^0 - FeSquare(&t2, &t1) // 5,4,3,2,1 - for i = 1; i < 5; i++ { // 9,8,7,6,5 - FeSquare(&t2, &t2) - } - FeMul(&t1, &t2, &t1) // 9,8,7,6,5,4,3,2,1,0 - FeSquare(&t2, &t1) // 10..1 - for i = 1; i < 10; i++ { // 19..10 - FeSquare(&t2, &t2) - } - FeMul(&t2, &t2, &t1) // 19..0 - FeSquare(&t3, &t2) // 20..1 - for i = 1; i < 20; i++ { // 39..20 - FeSquare(&t3, &t3) - } - FeMul(&t2, &t3, &t2) // 39..0 - FeSquare(&t2, &t2) // 40..1 - for i = 1; i < 10; i++ { // 49..10 - FeSquare(&t2, &t2) - } - FeMul(&t1, &t2, &t1) // 49..0 - FeSquare(&t2, &t1) // 50..1 - for i = 1; i < 50; i++ { // 99..50 - FeSquare(&t2, &t2) - } - FeMul(&t2, &t2, &t1) // 99..0 - FeSquare(&t3, &t2) // 100..1 - for i = 1; i < 100; i++ { // 199..100 - FeSquare(&t3, &t3) - } - FeMul(&t2, &t3, &t2) // 199..0 - FeSquare(&t2, &t2) // 200..1 - for i = 1; i < 50; i++ { // 249..50 - FeSquare(&t2, &t2) - } - FeMul(&t1, &t2, &t1) // 249..0 - FeSquare(&t1, &t1) // 250..1 - for i = 1; i < 5; i++ { // 254..5 - FeSquare(&t1, &t1) - } - FeMul(out, &t1, &t0) // 254..5,3,1,0 -} - -func fePow22523(out, z *FieldElement) { - var t0, t1, t2 FieldElement - var i int - - FeSquare(&t0, z) - for i = 1; i < 1; i++ { - FeSquare(&t0, &t0) - } - FeSquare(&t1, &t0) - for i = 1; i < 2; i++ { - FeSquare(&t1, &t1) - } - FeMul(&t1, z, &t1) - FeMul(&t0, &t0, &t1) - FeSquare(&t0, &t0) - for i = 1; i < 1; i++ { - FeSquare(&t0, &t0) - } - FeMul(&t0, &t1, &t0) - FeSquare(&t1, &t0) - for i = 1; i < 5; i++ { - FeSquare(&t1, &t1) - } - FeMul(&t0, &t1, &t0) - FeSquare(&t1, &t0) - for i = 1; i < 10; i++ { - FeSquare(&t1, &t1) - } - FeMul(&t1, &t1, &t0) - FeSquare(&t2, &t1) - for i = 1; i < 20; i++ { - FeSquare(&t2, &t2) - } - FeMul(&t1, &t2, &t1) - FeSquare(&t1, &t1) - for i = 1; i < 10; i++ { - FeSquare(&t1, &t1) - } - FeMul(&t0, &t1, &t0) - FeSquare(&t1, &t0) - for i = 1; i < 50; i++ { - FeSquare(&t1, &t1) - } - FeMul(&t1, &t1, &t0) - FeSquare(&t2, &t1) - for i = 1; i < 100; i++ { - FeSquare(&t2, &t2) - } - FeMul(&t1, &t2, &t1) - FeSquare(&t1, &t1) - for i = 1; i < 50; i++ { - FeSquare(&t1, &t1) - } - FeMul(&t0, &t1, &t0) - FeSquare(&t0, &t0) - for i = 1; i < 2; i++ { - FeSquare(&t0, &t0) - } - FeMul(out, &t0, z) -} - -// Group elements are members of the elliptic curve -x^2 + y^2 = 1 + d * x^2 * -// y^2 where d = -121665/121666. -// -// Several representations are used: -// ProjectiveGroupElement: (X:Y:Z) satisfying x=X/Z, y=Y/Z -// ExtendedGroupElement: (X:Y:Z:T) satisfying x=X/Z, y=Y/Z, XY=ZT -// CompletedGroupElement: ((X:Z),(Y:T)) satisfying x=X/Z, y=Y/T -// PreComputedGroupElement: (y+x,y-x,2dxy) - -type ProjectiveGroupElement struct { - X, Y, Z FieldElement -} - -type ExtendedGroupElement struct { - X, Y, Z, T FieldElement -} - -type CompletedGroupElement struct { - X, Y, Z, T FieldElement -} - -type PreComputedGroupElement struct { - yPlusX, yMinusX, xy2d FieldElement -} - -type CachedGroupElement struct { - yPlusX, yMinusX, Z, T2d FieldElement -} - -func (p *ProjectiveGroupElement) Zero() { - FeZero(&p.X) - FeOne(&p.Y) - FeOne(&p.Z) -} - -func (p *ProjectiveGroupElement) Double(r *CompletedGroupElement) { - var t0 FieldElement - - FeSquare(&r.X, &p.X) - FeSquare(&r.Z, &p.Y) - FeSquare2(&r.T, &p.Z) - FeAdd(&r.Y, &p.X, &p.Y) - FeSquare(&t0, &r.Y) - FeAdd(&r.Y, &r.Z, &r.X) - FeSub(&r.Z, &r.Z, &r.X) - FeSub(&r.X, &t0, &r.Y) - FeSub(&r.T, &r.T, &r.Z) -} - -func (p *ProjectiveGroupElement) ToBytes(s *[32]byte) { - var recip, x, y FieldElement - - FeInvert(&recip, &p.Z) - FeMul(&x, &p.X, &recip) - FeMul(&y, &p.Y, &recip) - FeToBytes(s, &y) - s[31] ^= FeIsNegative(&x) << 7 -} - -func (p *ExtendedGroupElement) Zero() { - FeZero(&p.X) - FeOne(&p.Y) - FeOne(&p.Z) - FeZero(&p.T) -} - -func (p *ExtendedGroupElement) Double(r *CompletedGroupElement) { - var q ProjectiveGroupElement - p.ToProjective(&q) - q.Double(r) -} - -func (p *ExtendedGroupElement) ToCached(r *CachedGroupElement) { - FeAdd(&r.yPlusX, &p.Y, &p.X) - FeSub(&r.yMinusX, &p.Y, &p.X) - FeCopy(&r.Z, &p.Z) - FeMul(&r.T2d, &p.T, &d2) -} - -func (p *ExtendedGroupElement) ToProjective(r *ProjectiveGroupElement) { - FeCopy(&r.X, &p.X) - FeCopy(&r.Y, &p.Y) - FeCopy(&r.Z, &p.Z) -} - -func (p *ExtendedGroupElement) ToBytes(s *[32]byte) { - var recip, x, y FieldElement - - FeInvert(&recip, &p.Z) - FeMul(&x, &p.X, &recip) - FeMul(&y, &p.Y, &recip) - FeToBytes(s, &y) - s[31] ^= FeIsNegative(&x) << 7 -} - -func (p *ExtendedGroupElement) FromBytes(s *[32]byte) bool { - var u, v, v3, vxx, check FieldElement - - FeFromBytes(&p.Y, s) - FeOne(&p.Z) - FeSquare(&u, &p.Y) - FeMul(&v, &u, &d) - FeSub(&u, &u, &p.Z) // y = y^2-1 - FeAdd(&v, &v, &p.Z) // v = dy^2+1 - - FeSquare(&v3, &v) - FeMul(&v3, &v3, &v) // v3 = v^3 - FeSquare(&p.X, &v3) - FeMul(&p.X, &p.X, &v) - FeMul(&p.X, &p.X, &u) // x = uv^7 - - fePow22523(&p.X, &p.X) // x = (uv^7)^((q-5)/8) - FeMul(&p.X, &p.X, &v3) - FeMul(&p.X, &p.X, &u) // x = uv^3(uv^7)^((q-5)/8) - - var tmpX, tmp2 [32]byte - - FeSquare(&vxx, &p.X) - FeMul(&vxx, &vxx, &v) - FeSub(&check, &vxx, &u) // vx^2-u - if FeIsNonZero(&check) == 1 { - FeAdd(&check, &vxx, &u) // vx^2+u - if FeIsNonZero(&check) == 1 { - return false - } - FeMul(&p.X, &p.X, &SqrtM1) - - FeToBytes(&tmpX, &p.X) - for i, v := range tmpX { - tmp2[31-i] = v - } - } - - if FeIsNegative(&p.X) != (s[31] >> 7) { - FeNeg(&p.X, &p.X) - } - - FeMul(&p.T, &p.X, &p.Y) - return true -} - -func (p *CompletedGroupElement) ToProjective(r *ProjectiveGroupElement) { - FeMul(&r.X, &p.X, &p.T) - FeMul(&r.Y, &p.Y, &p.Z) - FeMul(&r.Z, &p.Z, &p.T) -} - -func (p *CompletedGroupElement) ToExtended(r *ExtendedGroupElement) { - FeMul(&r.X, &p.X, &p.T) - FeMul(&r.Y, &p.Y, &p.Z) - FeMul(&r.Z, &p.Z, &p.T) - FeMul(&r.T, &p.X, &p.Y) -} - -func (p *PreComputedGroupElement) Zero() { - FeOne(&p.yPlusX) - FeOne(&p.yMinusX) - FeZero(&p.xy2d) -} - -func geAdd(r *CompletedGroupElement, p *ExtendedGroupElement, q *CachedGroupElement) { - var t0 FieldElement - - FeAdd(&r.X, &p.Y, &p.X) - FeSub(&r.Y, &p.Y, &p.X) - FeMul(&r.Z, &r.X, &q.yPlusX) - FeMul(&r.Y, &r.Y, &q.yMinusX) - FeMul(&r.T, &q.T2d, &p.T) - FeMul(&r.X, &p.Z, &q.Z) - FeAdd(&t0, &r.X, &r.X) - FeSub(&r.X, &r.Z, &r.Y) - FeAdd(&r.Y, &r.Z, &r.Y) - FeAdd(&r.Z, &t0, &r.T) - FeSub(&r.T, &t0, &r.T) -} - -func geSub(r *CompletedGroupElement, p *ExtendedGroupElement, q *CachedGroupElement) { - var t0 FieldElement - - FeAdd(&r.X, &p.Y, &p.X) - FeSub(&r.Y, &p.Y, &p.X) - FeMul(&r.Z, &r.X, &q.yMinusX) - FeMul(&r.Y, &r.Y, &q.yPlusX) - FeMul(&r.T, &q.T2d, &p.T) - FeMul(&r.X, &p.Z, &q.Z) - FeAdd(&t0, &r.X, &r.X) - FeSub(&r.X, &r.Z, &r.Y) - FeAdd(&r.Y, &r.Z, &r.Y) - FeSub(&r.Z, &t0, &r.T) - FeAdd(&r.T, &t0, &r.T) -} - -func geMixedAdd(r *CompletedGroupElement, p *ExtendedGroupElement, q *PreComputedGroupElement) { - var t0 FieldElement - - FeAdd(&r.X, &p.Y, &p.X) - FeSub(&r.Y, &p.Y, &p.X) - FeMul(&r.Z, &r.X, &q.yPlusX) - FeMul(&r.Y, &r.Y, &q.yMinusX) - FeMul(&r.T, &q.xy2d, &p.T) - FeAdd(&t0, &p.Z, &p.Z) - FeSub(&r.X, &r.Z, &r.Y) - FeAdd(&r.Y, &r.Z, &r.Y) - FeAdd(&r.Z, &t0, &r.T) - FeSub(&r.T, &t0, &r.T) -} - -func geMixedSub(r *CompletedGroupElement, p *ExtendedGroupElement, q *PreComputedGroupElement) { - var t0 FieldElement - - FeAdd(&r.X, &p.Y, &p.X) - FeSub(&r.Y, &p.Y, &p.X) - FeMul(&r.Z, &r.X, &q.yMinusX) - FeMul(&r.Y, &r.Y, &q.yPlusX) - FeMul(&r.T, &q.xy2d, &p.T) - FeAdd(&t0, &p.Z, &p.Z) - FeSub(&r.X, &r.Z, &r.Y) - FeAdd(&r.Y, &r.Z, &r.Y) - FeSub(&r.Z, &t0, &r.T) - FeAdd(&r.T, &t0, &r.T) -} - -func slide(r *[256]int8, a *[32]byte) { - for i := range r { - r[i] = int8(1 & (a[i>>3] >> uint(i&7))) - } - - for i := range r { - if r[i] != 0 { - for b := 1; b <= 6 && i+b < 256; b++ { - if r[i+b] != 0 { - if r[i]+(r[i+b]<= -15 { - r[i] -= r[i+b] << uint(b) - for k := i + b; k < 256; k++ { - if r[k] == 0 { - r[k] = 1 - break - } - r[k] = 0 - } - } else { - break - } - } - } - } - } -} - -// GeDoubleScalarMultVartime sets r = a*A + b*B -// where a = a[0]+256*a[1]+...+256^31 a[31]. -// and b = b[0]+256*b[1]+...+256^31 b[31]. -// B is the Ed25519 base point (x,4/5) with x positive. -func GeDoubleScalarMultVartime(r *ProjectiveGroupElement, a *[32]byte, A *ExtendedGroupElement, b *[32]byte) { - var aSlide, bSlide [256]int8 - var Ai [8]CachedGroupElement // A,3A,5A,7A,9A,11A,13A,15A - var t CompletedGroupElement - var u, A2 ExtendedGroupElement - var i int - - slide(&aSlide, a) - slide(&bSlide, b) - - A.ToCached(&Ai[0]) - A.Double(&t) - t.ToExtended(&A2) - - for i := 0; i < 7; i++ { - geAdd(&t, &A2, &Ai[i]) - t.ToExtended(&u) - u.ToCached(&Ai[i+1]) - } - - r.Zero() - - for i = 255; i >= 0; i-- { - if aSlide[i] != 0 || bSlide[i] != 0 { - break - } - } - - for ; i >= 0; i-- { - r.Double(&t) - - if aSlide[i] > 0 { - t.ToExtended(&u) - geAdd(&t, &u, &Ai[aSlide[i]/2]) - } else if aSlide[i] < 0 { - t.ToExtended(&u) - geSub(&t, &u, &Ai[(-aSlide[i])/2]) - } - - if bSlide[i] > 0 { - t.ToExtended(&u) - geMixedAdd(&t, &u, &bi[bSlide[i]/2]) - } else if bSlide[i] < 0 { - t.ToExtended(&u) - geMixedSub(&t, &u, &bi[(-bSlide[i])/2]) - } - - t.ToProjective(r) - } -} - -// equal returns 1 if b == c and 0 otherwise, assuming that b and c are -// non-negative. -func equal(b, c int32) int32 { - x := uint32(b ^ c) - x-- - return int32(x >> 31) -} - -// negative returns 1 if b < 0 and 0 otherwise. -func negative(b int32) int32 { - return (b >> 31) & 1 -} - -func PreComputedGroupElementCMove(t, u *PreComputedGroupElement, b int32) { - FeCMove(&t.yPlusX, &u.yPlusX, b) - FeCMove(&t.yMinusX, &u.yMinusX, b) - FeCMove(&t.xy2d, &u.xy2d, b) -} - -func selectPoint(t *PreComputedGroupElement, pos int32, b int32) { - var minusT PreComputedGroupElement - bNegative := negative(b) - bAbs := b - (((-bNegative) & b) << 1) - - t.Zero() - for i := int32(0); i < 8; i++ { - PreComputedGroupElementCMove(t, &base[pos][i], equal(bAbs, i+1)) - } - FeCopy(&minusT.yPlusX, &t.yMinusX) - FeCopy(&minusT.yMinusX, &t.yPlusX) - FeNeg(&minusT.xy2d, &t.xy2d) - PreComputedGroupElementCMove(t, &minusT, bNegative) -} - -// GeScalarMultBase computes h = a*B, where -// a = a[0]+256*a[1]+...+256^31 a[31] -// B is the Ed25519 base point (x,4/5) with x positive. -// -// Preconditions: -// a[31] <= 127 -func GeScalarMultBase(h *ExtendedGroupElement, a *[32]byte) { - var e [64]int8 - - for i, v := range a { - e[2*i] = int8(v & 15) - e[2*i+1] = int8((v >> 4) & 15) - } - - // each e[i] is between 0 and 15 and e[63] is between 0 and 7. - - carry := int8(0) - for i := 0; i < 63; i++ { - e[i] += carry - carry = (e[i] + 8) >> 4 - e[i] -= carry << 4 - } - e[63] += carry - // each e[i] is between -8 and 8. - - h.Zero() - var t PreComputedGroupElement - var r CompletedGroupElement - for i := int32(1); i < 64; i += 2 { - selectPoint(&t, i/2, int32(e[i])) - geMixedAdd(&r, h, &t) - r.ToExtended(h) - } - - var s ProjectiveGroupElement - - h.Double(&r) - r.ToProjective(&s) - s.Double(&r) - r.ToProjective(&s) - s.Double(&r) - r.ToProjective(&s) - s.Double(&r) - r.ToExtended(h) - - for i := int32(0); i < 64; i += 2 { - selectPoint(&t, i/2, int32(e[i])) - geMixedAdd(&r, h, &t) - r.ToExtended(h) - } -} - -// The scalars are GF(2^252 + 27742317777372353535851937790883648493). - -// Input: -// a[0]+256*a[1]+...+256^31*a[31] = a -// b[0]+256*b[1]+...+256^31*b[31] = b -// c[0]+256*c[1]+...+256^31*c[31] = c -// -// Output: -// s[0]+256*s[1]+...+256^31*s[31] = (ab+c) mod l -// where l = 2^252 + 27742317777372353535851937790883648493. -func ScMulAdd(s, a, b, c *[32]byte) { - a0 := 2097151 & load3(a[:]) - a1 := 2097151 & (load4(a[2:]) >> 5) - a2 := 2097151 & (load3(a[5:]) >> 2) - a3 := 2097151 & (load4(a[7:]) >> 7) - a4 := 2097151 & (load4(a[10:]) >> 4) - a5 := 2097151 & (load3(a[13:]) >> 1) - a6 := 2097151 & (load4(a[15:]) >> 6) - a7 := 2097151 & (load3(a[18:]) >> 3) - a8 := 2097151 & load3(a[21:]) - a9 := 2097151 & (load4(a[23:]) >> 5) - a10 := 2097151 & (load3(a[26:]) >> 2) - a11 := (load4(a[28:]) >> 7) - b0 := 2097151 & load3(b[:]) - b1 := 2097151 & (load4(b[2:]) >> 5) - b2 := 2097151 & (load3(b[5:]) >> 2) - b3 := 2097151 & (load4(b[7:]) >> 7) - b4 := 2097151 & (load4(b[10:]) >> 4) - b5 := 2097151 & (load3(b[13:]) >> 1) - b6 := 2097151 & (load4(b[15:]) >> 6) - b7 := 2097151 & (load3(b[18:]) >> 3) - b8 := 2097151 & load3(b[21:]) - b9 := 2097151 & (load4(b[23:]) >> 5) - b10 := 2097151 & (load3(b[26:]) >> 2) - b11 := (load4(b[28:]) >> 7) - c0 := 2097151 & load3(c[:]) - c1 := 2097151 & (load4(c[2:]) >> 5) - c2 := 2097151 & (load3(c[5:]) >> 2) - c3 := 2097151 & (load4(c[7:]) >> 7) - c4 := 2097151 & (load4(c[10:]) >> 4) - c5 := 2097151 & (load3(c[13:]) >> 1) - c6 := 2097151 & (load4(c[15:]) >> 6) - c7 := 2097151 & (load3(c[18:]) >> 3) - c8 := 2097151 & load3(c[21:]) - c9 := 2097151 & (load4(c[23:]) >> 5) - c10 := 2097151 & (load3(c[26:]) >> 2) - c11 := (load4(c[28:]) >> 7) - var carry [23]int64 - - s0 := c0 + a0*b0 - s1 := c1 + a0*b1 + a1*b0 - s2 := c2 + a0*b2 + a1*b1 + a2*b0 - s3 := c3 + a0*b3 + a1*b2 + a2*b1 + a3*b0 - s4 := c4 + a0*b4 + a1*b3 + a2*b2 + a3*b1 + a4*b0 - s5 := c5 + a0*b5 + a1*b4 + a2*b3 + a3*b2 + a4*b1 + a5*b0 - s6 := c6 + a0*b6 + a1*b5 + a2*b4 + a3*b3 + a4*b2 + a5*b1 + a6*b0 - s7 := c7 + a0*b7 + a1*b6 + a2*b5 + a3*b4 + a4*b3 + a5*b2 + a6*b1 + a7*b0 - s8 := c8 + a0*b8 + a1*b7 + a2*b6 + a3*b5 + a4*b4 + a5*b3 + a6*b2 + a7*b1 + a8*b0 - s9 := c9 + a0*b9 + a1*b8 + a2*b7 + a3*b6 + a4*b5 + a5*b4 + a6*b3 + a7*b2 + a8*b1 + a9*b0 - s10 := c10 + a0*b10 + a1*b9 + a2*b8 + a3*b7 + a4*b6 + a5*b5 + a6*b4 + a7*b3 + a8*b2 + a9*b1 + a10*b0 - s11 := c11 + a0*b11 + a1*b10 + a2*b9 + a3*b8 + a4*b7 + a5*b6 + a6*b5 + a7*b4 + a8*b3 + a9*b2 + a10*b1 + a11*b0 - s12 := a1*b11 + a2*b10 + a3*b9 + a4*b8 + a5*b7 + a6*b6 + a7*b5 + a8*b4 + a9*b3 + a10*b2 + a11*b1 - s13 := a2*b11 + a3*b10 + a4*b9 + a5*b8 + a6*b7 + a7*b6 + a8*b5 + a9*b4 + a10*b3 + a11*b2 - s14 := a3*b11 + a4*b10 + a5*b9 + a6*b8 + a7*b7 + a8*b6 + a9*b5 + a10*b4 + a11*b3 - s15 := a4*b11 + a5*b10 + a6*b9 + a7*b8 + a8*b7 + a9*b6 + a10*b5 + a11*b4 - s16 := a5*b11 + a6*b10 + a7*b9 + a8*b8 + a9*b7 + a10*b6 + a11*b5 - s17 := a6*b11 + a7*b10 + a8*b9 + a9*b8 + a10*b7 + a11*b6 - s18 := a7*b11 + a8*b10 + a9*b9 + a10*b8 + a11*b7 - s19 := a8*b11 + a9*b10 + a10*b9 + a11*b8 - s20 := a9*b11 + a10*b10 + a11*b9 - s21 := a10*b11 + a11*b10 - s22 := a11 * b11 - s23 := int64(0) - - carry[0] = (s0 + (1 << 20)) >> 21 - s1 += carry[0] - s0 -= carry[0] << 21 - carry[2] = (s2 + (1 << 20)) >> 21 - s3 += carry[2] - s2 -= carry[2] << 21 - carry[4] = (s4 + (1 << 20)) >> 21 - s5 += carry[4] - s4 -= carry[4] << 21 - carry[6] = (s6 + (1 << 20)) >> 21 - s7 += carry[6] - s6 -= carry[6] << 21 - carry[8] = (s8 + (1 << 20)) >> 21 - s9 += carry[8] - s8 -= carry[8] << 21 - carry[10] = (s10 + (1 << 20)) >> 21 - s11 += carry[10] - s10 -= carry[10] << 21 - carry[12] = (s12 + (1 << 20)) >> 21 - s13 += carry[12] - s12 -= carry[12] << 21 - carry[14] = (s14 + (1 << 20)) >> 21 - s15 += carry[14] - s14 -= carry[14] << 21 - carry[16] = (s16 + (1 << 20)) >> 21 - s17 += carry[16] - s16 -= carry[16] << 21 - carry[18] = (s18 + (1 << 20)) >> 21 - s19 += carry[18] - s18 -= carry[18] << 21 - carry[20] = (s20 + (1 << 20)) >> 21 - s21 += carry[20] - s20 -= carry[20] << 21 - carry[22] = (s22 + (1 << 20)) >> 21 - s23 += carry[22] - s22 -= carry[22] << 21 - - carry[1] = (s1 + (1 << 20)) >> 21 - s2 += carry[1] - s1 -= carry[1] << 21 - carry[3] = (s3 + (1 << 20)) >> 21 - s4 += carry[3] - s3 -= carry[3] << 21 - carry[5] = (s5 + (1 << 20)) >> 21 - s6 += carry[5] - s5 -= carry[5] << 21 - carry[7] = (s7 + (1 << 20)) >> 21 - s8 += carry[7] - s7 -= carry[7] << 21 - carry[9] = (s9 + (1 << 20)) >> 21 - s10 += carry[9] - s9 -= carry[9] << 21 - carry[11] = (s11 + (1 << 20)) >> 21 - s12 += carry[11] - s11 -= carry[11] << 21 - carry[13] = (s13 + (1 << 20)) >> 21 - s14 += carry[13] - s13 -= carry[13] << 21 - carry[15] = (s15 + (1 << 20)) >> 21 - s16 += carry[15] - s15 -= carry[15] << 21 - carry[17] = (s17 + (1 << 20)) >> 21 - s18 += carry[17] - s17 -= carry[17] << 21 - carry[19] = (s19 + (1 << 20)) >> 21 - s20 += carry[19] - s19 -= carry[19] << 21 - carry[21] = (s21 + (1 << 20)) >> 21 - s22 += carry[21] - s21 -= carry[21] << 21 - - s11 += s23 * 666643 - s12 += s23 * 470296 - s13 += s23 * 654183 - s14 -= s23 * 997805 - s15 += s23 * 136657 - s16 -= s23 * 683901 - s23 = 0 - - s10 += s22 * 666643 - s11 += s22 * 470296 - s12 += s22 * 654183 - s13 -= s22 * 997805 - s14 += s22 * 136657 - s15 -= s22 * 683901 - s22 = 0 - - s9 += s21 * 666643 - s10 += s21 * 470296 - s11 += s21 * 654183 - s12 -= s21 * 997805 - s13 += s21 * 136657 - s14 -= s21 * 683901 - s21 = 0 - - s8 += s20 * 666643 - s9 += s20 * 470296 - s10 += s20 * 654183 - s11 -= s20 * 997805 - s12 += s20 * 136657 - s13 -= s20 * 683901 - s20 = 0 - - s7 += s19 * 666643 - s8 += s19 * 470296 - s9 += s19 * 654183 - s10 -= s19 * 997805 - s11 += s19 * 136657 - s12 -= s19 * 683901 - s19 = 0 - - s6 += s18 * 666643 - s7 += s18 * 470296 - s8 += s18 * 654183 - s9 -= s18 * 997805 - s10 += s18 * 136657 - s11 -= s18 * 683901 - s18 = 0 - - carry[6] = (s6 + (1 << 20)) >> 21 - s7 += carry[6] - s6 -= carry[6] << 21 - carry[8] = (s8 + (1 << 20)) >> 21 - s9 += carry[8] - s8 -= carry[8] << 21 - carry[10] = (s10 + (1 << 20)) >> 21 - s11 += carry[10] - s10 -= carry[10] << 21 - carry[12] = (s12 + (1 << 20)) >> 21 - s13 += carry[12] - s12 -= carry[12] << 21 - carry[14] = (s14 + (1 << 20)) >> 21 - s15 += carry[14] - s14 -= carry[14] << 21 - carry[16] = (s16 + (1 << 20)) >> 21 - s17 += carry[16] - s16 -= carry[16] << 21 - - carry[7] = (s7 + (1 << 20)) >> 21 - s8 += carry[7] - s7 -= carry[7] << 21 - carry[9] = (s9 + (1 << 20)) >> 21 - s10 += carry[9] - s9 -= carry[9] << 21 - carry[11] = (s11 + (1 << 20)) >> 21 - s12 += carry[11] - s11 -= carry[11] << 21 - carry[13] = (s13 + (1 << 20)) >> 21 - s14 += carry[13] - s13 -= carry[13] << 21 - carry[15] = (s15 + (1 << 20)) >> 21 - s16 += carry[15] - s15 -= carry[15] << 21 - - s5 += s17 * 666643 - s6 += s17 * 470296 - s7 += s17 * 654183 - s8 -= s17 * 997805 - s9 += s17 * 136657 - s10 -= s17 * 683901 - s17 = 0 - - s4 += s16 * 666643 - s5 += s16 * 470296 - s6 += s16 * 654183 - s7 -= s16 * 997805 - s8 += s16 * 136657 - s9 -= s16 * 683901 - s16 = 0 - - s3 += s15 * 666643 - s4 += s15 * 470296 - s5 += s15 * 654183 - s6 -= s15 * 997805 - s7 += s15 * 136657 - s8 -= s15 * 683901 - s15 = 0 - - s2 += s14 * 666643 - s3 += s14 * 470296 - s4 += s14 * 654183 - s5 -= s14 * 997805 - s6 += s14 * 136657 - s7 -= s14 * 683901 - s14 = 0 - - s1 += s13 * 666643 - s2 += s13 * 470296 - s3 += s13 * 654183 - s4 -= s13 * 997805 - s5 += s13 * 136657 - s6 -= s13 * 683901 - s13 = 0 - - s0 += s12 * 666643 - s1 += s12 * 470296 - s2 += s12 * 654183 - s3 -= s12 * 997805 - s4 += s12 * 136657 - s5 -= s12 * 683901 - s12 = 0 - - carry[0] = (s0 + (1 << 20)) >> 21 - s1 += carry[0] - s0 -= carry[0] << 21 - carry[2] = (s2 + (1 << 20)) >> 21 - s3 += carry[2] - s2 -= carry[2] << 21 - carry[4] = (s4 + (1 << 20)) >> 21 - s5 += carry[4] - s4 -= carry[4] << 21 - carry[6] = (s6 + (1 << 20)) >> 21 - s7 += carry[6] - s6 -= carry[6] << 21 - carry[8] = (s8 + (1 << 20)) >> 21 - s9 += carry[8] - s8 -= carry[8] << 21 - carry[10] = (s10 + (1 << 20)) >> 21 - s11 += carry[10] - s10 -= carry[10] << 21 - - carry[1] = (s1 + (1 << 20)) >> 21 - s2 += carry[1] - s1 -= carry[1] << 21 - carry[3] = (s3 + (1 << 20)) >> 21 - s4 += carry[3] - s3 -= carry[3] << 21 - carry[5] = (s5 + (1 << 20)) >> 21 - s6 += carry[5] - s5 -= carry[5] << 21 - carry[7] = (s7 + (1 << 20)) >> 21 - s8 += carry[7] - s7 -= carry[7] << 21 - carry[9] = (s9 + (1 << 20)) >> 21 - s10 += carry[9] - s9 -= carry[9] << 21 - carry[11] = (s11 + (1 << 20)) >> 21 - s12 += carry[11] - s11 -= carry[11] << 21 - - s0 += s12 * 666643 - s1 += s12 * 470296 - s2 += s12 * 654183 - s3 -= s12 * 997805 - s4 += s12 * 136657 - s5 -= s12 * 683901 - s12 = 0 - - carry[0] = s0 >> 21 - s1 += carry[0] - s0 -= carry[0] << 21 - carry[1] = s1 >> 21 - s2 += carry[1] - s1 -= carry[1] << 21 - carry[2] = s2 >> 21 - s3 += carry[2] - s2 -= carry[2] << 21 - carry[3] = s3 >> 21 - s4 += carry[3] - s3 -= carry[3] << 21 - carry[4] = s4 >> 21 - s5 += carry[4] - s4 -= carry[4] << 21 - carry[5] = s5 >> 21 - s6 += carry[5] - s5 -= carry[5] << 21 - carry[6] = s6 >> 21 - s7 += carry[6] - s6 -= carry[6] << 21 - carry[7] = s7 >> 21 - s8 += carry[7] - s7 -= carry[7] << 21 - carry[8] = s8 >> 21 - s9 += carry[8] - s8 -= carry[8] << 21 - carry[9] = s9 >> 21 - s10 += carry[9] - s9 -= carry[9] << 21 - carry[10] = s10 >> 21 - s11 += carry[10] - s10 -= carry[10] << 21 - carry[11] = s11 >> 21 - s12 += carry[11] - s11 -= carry[11] << 21 - - s0 += s12 * 666643 - s1 += s12 * 470296 - s2 += s12 * 654183 - s3 -= s12 * 997805 - s4 += s12 * 136657 - s5 -= s12 * 683901 - s12 = 0 - - carry[0] = s0 >> 21 - s1 += carry[0] - s0 -= carry[0] << 21 - carry[1] = s1 >> 21 - s2 += carry[1] - s1 -= carry[1] << 21 - carry[2] = s2 >> 21 - s3 += carry[2] - s2 -= carry[2] << 21 - carry[3] = s3 >> 21 - s4 += carry[3] - s3 -= carry[3] << 21 - carry[4] = s4 >> 21 - s5 += carry[4] - s4 -= carry[4] << 21 - carry[5] = s5 >> 21 - s6 += carry[5] - s5 -= carry[5] << 21 - carry[6] = s6 >> 21 - s7 += carry[6] - s6 -= carry[6] << 21 - carry[7] = s7 >> 21 - s8 += carry[7] - s7 -= carry[7] << 21 - carry[8] = s8 >> 21 - s9 += carry[8] - s8 -= carry[8] << 21 - carry[9] = s9 >> 21 - s10 += carry[9] - s9 -= carry[9] << 21 - carry[10] = s10 >> 21 - s11 += carry[10] - s10 -= carry[10] << 21 - - s[0] = byte(s0 >> 0) - s[1] = byte(s0 >> 8) - s[2] = byte((s0 >> 16) | (s1 << 5)) - s[3] = byte(s1 >> 3) - s[4] = byte(s1 >> 11) - s[5] = byte((s1 >> 19) | (s2 << 2)) - s[6] = byte(s2 >> 6) - s[7] = byte((s2 >> 14) | (s3 << 7)) - s[8] = byte(s3 >> 1) - s[9] = byte(s3 >> 9) - s[10] = byte((s3 >> 17) | (s4 << 4)) - s[11] = byte(s4 >> 4) - s[12] = byte(s4 >> 12) - s[13] = byte((s4 >> 20) | (s5 << 1)) - s[14] = byte(s5 >> 7) - s[15] = byte((s5 >> 15) | (s6 << 6)) - s[16] = byte(s6 >> 2) - s[17] = byte(s6 >> 10) - s[18] = byte((s6 >> 18) | (s7 << 3)) - s[19] = byte(s7 >> 5) - s[20] = byte(s7 >> 13) - s[21] = byte(s8 >> 0) - s[22] = byte(s8 >> 8) - s[23] = byte((s8 >> 16) | (s9 << 5)) - s[24] = byte(s9 >> 3) - s[25] = byte(s9 >> 11) - s[26] = byte((s9 >> 19) | (s10 << 2)) - s[27] = byte(s10 >> 6) - s[28] = byte((s10 >> 14) | (s11 << 7)) - s[29] = byte(s11 >> 1) - s[30] = byte(s11 >> 9) - s[31] = byte(s11 >> 17) -} - -// Input: -// s[0]+256*s[1]+...+256^63*s[63] = s -// -// Output: -// s[0]+256*s[1]+...+256^31*s[31] = s mod l -// where l = 2^252 + 27742317777372353535851937790883648493. -func ScReduce(out *[32]byte, s *[64]byte) { - s0 := 2097151 & load3(s[:]) - s1 := 2097151 & (load4(s[2:]) >> 5) - s2 := 2097151 & (load3(s[5:]) >> 2) - s3 := 2097151 & (load4(s[7:]) >> 7) - s4 := 2097151 & (load4(s[10:]) >> 4) - s5 := 2097151 & (load3(s[13:]) >> 1) - s6 := 2097151 & (load4(s[15:]) >> 6) - s7 := 2097151 & (load3(s[18:]) >> 3) - s8 := 2097151 & load3(s[21:]) - s9 := 2097151 & (load4(s[23:]) >> 5) - s10 := 2097151 & (load3(s[26:]) >> 2) - s11 := 2097151 & (load4(s[28:]) >> 7) - s12 := 2097151 & (load4(s[31:]) >> 4) - s13 := 2097151 & (load3(s[34:]) >> 1) - s14 := 2097151 & (load4(s[36:]) >> 6) - s15 := 2097151 & (load3(s[39:]) >> 3) - s16 := 2097151 & load3(s[42:]) - s17 := 2097151 & (load4(s[44:]) >> 5) - s18 := 2097151 & (load3(s[47:]) >> 2) - s19 := 2097151 & (load4(s[49:]) >> 7) - s20 := 2097151 & (load4(s[52:]) >> 4) - s21 := 2097151 & (load3(s[55:]) >> 1) - s22 := 2097151 & (load4(s[57:]) >> 6) - s23 := (load4(s[60:]) >> 3) - - s11 += s23 * 666643 - s12 += s23 * 470296 - s13 += s23 * 654183 - s14 -= s23 * 997805 - s15 += s23 * 136657 - s16 -= s23 * 683901 - s23 = 0 - - s10 += s22 * 666643 - s11 += s22 * 470296 - s12 += s22 * 654183 - s13 -= s22 * 997805 - s14 += s22 * 136657 - s15 -= s22 * 683901 - s22 = 0 - - s9 += s21 * 666643 - s10 += s21 * 470296 - s11 += s21 * 654183 - s12 -= s21 * 997805 - s13 += s21 * 136657 - s14 -= s21 * 683901 - s21 = 0 - - s8 += s20 * 666643 - s9 += s20 * 470296 - s10 += s20 * 654183 - s11 -= s20 * 997805 - s12 += s20 * 136657 - s13 -= s20 * 683901 - s20 = 0 - - s7 += s19 * 666643 - s8 += s19 * 470296 - s9 += s19 * 654183 - s10 -= s19 * 997805 - s11 += s19 * 136657 - s12 -= s19 * 683901 - s19 = 0 - - s6 += s18 * 666643 - s7 += s18 * 470296 - s8 += s18 * 654183 - s9 -= s18 * 997805 - s10 += s18 * 136657 - s11 -= s18 * 683901 - s18 = 0 - - var carry [17]int64 - - carry[6] = (s6 + (1 << 20)) >> 21 - s7 += carry[6] - s6 -= carry[6] << 21 - carry[8] = (s8 + (1 << 20)) >> 21 - s9 += carry[8] - s8 -= carry[8] << 21 - carry[10] = (s10 + (1 << 20)) >> 21 - s11 += carry[10] - s10 -= carry[10] << 21 - carry[12] = (s12 + (1 << 20)) >> 21 - s13 += carry[12] - s12 -= carry[12] << 21 - carry[14] = (s14 + (1 << 20)) >> 21 - s15 += carry[14] - s14 -= carry[14] << 21 - carry[16] = (s16 + (1 << 20)) >> 21 - s17 += carry[16] - s16 -= carry[16] << 21 - - carry[7] = (s7 + (1 << 20)) >> 21 - s8 += carry[7] - s7 -= carry[7] << 21 - carry[9] = (s9 + (1 << 20)) >> 21 - s10 += carry[9] - s9 -= carry[9] << 21 - carry[11] = (s11 + (1 << 20)) >> 21 - s12 += carry[11] - s11 -= carry[11] << 21 - carry[13] = (s13 + (1 << 20)) >> 21 - s14 += carry[13] - s13 -= carry[13] << 21 - carry[15] = (s15 + (1 << 20)) >> 21 - s16 += carry[15] - s15 -= carry[15] << 21 - - s5 += s17 * 666643 - s6 += s17 * 470296 - s7 += s17 * 654183 - s8 -= s17 * 997805 - s9 += s17 * 136657 - s10 -= s17 * 683901 - s17 = 0 - - s4 += s16 * 666643 - s5 += s16 * 470296 - s6 += s16 * 654183 - s7 -= s16 * 997805 - s8 += s16 * 136657 - s9 -= s16 * 683901 - s16 = 0 - - s3 += s15 * 666643 - s4 += s15 * 470296 - s5 += s15 * 654183 - s6 -= s15 * 997805 - s7 += s15 * 136657 - s8 -= s15 * 683901 - s15 = 0 - - s2 += s14 * 666643 - s3 += s14 * 470296 - s4 += s14 * 654183 - s5 -= s14 * 997805 - s6 += s14 * 136657 - s7 -= s14 * 683901 - s14 = 0 - - s1 += s13 * 666643 - s2 += s13 * 470296 - s3 += s13 * 654183 - s4 -= s13 * 997805 - s5 += s13 * 136657 - s6 -= s13 * 683901 - s13 = 0 - - s0 += s12 * 666643 - s1 += s12 * 470296 - s2 += s12 * 654183 - s3 -= s12 * 997805 - s4 += s12 * 136657 - s5 -= s12 * 683901 - s12 = 0 - - carry[0] = (s0 + (1 << 20)) >> 21 - s1 += carry[0] - s0 -= carry[0] << 21 - carry[2] = (s2 + (1 << 20)) >> 21 - s3 += carry[2] - s2 -= carry[2] << 21 - carry[4] = (s4 + (1 << 20)) >> 21 - s5 += carry[4] - s4 -= carry[4] << 21 - carry[6] = (s6 + (1 << 20)) >> 21 - s7 += carry[6] - s6 -= carry[6] << 21 - carry[8] = (s8 + (1 << 20)) >> 21 - s9 += carry[8] - s8 -= carry[8] << 21 - carry[10] = (s10 + (1 << 20)) >> 21 - s11 += carry[10] - s10 -= carry[10] << 21 - - carry[1] = (s1 + (1 << 20)) >> 21 - s2 += carry[1] - s1 -= carry[1] << 21 - carry[3] = (s3 + (1 << 20)) >> 21 - s4 += carry[3] - s3 -= carry[3] << 21 - carry[5] = (s5 + (1 << 20)) >> 21 - s6 += carry[5] - s5 -= carry[5] << 21 - carry[7] = (s7 + (1 << 20)) >> 21 - s8 += carry[7] - s7 -= carry[7] << 21 - carry[9] = (s9 + (1 << 20)) >> 21 - s10 += carry[9] - s9 -= carry[9] << 21 - carry[11] = (s11 + (1 << 20)) >> 21 - s12 += carry[11] - s11 -= carry[11] << 21 - - s0 += s12 * 666643 - s1 += s12 * 470296 - s2 += s12 * 654183 - s3 -= s12 * 997805 - s4 += s12 * 136657 - s5 -= s12 * 683901 - s12 = 0 - - carry[0] = s0 >> 21 - s1 += carry[0] - s0 -= carry[0] << 21 - carry[1] = s1 >> 21 - s2 += carry[1] - s1 -= carry[1] << 21 - carry[2] = s2 >> 21 - s3 += carry[2] - s2 -= carry[2] << 21 - carry[3] = s3 >> 21 - s4 += carry[3] - s3 -= carry[3] << 21 - carry[4] = s4 >> 21 - s5 += carry[4] - s4 -= carry[4] << 21 - carry[5] = s5 >> 21 - s6 += carry[5] - s5 -= carry[5] << 21 - carry[6] = s6 >> 21 - s7 += carry[6] - s6 -= carry[6] << 21 - carry[7] = s7 >> 21 - s8 += carry[7] - s7 -= carry[7] << 21 - carry[8] = s8 >> 21 - s9 += carry[8] - s8 -= carry[8] << 21 - carry[9] = s9 >> 21 - s10 += carry[9] - s9 -= carry[9] << 21 - carry[10] = s10 >> 21 - s11 += carry[10] - s10 -= carry[10] << 21 - carry[11] = s11 >> 21 - s12 += carry[11] - s11 -= carry[11] << 21 - - s0 += s12 * 666643 - s1 += s12 * 470296 - s2 += s12 * 654183 - s3 -= s12 * 997805 - s4 += s12 * 136657 - s5 -= s12 * 683901 - s12 = 0 - - carry[0] = s0 >> 21 - s1 += carry[0] - s0 -= carry[0] << 21 - carry[1] = s1 >> 21 - s2 += carry[1] - s1 -= carry[1] << 21 - carry[2] = s2 >> 21 - s3 += carry[2] - s2 -= carry[2] << 21 - carry[3] = s3 >> 21 - s4 += carry[3] - s3 -= carry[3] << 21 - carry[4] = s4 >> 21 - s5 += carry[4] - s4 -= carry[4] << 21 - carry[5] = s5 >> 21 - s6 += carry[5] - s5 -= carry[5] << 21 - carry[6] = s6 >> 21 - s7 += carry[6] - s6 -= carry[6] << 21 - carry[7] = s7 >> 21 - s8 += carry[7] - s7 -= carry[7] << 21 - carry[8] = s8 >> 21 - s9 += carry[8] - s8 -= carry[8] << 21 - carry[9] = s9 >> 21 - s10 += carry[9] - s9 -= carry[9] << 21 - carry[10] = s10 >> 21 - s11 += carry[10] - s10 -= carry[10] << 21 - - out[0] = byte(s0 >> 0) - out[1] = byte(s0 >> 8) - out[2] = byte((s0 >> 16) | (s1 << 5)) - out[3] = byte(s1 >> 3) - out[4] = byte(s1 >> 11) - out[5] = byte((s1 >> 19) | (s2 << 2)) - out[6] = byte(s2 >> 6) - out[7] = byte((s2 >> 14) | (s3 << 7)) - out[8] = byte(s3 >> 1) - out[9] = byte(s3 >> 9) - out[10] = byte((s3 >> 17) | (s4 << 4)) - out[11] = byte(s4 >> 4) - out[12] = byte(s4 >> 12) - out[13] = byte((s4 >> 20) | (s5 << 1)) - out[14] = byte(s5 >> 7) - out[15] = byte((s5 >> 15) | (s6 << 6)) - out[16] = byte(s6 >> 2) - out[17] = byte(s6 >> 10) - out[18] = byte((s6 >> 18) | (s7 << 3)) - out[19] = byte(s7 >> 5) - out[20] = byte(s7 >> 13) - out[21] = byte(s8 >> 0) - out[22] = byte(s8 >> 8) - out[23] = byte((s8 >> 16) | (s9 << 5)) - out[24] = byte(s9 >> 3) - out[25] = byte(s9 >> 11) - out[26] = byte((s9 >> 19) | (s10 << 2)) - out[27] = byte(s10 >> 6) - out[28] = byte((s10 >> 14) | (s11 << 7)) - out[29] = byte(s11 >> 1) - out[30] = byte(s11 >> 9) - out[31] = byte(s11 >> 17) -} From 08a76875a44c4f72e1c690633bc2f27404d2be9b Mon Sep 17 00:00:00 2001 From: George Tankersley Date: Fri, 7 Jul 2017 00:00:00 +0000 Subject: [PATCH 14/46] Cache the field representation of d --- ed25519.go | 8 ++++---- internal/group/const.go | 11 +++++++++++ internal/group/ge.go | 36 ++++++++++++++++++------------------ internal/radix51/const.go | 3 --- 4 files changed, 33 insertions(+), 25 deletions(-) create mode 100644 internal/group/const.go diff --git a/ed25519.go b/ed25519.go index 05a23d6..0b06b4c 100644 --- a/ed25519.go +++ b/ed25519.go @@ -51,18 +51,18 @@ func (curve ed25519Curve) Params() *elliptic.CurveParams { } // IsOnCurve reports whether the given (x,y) lies on the curve by checking that -// -x^2 + y^2 - 1 - dx^2y^2 = 0 (mod p). +// -x^2 + y^2 - 1 - dx^2y^2 = 0 (mod p). This function uses a hardcoded value +// of d. func (curve ed25519Curve) IsOnCurve(x, y *big.Int) bool { - var feX, feY, feB field.FieldElement + var feX, feY field.FieldElement field.FeFromBig(&feX, x) field.FeFromBig(&feY, y) - field.FeFromBig(&feB, curve.B) var lh, y2, rh field.FieldElement field.FeSquare(&lh, &feX) // x^2 field.FeSquare(&y2, &feY) // y^2 field.FeMul(&rh, &lh, &y2) // x^2*y^2 - field.FeMul(&rh, &rh, &feB) // d*x^2*y^2 + field.FeMul(&rh, &rh, &group.D) // d*x^2*y^2 field.FeAdd(&rh, &rh, &field.FieldOne) // 1 + d*x^2*y^2 field.FeNeg(&lh, &lh) // -x^2 field.FeAdd(&lh, &lh, &y2) // -x^2 + y^2 diff --git a/internal/group/const.go b/internal/group/const.go new file mode 100644 index 0000000..e32bc9d --- /dev/null +++ b/internal/group/const.go @@ -0,0 +1,11 @@ +package group + +import "github.com/gtank/ed25519/internal/radix51" + +var ( + // d, a constant in the curve equation + D radix51.FieldElement = [5]uint64{929955233495203, 466365720129213, 1662059464998953, 2033849074728123, 1442794654840575} + + // 2*d, used in addition formula + D2 radix51.FieldElement = [5]uint64{1859910466990425, 932731440258426, 1072319116312658, 1815898335770999, 633789495995903} +) diff --git a/internal/group/ge.go b/internal/group/ge.go index 99cb31f..003cd40 100644 --- a/internal/group/ge.go +++ b/internal/group/ge.go @@ -76,24 +76,24 @@ func (v *ExtendedGroupElement) Zero() *ExtendedGroupElement { // TODO We know Z1=1 and Z2=1 here, so mmadd-2008-hwcd-3 (6M + 1S + 1*k + 9add) could apply func (v *ExtendedGroupElement) Add(p1, p2 *ExtendedGroupElement) *ExtendedGroupElement { var tmp1, tmp2, A, B, C, D, E, F, G, H field.FieldElement - field.FeSub(&tmp1, &p1.Y, &p1.X) // tmp1 <-- Y1-X1 - field.FeSub(&tmp2, &p2.Y, &p2.X) // tmp2 <-- Y2-X2 - field.FeMul(&A, &tmp1, &tmp2) // A <-- tmp1*tmp2 = (Y1-X1)*(Y2-X2) - field.FeAdd(&tmp1, &p1.Y, &p1.X) // tmp1 <-- Y1+X1 - field.FeAdd(&tmp2, &p2.Y, &p2.X) // tmp2 <-- Y2+X2 - field.FeMul(&B, &tmp1, &tmp2) // B <-- tmp1*tmp2 = (Y1+X1)*(Y2+X2) - field.FeMul(&tmp1, &p1.T, &p2.T) // tmp1 <-- T1*T2 - field.FeMul(&C, &tmp1, &field.D2) // C <-- tmp1*2d = T1*2d*T2 - field.FeMul(&tmp1, &p1.Z, &p2.Z) // tmp1 <-- Z1*Z2 - field.FeAdd(&D, &tmp1, &tmp1) // D <-- tmp1 + tmp1 = 2*Z1*Z2 - field.FeSub(&E, &B, &A) // E <-- B-A - field.FeSub(&F, &D, &C) // F <-- D-C - field.FeAdd(&G, &D, &C) // G <-- D+C - field.FeAdd(&H, &B, &A) // H <-- B+A - field.FeMul(&v.X, &E, &F) // X3 <-- E*F - field.FeMul(&v.Y, &G, &H) // Y3 <-- G*H - field.FeMul(&v.T, &E, &H) // T3 <-- E*H - field.FeMul(&v.Z, &F, &G) // Z3 <-- F*G + field.FeSub(&tmp1, &p1.Y, &p1.X) // tmp1 <-- Y1-X1 + field.FeSub(&tmp2, &p2.Y, &p2.X) // tmp2 <-- Y2-X2 + field.FeMul(&A, &tmp1, &tmp2) // A <-- tmp1*tmp2 = (Y1-X1)*(Y2-X2) + field.FeAdd(&tmp1, &p1.Y, &p1.X) // tmp1 <-- Y1+X1 + field.FeAdd(&tmp2, &p2.Y, &p2.X) // tmp2 <-- Y2+X2 + field.FeMul(&B, &tmp1, &tmp2) // B <-- tmp1*tmp2 = (Y1+X1)*(Y2+X2) + field.FeMul(&tmp1, &p1.T, &p2.T) // tmp1 <-- T1*T2 + field.FeMul(&C, &tmp1, &D2) // C <-- tmp1*2d = T1*2d*T2 + field.FeMul(&tmp1, &p1.Z, &p2.Z) // tmp1 <-- Z1*Z2 + field.FeAdd(&D, &tmp1, &tmp1) // D <-- tmp1 + tmp1 = 2*Z1*Z2 + field.FeSub(&E, &B, &A) // E <-- B-A + field.FeSub(&F, &D, &C) // F <-- D-C + field.FeAdd(&G, &D, &C) // G <-- D+C + field.FeAdd(&H, &B, &A) // H <-- B+A + field.FeMul(&v.X, &E, &F) // X3 <-- E*F + field.FeMul(&v.Y, &G, &H) // Y3 <-- G*H + field.FeMul(&v.T, &E, &H) // T3 <-- E*H + field.FeMul(&v.Z, &F, &G) // Z3 <-- F*G return v } diff --git a/internal/radix51/const.go b/internal/radix51/const.go index fa6e7b1..51b6190 100644 --- a/internal/radix51/const.go +++ b/internal/radix51/const.go @@ -12,7 +12,4 @@ var ( FieldZero FieldElement = [5]uint64{0, 0, 0, 0, 0} FieldOne FieldElement = [5]uint64{1, 0, 0, 0, 0} FieldTwo FieldElement = [5]uint64{2, 0, 0, 0, 0} - - // 2*d, used in addition formula - D2 FieldElement = [5]uint64{1859910466990425, 932731440258426, 1072319116312658, 1815898335770999, 633789495995903} ) From 0a030f62c035e01c406d2d85a69daa05894d7e11 Mon Sep 17 00:00:00 2001 From: George Tankersley Date: Tue, 25 Jul 2017 00:00:00 +0000 Subject: [PATCH 15/46] make things more generally presentable --- README | 11 +++++++++++ ed25519.go | 19 ++++++++----------- ed25519_test.go | 20 +------------------- internal/radix51/const.go | 4 +++- internal/radix51/fe.go | 4 +++- internal/radix51/fe_mul.go | 4 ++++ internal/radix51/fe_mul_amd64.go | 4 ++++ internal/radix51/fe_mul_amd64.s | 4 ++++ internal/radix51/fe_square.go | 4 ++++ internal/radix51/fe_square_amd64.go | 4 ++++ internal/radix51/fe_square_amd64.s | 4 ++++ internal/radix51/fe_test.go | 4 ++++ internal/radix51/mul.go | 4 ++++ 13 files changed, 58 insertions(+), 32 deletions(-) create mode 100644 README diff --git a/README b/README new file mode 100644 index 0000000..0a1103a --- /dev/null +++ b/README @@ -0,0 +1,11 @@ +This is foremost an implementation of the curve Ed25519 that satisfies Go's +elliptic.Curve interface. It targets amd64 systems, and is eventually intended +to be a legible general-purpose library along the lines of curve25519-dalek. + +It is also an implementation of GF(2^255-19) field operations in a 64-bit +representation, in both pure Go and plan9 assembly for amd64. This code is +currently package-internal. + +The library is a WORK IN PROGRESS. Everything will change dramatically as +development continues. There are no guarantees of stability, functionality, +correctness, or safety. We aren't open yet, come back later! diff --git a/ed25519.go b/ed25519.go index 0b06b4c..52bc9cc 100644 --- a/ed25519.go +++ b/ed25519.go @@ -94,7 +94,7 @@ func (curve ed25519Curve) Double(x1, y1 *big.Int) (x, y *big.Int) { // ScalarMult returns k*(Bx,By) where k is a number in big-endian form. func (curve ed25519Curve) ScalarMult(x1, y1 *big.Int, k []byte) (x, y *big.Int) { - // if either coordinate is nil, return the point at infinity + // if either coordinate is nil, return the identity point if x1 == nil || y1 == nil { x = new(big.Int).Set(bigZero) y = new(big.Int).Set(bigOne) @@ -144,13 +144,10 @@ func (curve ed25519Curve) scalarFromBytes(out *[32]byte, in []byte) { } } -// // ScalarBaseMult returns k*G, where G is the base point of the group and k is -// // an integer in big-endian form. -// func (curve ed25519Curve) ScalarBaseMult(k []byte) (x, y *big.Int) { -// var p edwards25519.ExtendedGroupElement -// var scBytes [32]byte - -// curve.scalarFromBytes(&scBytes, k) -// edwards25519.GeScalarMultBase(&p, &scBytes) -// return extendedToAffine(&p) -// } +// ScalarBaseMult returns k*G, where G is the base point of the curve and k is +// an integer in big-endian form. The difference between this and +// arbitrary-point ScalarMult is the availability of precomputed multiples of +// the base point. +func (curve ed25519Curve) ScalarBaseMult(k []byte) (x, y *big.Int) { + panic("not yet implemented") +} diff --git a/ed25519_test.go b/ed25519_test.go index 3808847..c8a28b3 100644 --- a/ed25519_test.go +++ b/ed25519_test.go @@ -369,24 +369,7 @@ func BenchmarkScalarMult(b *testing.B) { // } // } -// // BENCHMARKS - -// func BenchmarkScalarBaseMult(b *testing.B) { -// ed := Ed25519() - -// var k [32]byte -// _, err := io.ReadFull(rand.Reader, k[:]) -// if err != nil { -// b.Fatal(err) -// } -// k[0] &= 248 -// k[31] &= 127 -// k[31] |= 64 - -// for i := 0; i < b.N; i++ { -// _, _ = ed.ScalarBaseMult(k[:]) -// } -// } +// COMPARATIVE FIELD BENCHMARKS var radix51A = field.FieldElement{ 486662, 0, 0, 0, 0, @@ -418,7 +401,6 @@ func BenchmarkFeFromBig(b *testing.B) { var feOnes field.FieldElement = [5]uint64{1, 1, 1, 1, 1} -//func FeToBig(h *FieldElement) *big.Int { func BenchmarkFeToBig(b *testing.B) { for i := 0; i < b.N; i++ { _ = field.FeToBig(&feOnes) diff --git a/internal/radix51/const.go b/internal/radix51/const.go index 51b6190..b650ff8 100644 --- a/internal/radix51/const.go +++ b/internal/radix51/const.go @@ -1,4 +1,6 @@ -// Copyright 2017 George Tankersley. All rights reserved. +// Copyright (c) 2017 George Tankersley. All rights reserved. +// Use of this source code is governed by a BSD-style +// license that can be found in the LICENSE file. // Constants used in the implementation of GF(2^255-19) field arithmetic. package radix51 diff --git a/internal/radix51/fe.go b/internal/radix51/fe.go index d78922f..66a7093 100644 --- a/internal/radix51/fe.go +++ b/internal/radix51/fe.go @@ -1,4 +1,6 @@ -// Copyright 2017 George Tankersley. All rights reserved. +// Copyright (c) 2017 George Tankersley. All rights reserved. +// Use of this source code is governed by a BSD-style +// license that can be found in the LICENSE file. // Field arithmetic in radix 2^51 representation. This code is a port of the // public domain amd64-51-30k version of ed25519 from SUPERCOP. diff --git a/internal/radix51/fe_mul.go b/internal/radix51/fe_mul.go index 68b6849..edc7bd2 100644 --- a/internal/radix51/fe_mul.go +++ b/internal/radix51/fe_mul.go @@ -1,3 +1,7 @@ +// Copyright (c) 2017 George Tankersley. All rights reserved. +// Use of this source code is governed by a BSD-style +// license that can be found in the LICENSE file. + // +build !amd64 noasm package radix51 diff --git a/internal/radix51/fe_mul_amd64.go b/internal/radix51/fe_mul_amd64.go index c76b4db..b0d38be 100644 --- a/internal/radix51/fe_mul_amd64.go +++ b/internal/radix51/fe_mul_amd64.go @@ -1,3 +1,7 @@ +// Copyright (c) 2017 George Tankersley. All rights reserved. +// Use of this source code is governed by a BSD-style +// license that can be found in the LICENSE file. + // +build amd64,!noasm package radix51 diff --git a/internal/radix51/fe_mul_amd64.s b/internal/radix51/fe_mul_amd64.s index 07ebb0c..e246262 100644 --- a/internal/radix51/fe_mul_amd64.s +++ b/internal/radix51/fe_mul_amd64.s @@ -1,3 +1,7 @@ +// Copyright (c) 2017 George Tankersley. All rights reserved. +// Use of this source code is governed by a BSD-style +// license that can be found in the LICENSE file. + // Based on assembly generated by PeachPy. Equivalent to the Go in fe_mul.go, // which was originally based on the amd64-51-30k assembly in SUPERCOP. diff --git a/internal/radix51/fe_square.go b/internal/radix51/fe_square.go index fa738eb..ec5bd62 100644 --- a/internal/radix51/fe_square.go +++ b/internal/radix51/fe_square.go @@ -1,3 +1,7 @@ +// Copyright (c) 2017 George Tankersley. All rights reserved. +// Use of this source code is governed by a BSD-style +// license that can be found in the LICENSE file. + // +build !amd64 noasm package radix51 diff --git a/internal/radix51/fe_square_amd64.go b/internal/radix51/fe_square_amd64.go index c089146..8c38356 100644 --- a/internal/radix51/fe_square_amd64.go +++ b/internal/radix51/fe_square_amd64.go @@ -1,3 +1,7 @@ +// Copyright (c) 2017 George Tankersley. All rights reserved. +// Use of this source code is governed by a BSD-style +// license that can be found in the LICENSE file. + // +build amd64,!noasm package radix51 diff --git a/internal/radix51/fe_square_amd64.s b/internal/radix51/fe_square_amd64.s index f9af5c8..fa01331 100644 --- a/internal/radix51/fe_square_amd64.s +++ b/internal/radix51/fe_square_amd64.s @@ -1,3 +1,7 @@ +// Copyright (c) 2017 George Tankersley. All rights reserved. +// Use of this source code is governed by a BSD-style +// license that can be found in the LICENSE file. + // +build amd64,!noasm // func FeSquare(outp *uint64, xp *uint64) diff --git a/internal/radix51/fe_test.go b/internal/radix51/fe_test.go index 15a5a06..3450871 100644 --- a/internal/radix51/fe_test.go +++ b/internal/radix51/fe_test.go @@ -1,3 +1,7 @@ +// Copyright (c) 2017 George Tankersley. All rights reserved. +// Use of this source code is governed by a BSD-style +// license that can be found in the LICENSE file. + package radix51 import ( diff --git a/internal/radix51/mul.go b/internal/radix51/mul.go index dbea36c..c196d9d 100644 --- a/internal/radix51/mul.go +++ b/internal/radix51/mul.go @@ -1,3 +1,7 @@ +// Copyright (c) 2017 George Tankersley. All rights reserved. +// Use of this source code is governed by a BSD-style +// license that can be found in the LICENSE file. + package radix51 import "unsafe" From 0e8b4dbb24263feddc0dc26ab4d09b3316ed8ca1 Mon Sep 17 00:00:00 2001 From: Filippo Valsorda Date: Sun, 20 Jan 2019 18:13:25 -0500 Subject: [PATCH 16/46] internal/radix51: remove FeCSwap It's unused and it does not do what it says in the docs. --- internal/radix51/fe.go | 25 ------------------------- 1 file changed, 25 deletions(-) diff --git a/internal/radix51/fe.go b/internal/radix51/fe.go index 66a7093..608a083 100644 --- a/internal/radix51/fe.go +++ b/internal/radix51/fe.go @@ -192,31 +192,6 @@ func FeInvert(out, z *FieldElement) { FeMul(out, &t, &z11) // 2^255 - 21 } -// FeCSwap implements a constant-time conditional swap. That is: -// (a, b) = (b, a) if c = 1 -// (a, b) = (a, b) if c = 0 -func FeCSwap(a, b *FieldElement, c uint64) { - var t FieldElement - - t[0] ^= c & (a[0] ^ b[0]) - t[1] ^= c & (a[1] ^ b[1]) - t[2] ^= c & (a[2] ^ b[2]) - t[3] ^= c & (a[3] ^ b[3]) - t[4] ^= c & (a[4] ^ b[4]) - - a[0] ^= t[0] - a[1] ^= t[1] - a[2] ^= t[2] - a[3] ^= t[3] - a[4] ^= t[4] - - b[0] ^= t[0] - b[1] ^= t[1] - b[2] ^= t[2] - b[3] ^= t[3] - b[4] ^= t[4] -} - // Returns true if two field elements are equal. func FeEqual(a, b *FieldElement) bool { var result uint64 From 7ec609da40c073af5eb68b2c4770a33299bf2cbd Mon Sep 17 00:00:00 2001 From: Filippo Valsorda Date: Mon, 21 Jan 2019 17:48:58 -0500 Subject: [PATCH 17/46] internal/radix51: remove FeEqual It does not normalize the representative before comparing them so it will have false negatives. --- internal/radix51/fe.go | 9 --------- 1 file changed, 9 deletions(-) diff --git a/internal/radix51/fe.go b/internal/radix51/fe.go index 608a083..bf9ccd0 100644 --- a/internal/radix51/fe.go +++ b/internal/radix51/fe.go @@ -192,15 +192,6 @@ func FeInvert(out, z *FieldElement) { FeMul(out, &t, &z11) // 2^255 - 21 } -// Returns true if two field elements are equal. -func FeEqual(a, b *FieldElement) bool { - var result uint64 - for i := 0; i < 5; i++ { - result |= a[i] ^ b[i] - } - return result == 0 -} - func FeCopy(out, in *FieldElement) { copy(out[:], in[:]) } From fe6e0954e0ad069aa87da23607c1e05a3151d0a2 Mon Sep 17 00:00:00 2001 From: Filippo Valsorda Date: Mon, 21 Jan 2019 17:51:39 -0500 Subject: [PATCH 18/46] internal/radix51: add constant time field operations --- LICENSE | 1 + internal/radix51/fe.go | 48 ++++++++++++++++++++++++++++++++++++++++++ 2 files changed, 49 insertions(+) diff --git a/LICENSE b/LICENSE index 10f0ec1..22a10c8 100644 --- a/LICENSE +++ b/LICENSE @@ -1,4 +1,5 @@ Copyright 2017 George Tankersley. All rights reserved. +Copyright 2019 The Go Authors. All rights reserved. Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: diff --git a/internal/radix51/fe.go b/internal/radix51/fe.go index bf9ccd0..0756e87 100644 --- a/internal/radix51/fe.go +++ b/internal/radix51/fe.go @@ -1,4 +1,5 @@ // Copyright (c) 2017 George Tankersley. All rights reserved. +// Copyright (c) 2019 The Go Authors. All rights reserved. // Use of this source code is governed by a BSD-style // license that can be found in the LICENSE file. @@ -7,6 +8,7 @@ package radix51 import ( + "crypto/subtle" "math/big" "math/bits" ) @@ -330,3 +332,49 @@ func FeToBig(h *FieldElement) *big.Int { out := new(big.Int) return out.SetBits(words) } + +// FeEqual returns 1 if a and b are equal, and 0 otherwise. +func FeEqual(a, b *FieldElement) int { + var sa, sb [32]byte + FeToBytes(&sa, a) + FeToBytes(&sb, b) + return subtle.ConstantTimeCompare(sa[:], sb[:]) +} + +// FeSelect sets out to v if cond == 1, and to u if cond == 0. +// out, v and u are allowed to overlap. +func FeSelect(out, v, u *FieldElement, cond int) { + b := uint64(cond) * 0xffffffffffffffff + out[0] = (b & v[0]) | (^b & u[0]) + out[1] = (b & v[1]) | (^b & u[1]) + out[2] = (b & v[2]) | (^b & u[2]) + out[3] = (b & v[3]) | (^b & u[3]) + out[4] = (b & v[4]) | (^b & u[4]) +} + +// FeCondNeg sets u to -u if cond == 1, and to u if cond == 0. +func FeCondNeg(u *FieldElement, cond int) { + var neg FieldElement + FeNeg(&neg, u) + + b := uint64(cond) * 0xffffffffffffffff + u[0] ^= b & (u[0] ^ neg[0]) + u[1] ^= b & (u[1] ^ neg[1]) + u[2] ^= b & (u[2] ^ neg[2]) + u[3] ^= b & (u[3] ^ neg[3]) + u[4] ^= b & (u[4] ^ neg[4]) +} + +// FeIsNegative returns 1 if u is negative, and 0 otherwise. +func FeIsNegative(u *FieldElement) int { + var b [32]byte + FeToBytes(&b, u) + return int(b[0] & 1) +} + +// FeAbs sets out to |u|. out and u are allowed to overlap. +func FeAbs(out, u *FieldElement) { + var neg FieldElement + FeNeg(&neg, u) + FeSelect(out, &neg, u, FeIsNegative(u)) +} From 930ef44224cb8012e304c87098f11964459633a1 Mon Sep 17 00:00:00 2001 From: Filippo Valsorda Date: Sat, 26 Jan 2019 19:54:29 -0500 Subject: [PATCH 19/46] all: refactor field API to be methods based --- doc.go | 8 - ed25519.go | 37 ++-- ed25519_test.go | 13 +- internal/group/const.go | 11 -- internal/group/ge.go | 193 ++++++++++---------- internal/radix51/const.go | 17 -- internal/radix51/fe.go | 271 ++++++++++++++-------------- internal/radix51/fe_mul.go | 14 +- internal/radix51/fe_mul_amd64.go | 7 +- internal/radix51/fe_mul_amd64.s | 4 +- internal/radix51/fe_square.go | 14 +- internal/radix51/fe_square_amd64.go | 7 +- internal/radix51/fe_square_amd64.s | 4 +- internal/radix51/fe_test.go | 40 ++-- 14 files changed, 315 insertions(+), 325 deletions(-) delete mode 100644 doc.go delete mode 100644 internal/group/const.go delete mode 100644 internal/radix51/const.go diff --git a/doc.go b/doc.go deleted file mode 100644 index f2a98d5..0000000 --- a/doc.go +++ /dev/null @@ -1,8 +0,0 @@ -// Package ed25519 implements an elliptic.Curve interface on top of the twisted -// Edwards curve -x^2 + y^2 = 1 + -(121665/121666)*x^2*y^2. This is better -// known as the Edwards curve equivalent to curve25519, and is the curve used -// by the Ed25519 signature scheme. -// -// Because of the Curve interface, this package takes input in affine (x,y) -// pairs instead of the more standard compressed Edwards y. -package ed25519 diff --git a/ed25519.go b/ed25519.go index 52bc9cc..c3277f4 100644 --- a/ed25519.go +++ b/ed25519.go @@ -2,6 +2,13 @@ // Use of this source code is governed by a BSD-style // license that can be found in the LICENSE file. +// Package ed25519 implements an elliptic.Curve interface on top of the twisted +// Edwards curve -x^2 + y^2 = 1 + -(121665/121666)*x^2*y^2. This is better +// known as the Edwards curve equivalent to curve25519, and is the curve used +// by the Ed25519 signature scheme. +// +// Because of the Curve interface, this package takes input in affine (x,y) +// pairs instead of the more standard compressed Edwards y. package ed25519 import ( @@ -10,7 +17,7 @@ import ( "sync" "github.com/gtank/ed25519/internal/group" - field "github.com/gtank/ed25519/internal/radix51" + "github.com/gtank/ed25519/internal/radix51" ) var bigZero *big.Int @@ -54,22 +61,22 @@ func (curve ed25519Curve) Params() *elliptic.CurveParams { // -x^2 + y^2 - 1 - dx^2y^2 = 0 (mod p). This function uses a hardcoded value // of d. func (curve ed25519Curve) IsOnCurve(x, y *big.Int) bool { - var feX, feY field.FieldElement - field.FeFromBig(&feX, x) - field.FeFromBig(&feY, y) + var feX, feY radix51.FieldElement + feX.FromBig(x) + feY.FromBig(y) - var lh, y2, rh field.FieldElement - field.FeSquare(&lh, &feX) // x^2 - field.FeSquare(&y2, &feY) // y^2 - field.FeMul(&rh, &lh, &y2) // x^2*y^2 - field.FeMul(&rh, &rh, &group.D) // d*x^2*y^2 - field.FeAdd(&rh, &rh, &field.FieldOne) // 1 + d*x^2*y^2 - field.FeNeg(&lh, &lh) // -x^2 - field.FeAdd(&lh, &lh, &y2) // -x^2 + y^2 - field.FeSub(&lh, &lh, &rh) // -x^2 + y^2 - 1 - dx^2y^2 - field.FeReduce(&lh, &lh) // mod p + var lh, y2, rh radix51.FieldElement + lh.Square(&feX) // x^2 + y2.Square(&feY) // y^2 + rh.Mul(&lh, &y2) // x^2*y^2 + rh.Mul(&rh, group.D) // d*x^2*y^2 + rh.Add(&rh, radix51.One) // 1 + d*x^2*y^2 + lh.Neg(&lh) // -x^2 + lh.Add(&lh, &y2) // -x^2 + y^2 + lh.Sub(&lh, &rh) // -x^2 + y^2 - 1 - dx^2y^2 + lh.Reduce() // mod p - return field.FeEqual(&lh, &field.FieldZero) + return lh.Equal(radix51.Zero) == 1 } // Add returns the sum of (x1, y1) and (x2, y2). diff --git a/ed25519_test.go b/ed25519_test.go index c8a28b3..1949df1 100644 --- a/ed25519_test.go +++ b/ed25519_test.go @@ -27,8 +27,8 @@ func TestRadixRoundTrip(t *testing.T) { if err != nil { t.Fatal(err) } - field.FeFromBig(&fe, n) - out = field.FeToBig(&fe) + fe.FromBig(n) + out = fe.ToBig() if n.Cmp(out) != 0 { t.Errorf("fe<>bn failed for %x", n.Bytes()) } @@ -378,14 +378,14 @@ var radix51A = field.FieldElement{ func BenchmarkFeMul51(b *testing.B) { var h field.FieldElement for i := 0; i < b.N; i++ { - field.FeMul(&h, &radix51A, &radix51A) + h.Mul(&radix51A, &radix51A) } } func BenchmarkFeSquare51(b *testing.B) { var h field.FieldElement for i := 0; i < b.N; i++ { - field.FeSquare(&h, &radix51A) + h.Square(&radix51A) } } @@ -395,14 +395,15 @@ var randFieldInt, _ = rand.Int(rand.Reader, concreteCurve.P) func BenchmarkFeFromBig(b *testing.B) { var fe field.FieldElement for i := 0; i < b.N; i++ { - field.FeFromBig(&fe, randFieldInt) + fe.FromBig(randFieldInt) } } var feOnes field.FieldElement = [5]uint64{1, 1, 1, 1, 1} +var sink *big.Int func BenchmarkFeToBig(b *testing.B) { for i := 0; i < b.N; i++ { - _ = field.FeToBig(&feOnes) + sink = feOnes.ToBig() } } diff --git a/internal/group/const.go b/internal/group/const.go deleted file mode 100644 index e32bc9d..0000000 --- a/internal/group/const.go +++ /dev/null @@ -1,11 +0,0 @@ -package group - -import "github.com/gtank/ed25519/internal/radix51" - -var ( - // d, a constant in the curve equation - D radix51.FieldElement = [5]uint64{929955233495203, 466365720129213, 1662059464998953, 2033849074728123, 1442794654840575} - - // 2*d, used in addition formula - D2 radix51.FieldElement = [5]uint64{1859910466990425, 932731440258426, 1072319116312658, 1815898335770999, 633789495995903} -) diff --git a/internal/group/ge.go b/internal/group/ge.go index 003cd40..eb99ce7 100644 --- a/internal/group/ge.go +++ b/internal/group/ge.go @@ -1,13 +1,20 @@ -// Implements group logic for the Ed25519 curve. +// Copyright (c) 2017 George Tankersley. All rights reserved. +// Use of this source code is governed by a BSD-style +// license that can be found in the LICENSE file. +// Package group mplements group logic for the Ed25519 curve. package group import ( "math/big" - field "github.com/gtank/ed25519/internal/radix51" + "github.com/gtank/ed25519/internal/radix51" ) +// D is a constant in the curve equation. +var D = &radix51.FieldElement{929955233495203, 466365720129213, + 1662059464998953, 2033849074728123, 1442794654840575} + // From EFD https://hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html // An elliptic curve in twisted Edwards form has parameters a, d and coordinates // x, y satisfying the following equations: @@ -24,7 +31,7 @@ import ( // This representation was introduced in the HisilWongCarterDawson paper "Twisted // Edwards curves revisited" (Asiacrypt 2008). type ExtendedGroupElement struct { - X, Y, Z, T field.FieldElement + X, Y, Z, T radix51.FieldElement } // Converts (x,y) to (X:Y:T:Z) extended coordinates, or "P3" in ref10. As @@ -32,10 +39,10 @@ type ExtendedGroupElement struct { // 2008, Section 3.1 (https://eprint.iacr.org/2008/522.pdf) // See also https://hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html#addition-add-2008-hwcd-3 func (v *ExtendedGroupElement) FromAffine(x, y *big.Int) { - field.FeFromBig(&v.X, x) - field.FeFromBig(&v.Y, y) - field.FeMul(&v.T, &v.X, &v.Y) - field.FeOne(&v.Z) + v.X.FromBig(x) + v.Y.FromBig(y) + v.T.Mul(&v.X, &v.Y) + v.Z.One() } // Extended coordinates are XYZT with x = X/Z, y = Y/Z, or the "P3" @@ -43,57 +50,60 @@ func (v *ExtendedGroupElement) FromAffine(x, y *big.Int) { // from projective to affine. Per HWCD, it is safe to move from extended to // projective by simply ignoring T. func (v *ExtendedGroupElement) ToAffine() (*big.Int, *big.Int) { - var x, y, zinv field.FieldElement + var x, y, zinv radix51.FieldElement - field.FeInvert(&zinv, &v.Z) - field.FeMul(&x, &v.X, &zinv) - field.FeMul(&y, &v.Y, &zinv) + zinv.Invert(&v.Z) + x.Mul(&v.X, &zinv) + y.Mul(&v.Y, &zinv) - return field.FeToBig(&x), field.FeToBig(&y) + return x.ToBig(), y.ToBig() } // Per HWCD, it is safe to move from extended to projective by simply ignoring T. func (v *ExtendedGroupElement) ToProjective() *ProjectiveGroupElement { var p ProjectiveGroupElement - field.FeCopy(&p.X, &v.X) - field.FeCopy(&p.Y, &v.Y) - field.FeCopy(&p.Z, &v.Z) + p.X.Set(&v.X) + p.Y.Set(&v.Y) + p.Z.Set(&v.Z) return &p } func (v *ExtendedGroupElement) Zero() *ExtendedGroupElement { - field.FeZero(&v.X) - field.FeOne(&v.Y) - field.FeOne(&v.Z) - field.FeZero(&v.T) + v.X.Zero() + v.Y.One() + v.Z.One() + v.T.Zero() return v } +var twoD = &radix51.FieldElement{1859910466990425, 932731440258426, + 1072319116312658, 1815898335770999, 633789495995903} + // This is the same addition formula everyone uses, "add-2008-hwcd-3". // https://hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html#addition-add-2008-hwcd-3 // TODO We know Z1=1 and Z2=1 here, so mmadd-2008-hwcd-3 (6M + 1S + 1*k + 9add) could apply func (v *ExtendedGroupElement) Add(p1, p2 *ExtendedGroupElement) *ExtendedGroupElement { - var tmp1, tmp2, A, B, C, D, E, F, G, H field.FieldElement - field.FeSub(&tmp1, &p1.Y, &p1.X) // tmp1 <-- Y1-X1 - field.FeSub(&tmp2, &p2.Y, &p2.X) // tmp2 <-- Y2-X2 - field.FeMul(&A, &tmp1, &tmp2) // A <-- tmp1*tmp2 = (Y1-X1)*(Y2-X2) - field.FeAdd(&tmp1, &p1.Y, &p1.X) // tmp1 <-- Y1+X1 - field.FeAdd(&tmp2, &p2.Y, &p2.X) // tmp2 <-- Y2+X2 - field.FeMul(&B, &tmp1, &tmp2) // B <-- tmp1*tmp2 = (Y1+X1)*(Y2+X2) - field.FeMul(&tmp1, &p1.T, &p2.T) // tmp1 <-- T1*T2 - field.FeMul(&C, &tmp1, &D2) // C <-- tmp1*2d = T1*2d*T2 - field.FeMul(&tmp1, &p1.Z, &p2.Z) // tmp1 <-- Z1*Z2 - field.FeAdd(&D, &tmp1, &tmp1) // D <-- tmp1 + tmp1 = 2*Z1*Z2 - field.FeSub(&E, &B, &A) // E <-- B-A - field.FeSub(&F, &D, &C) // F <-- D-C - field.FeAdd(&G, &D, &C) // G <-- D+C - field.FeAdd(&H, &B, &A) // H <-- B+A - field.FeMul(&v.X, &E, &F) // X3 <-- E*F - field.FeMul(&v.Y, &G, &H) // Y3 <-- G*H - field.FeMul(&v.T, &E, &H) // T3 <-- E*H - field.FeMul(&v.Z, &F, &G) // Z3 <-- F*G + var tmp1, tmp2, A, B, C, D, E, F, G, H radix51.FieldElement + tmp1.Sub(&p1.Y, &p1.X) // tmp1 <-- Y1-X1 + tmp2.Sub(&p2.Y, &p2.X) // tmp2 <-- Y2-X2 + A.Mul(&tmp1, &tmp2) // A <-- tmp1*tmp2 = (Y1-X1)*(Y2-X2) + tmp1.Add(&p1.Y, &p1.X) // tmp1 <-- Y1+X1 + tmp2.Add(&p2.Y, &p2.X) // tmp2 <-- Y2+X2 + B.Mul(&tmp1, &tmp2) // B <-- tmp1*tmp2 = (Y1+X1)*(Y2+X2) + tmp1.Mul(&p1.T, &p2.T) // tmp1 <-- T1*T2 + C.Mul(&tmp1, twoD) // C <-- tmp1*2d = T1*2*d*T2 + tmp1.Mul(&p1.Z, &p2.Z) // tmp1 <-- Z1*Z2 + D.Add(&tmp1, &tmp1) // D <-- tmp1 + tmp1 = 2*Z1*Z2 + E.Sub(&B, &A) // E <-- B-A + F.Sub(&D, &C) // F <-- D-C + G.Add(&D, &C) // G <-- D+C + H.Add(&B, &A) // H <-- B+A + v.X.Mul(&E, &F) // X3 <-- E*F + v.Y.Mul(&G, &H) // Y3 <-- G*H + v.T.Mul(&E, &H) // T3 <-- E*H + v.Z.Mul(&F, &G) // Z3 <-- F*G return v } @@ -126,40 +136,33 @@ func (v *ExtendedGroupElement) Double() *ExtendedGroupElement { // *v = *(v.ToProjective().Double().ToExtended()) // return v - var A, B, C, D, E, F, G, H field.FieldElement + var A, B, C, D, E, F, G, H radix51.FieldElement // A ← X1^2, B ← Y1^2 - field.FeSquare(&A, &v.X) - field.FeSquare(&B, &v.Y) + A.Square(&v.X) + B.Square(&v.Y) // C ← 2*Z1^2 - field.FeSquare(&C, &v.Z) - field.FeAdd(&C, &C, &C) // TODO should probably implement FeSquare2 + C.Square(&v.Z) + C.Add(&C, &C) // TODO should probably implement FeSquare2 // D ← -1*A - field.FeNeg(&D, &A) // implemented as substraction + D.Neg(&A) // implemented as substraction // E ← (X1+Y1)^2 − A − B - var t0 field.FieldElement - field.FeAdd(&t0, &v.X, &v.Y) - field.FeSquare(&t0, &t0) - field.FeSub(&E, &t0, &A) - field.FeSub(&E, &E, &B) + var t0 radix51.FieldElement + t0.Add(&v.X, &v.Y) + t0.Square(&t0) + E.Sub(&t0, &A) + E.Sub(&E, &B) - // G ← D+B - field.FeAdd(&G, &D, &B) - // F ← G−C - field.FeSub(&F, &G, &C) - // H ← D−B - field.FeSub(&H, &D, &B) - // X3 ← E*F - field.FeMul(&v.X, &E, &F) - // Y3 ← G*H - field.FeMul(&v.Y, &G, &H) - // T3 ← E*H - field.FeMul(&v.T, &E, &H) - // Z3 ← F*G - field.FeMul(&v.Z, &F, &G) + G.Add(&D, &B) // G ← D+B + F.Sub(&G, &C) // F ← G−C + H.Sub(&D, &B) // H ← D−B + v.X.Mul(&E, &F) // X3 ← E*F + v.Y.Mul(&G, &H) // Y3 ← G*H + v.T.Mul(&E, &H) // T3 ← E*H + v.Z.Mul(&F, &G) // Z3 ← F*G return v } @@ -168,23 +171,23 @@ func (v *ExtendedGroupElement) Double() *ExtendedGroupElement { // representation in ref10. This representation has a cheaper doubling formula // than extended coordinates. type ProjectiveGroupElement struct { - X, Y, Z field.FieldElement + X, Y, Z radix51.FieldElement } func (v *ProjectiveGroupElement) FromAffine(x, y *big.Int) { - field.FeFromBig(&v.X, x) - field.FeFromBig(&v.Y, y) - field.FeOne(&v.Z) + v.X.FromBig(x) + v.Y.FromBig(y) + v.Z.Zero() } func (v *ProjectiveGroupElement) ToAffine() (*big.Int, *big.Int) { - var x, y, zinv field.FieldElement + var x, y, zinv radix51.FieldElement - field.FeInvert(&zinv, &v.Z) - field.FeMul(&x, &v.X, &zinv) - field.FeMul(&y, &v.Y, &zinv) + zinv.Invert(&v.Z) + x.Mul(&v.X, &zinv) + y.Mul(&v.Y, &zinv) - return field.FeToBig(&x), field.FeToBig(&y) + return x.ToBig(), y.ToBig() } // HWCD Section 3: "Given (X : Y : Z) in [projective coordinates] passing to @@ -193,18 +196,18 @@ func (v *ProjectiveGroupElement) ToAffine() (*big.Int, *big.Int) { func (v *ProjectiveGroupElement) ToExtended() *ExtendedGroupElement { var r ExtendedGroupElement - field.FeMul(&r.X, &v.X, &v.Z) - field.FeMul(&r.Y, &v.Y, &v.Z) - field.FeMul(&r.T, &v.X, &v.Y) - field.FeSquare(&r.Z, &v.Z) + r.X.Mul(&v.X, &v.Z) + r.Y.Mul(&v.Y, &v.Z) + r.T.Mul(&v.X, &v.Y) + r.Z.Square(&v.Z) return &r } func (v *ProjectiveGroupElement) Zero() *ProjectiveGroupElement { - field.FeZero(&v.X) - field.FeOne(&v.Y) - field.FeOne(&v.Z) + v.X.Zero() + v.Y.One() + v.Z.One() return v } @@ -235,38 +238,38 @@ func (v *ProjectiveGroupElement) DoubleZ1() *ProjectiveGroupElement { // because it is always called on ephemeral intermediate values, but should // fix. var p, q ProjectiveGroupElement - var t0, t1 field.FieldElement + var t0, t1 radix51.FieldElement p = *v // C = X1^2, D = Y1^2 - field.FeSquare(&t0, &p.X) - field.FeSquare(&t1, &p.Y) + t0.Square(&p.X) + t1.Square(&p.Y) // B = (X1+Y1)^2 - field.FeAdd(&p.Z, &p.X, &p.Y) // Z is irrelevant but already allocated - field.FeSquare(&q.X, &p.Z) + p.Z.Add(&p.X, &p.Y) // Z is irrelevant but already allocated + q.X.Square(&p.Z) // E = a*C where a = -1 - field.FeNeg(&q.Z, &t0) + q.Z.Neg(&t0) // F = E + D - field.FeAdd(&p.X, &q.Z, &t1) + p.X.Add(&q.Z, &t1) // X3 = (B-C-D)*(F-2) - field.FeSub(&p.Y, &q.X, &t0) - field.FeSub(&p.Y, &p.Y, &t1) - field.FeSub(&p.Z, &p.X, &field.FieldTwo) - field.FeMul(&q.X, &p.Y, &p.Z) + p.Y.Sub(&q.X, &t0) + p.Y.Sub(&p.Y, &t1) + p.Z.Sub(&p.X, radix51.Two) + q.X.Mul(&p.Y, &p.Z) // Y3 = F*(E-D) - field.FeSub(&p.Y, &q.Z, &t1) - field.FeMul(&q.Y, &p.X, &p.Y) + p.Y.Sub(&q.Z, &t1) + q.Y.Mul(&p.X, &p.Y) // Z3 = F^2 - 2*F - field.FeSquare(&q.Z, &p.X) - field.FeSub(&q.Z, &q.Z, &p.X) - field.FeSub(&q.Z, &q.Z, &p.X) + q.Z.Square(&p.X) + q.Z.Sub(&q.Z, &p.X) + q.Z.Sub(&q.Z, &p.X) return &q } diff --git a/internal/radix51/const.go b/internal/radix51/const.go deleted file mode 100644 index b650ff8..0000000 --- a/internal/radix51/const.go +++ /dev/null @@ -1,17 +0,0 @@ -// Copyright (c) 2017 George Tankersley. All rights reserved. -// Use of this source code is governed by a BSD-style -// license that can be found in the LICENSE file. - -// Constants used in the implementation of GF(2^255-19) field arithmetic. -package radix51 - -const ( - // The vaule 2^51-1, used in carry propagation - maskLow51Bits = uint64(1)<<51 - 1 -) - -var ( - FieldZero FieldElement = [5]uint64{0, 0, 0, 0, 0} - FieldOne FieldElement = [5]uint64{1, 0, 0, 0, 0} - FieldTwo FieldElement = [5]uint64{2, 0, 0, 0, 0} -) diff --git a/internal/radix51/fe.go b/internal/radix51/fe.go index 0756e87..d378d02 100644 --- a/internal/radix51/fe.go +++ b/internal/radix51/fe.go @@ -15,10 +15,21 @@ import ( // FieldElement represents an element of the field GF(2^255-19). An element t // represents the integer t[0] + t[1]*2^51 + t[2]*2^102 + t[3]*2^153 + -// t[4]*2^204. +// t[4]*2^204. The zero value is a valid zero element. type FieldElement [5]uint64 -func FeZero(v *FieldElement) { +const ( + // The vaule 2^51-1, used in carry propagation + maskLow51Bits = uint64(1)<<51 - 1 +) + +var ( + Zero = &FieldElement{0, 0, 0, 0, 0} + One = &FieldElement{1, 0, 0, 0, 0} + Two = &FieldElement{2, 0, 0, 0, 0} +) + +func (v *FieldElement) Zero() { v[0] = 0 v[1] = 0 v[2] = 0 @@ -26,7 +37,7 @@ func FeZero(v *FieldElement) { v[4] = 0 } -func FeOne(v *FieldElement) { +func (v *FieldElement) One() { v[0] = 1 v[1] = 0 v[2] = 0 @@ -35,7 +46,7 @@ func FeOne(v *FieldElement) { } // SetInt sets the receiving FieldElement to the specified small integer. -func SetInt(v *FieldElement, x uint64) { +func (v *FieldElement) SetInt(x uint64) { v[0] = x v[1] = 0 v[2] = 0 @@ -43,62 +54,58 @@ func SetInt(v *FieldElement, x uint64) { v[4] = 0 } -func FeReduce(t, v *FieldElement) { - // Copy v - *t = *v - +func (v *FieldElement) Reduce() { // Lev v = v[0] + v[1]*2^51 + v[2]*2^102 + v[3]*2^153 + v[4]*2^204 // Reduce each limb below 2^51, propagating carries. - t[1] += t[0] >> 51 - t[0] = t[0] & maskLow51Bits - t[2] += t[1] >> 51 - t[1] = t[1] & maskLow51Bits - t[3] += t[2] >> 51 - t[2] = t[2] & maskLow51Bits - t[4] += t[3] >> 51 - t[3] = t[3] & maskLow51Bits - t[0] += (t[4] >> 51) * 19 - t[4] = t[4] & maskLow51Bits + v[1] += v[0] >> 51 + v[0] = v[0] & maskLow51Bits + v[2] += v[1] >> 51 + v[1] = v[1] & maskLow51Bits + v[3] += v[2] >> 51 + v[2] = v[2] & maskLow51Bits + v[4] += v[3] >> 51 + v[3] = v[3] & maskLow51Bits + v[0] += (v[4] >> 51) * 19 + v[4] = v[4] & maskLow51Bits - // We now hate a field element t < 2^255, but need t <= 2^255-19 + // We now hate a field element v < 2^255, but need v <= 2^255-19 // TODO Document why this works. It's the elaborate comment about r = h-pq etc etc. // Get the carry bit - c := (t[0] + 19) >> 51 - c = (t[1] + c) >> 51 - c = (t[2] + c) >> 51 - c = (t[3] + c) >> 51 - c = (t[4] + c) >> 51 + c := (v[0] + 19) >> 51 + c = (v[1] + c) >> 51 + c = (v[2] + c) >> 51 + c = (v[3] + c) >> 51 + c = (v[4] + c) >> 51 - t[0] += 19 * c + v[0] += 19 * c - t[1] += t[0] >> 51 - t[0] = t[0] & maskLow51Bits - t[2] += t[1] >> 51 - t[1] = t[1] & maskLow51Bits - t[3] += t[2] >> 51 - t[2] = t[2] & maskLow51Bits - t[4] += t[3] >> 51 - t[3] = t[3] & maskLow51Bits + v[1] += v[0] >> 51 + v[0] = v[0] & maskLow51Bits + v[2] += v[1] >> 51 + v[1] = v[1] & maskLow51Bits + v[3] += v[2] >> 51 + v[2] = v[2] & maskLow51Bits + v[4] += v[3] >> 51 + v[3] = v[3] & maskLow51Bits // no additional carry - t[4] = t[4] & maskLow51Bits + v[4] = v[4] & maskLow51Bits } -// FeAdd sets out = a + b. Long sequences of additions without reduction that +// Add sets v = a + b. Long sequences of additions without reduction that // let coefficients grow larger than 54 bits would be a problem. Paper // cautions: "do not have such sequences of additions". -func FeAdd(out, a, b *FieldElement) { - out[0] = a[0] + b[0] - out[1] = a[1] + b[1] - out[2] = a[2] + b[2] - out[3] = a[3] + b[3] - out[4] = a[4] + b[4] +func (v *FieldElement) Add(a, b *FieldElement) { + v[0] = a[0] + b[0] + v[1] = a[1] + b[1] + v[2] = a[2] + b[2] + v[3] = a[3] + b[3] + v[4] = a[4] + b[4] } -// FeSub sets out = a - b -func FeSub(out, a, b *FieldElement) { - var t FieldElement - t = *b +// Sub sets v = a - b. +func (v *FieldElement) Sub(a, b *FieldElement) { + t := *b // Reduce each limb below 2^51, propagating carries. Ensures that results // fit within the limbs. This would not be required for reduced input. @@ -115,90 +122,88 @@ func FeSub(out, a, b *FieldElement) { // This is slightly more complicated. Because we use unsigned coefficients, we // first add a multiple of p and then subtract. - out[0] = (a[0] + 0xFFFFFFFFFFFDA) - t[0] - out[1] = (a[1] + 0xFFFFFFFFFFFFE) - t[1] - out[2] = (a[2] + 0xFFFFFFFFFFFFE) - t[2] - out[3] = (a[3] + 0xFFFFFFFFFFFFE) - t[3] - out[4] = (a[4] + 0xFFFFFFFFFFFFE) - t[4] + v[0] = (a[0] + 0xFFFFFFFFFFFDA) - t[0] + v[1] = (a[1] + 0xFFFFFFFFFFFFE) - t[1] + v[2] = (a[2] + 0xFFFFFFFFFFFFE) - t[2] + v[3] = (a[3] + 0xFFFFFFFFFFFFE) - t[3] + v[4] = (a[4] + 0xFFFFFFFFFFFFE) - t[4] } -// FeNeg sets out = -a -func FeNeg(out, a *FieldElement) { - var t FieldElement - FeZero(&t) - FeSub(out, &t, a) +// Neg sets v = -a. +func (v *FieldElement) Neg(a *FieldElement) { + v.Sub(&FieldElement{}, a) } -// FeInvert sets out = 1/z mod p by calculating z^(p-2), p-2 = 2^255 - 21. -func FeInvert(out, z *FieldElement) { +// Invert sets v = 1/z mod p by calculating z^(p-2), p-2 = 2^255 - 21. +func (v *FieldElement) Invert(z *FieldElement) { // Inversion is implemented as exponentiation with exponent p − 2. It uses the // same sequence of 255 squarings and 11 multiplications as [Curve25519]. var z2, z9, z11, z2_5_0, z2_10_0, z2_20_0, z2_50_0, z2_100_0, t FieldElement - FeSquare(&z2, z) // 2 - FeSquare(&t, &z2) // 4 - FeSquare(&t, &t) // 8 - FeMul(&z9, &t, z) // 9 - FeMul(&z11, &z9, &z2) // 11 - FeSquare(&t, &z11) // 22 - FeMul(&z2_5_0, &t, &z9) // 2^5 - 2^0 = 31 + z2.Square(z) // 2 + t.Square(&z2) // 4 + t.Square(&t) // 8 + z9.Mul(&t, z) // 9 + z11.Mul(&z9, &z2) // 11 + t.Square(&z11) // 22 + z2_5_0.Mul(&t, &z9) // 2^5 - 2^0 = 31 - FeSquare(&t, &z2_5_0) // 2^6 - 2^1 + t.Square(&z2_5_0) // 2^6 - 2^1 for i := 0; i < 4; i++ { - FeSquare(&t, &t) // 2^10 - 2^5 + t.Square(&t) // 2^10 - 2^5 } - FeMul(&z2_10_0, &t, &z2_5_0) // 2^10 - 2^0 + z2_10_0.Mul(&t, &z2_5_0) // 2^10 - 2^0 - FeSquare(&t, &z2_10_0) // 2^11 - 2^1 + t.Square(&z2_10_0) // 2^11 - 2^1 for i := 0; i < 9; i++ { - FeSquare(&t, &t) // 2^20 - 2^10 + t.Square(&t) // 2^20 - 2^10 } - FeMul(&z2_20_0, &t, &z2_10_0) // 2^20 - 2^0 + z2_20_0.Mul(&t, &z2_10_0) // 2^20 - 2^0 - FeSquare(&t, &z2_20_0) // 2^21 - 2^1 + t.Square(&z2_20_0) // 2^21 - 2^1 for i := 0; i < 19; i++ { - FeSquare(&t, &t) // 2^40 - 2^20 + t.Square(&t) // 2^40 - 2^20 } - FeMul(&t, &t, &z2_20_0) // 2^40 - 2^0 + t.Mul(&t, &z2_20_0) // 2^40 - 2^0 - FeSquare(&t, &t) // 2^41 - 2^1 + t.Square(&t) // 2^41 - 2^1 for i := 0; i < 9; i++ { - FeSquare(&t, &t) // 2^50 - 2^10 + t.Square(&t) // 2^50 - 2^10 } - FeMul(&z2_50_0, &t, &z2_10_0) // 2^50 - 2^0 + z2_50_0.Mul(&t, &z2_10_0) // 2^50 - 2^0 - FeSquare(&t, &z2_50_0) // 2^51 - 2^1 + t.Square(&z2_50_0) // 2^51 - 2^1 for i := 0; i < 49; i++ { - FeSquare(&t, &t) // 2^100 - 2^50 + t.Square(&t) // 2^100 - 2^50 } - FeMul(&z2_100_0, &t, &z2_50_0) // 2^100 - 2^0 + z2_100_0.Mul(&t, &z2_50_0) // 2^100 - 2^0 - FeSquare(&t, &z2_100_0) // 2^101 - 2^1 + t.Square(&z2_100_0) // 2^101 - 2^1 for i := 0; i < 99; i++ { - FeSquare(&t, &t) // 2^200 - 2^100 + t.Square(&t) // 2^200 - 2^100 } - FeMul(&t, &t, &z2_100_0) // 2^200 - 2^0 + t.Mul(&t, &z2_100_0) // 2^200 - 2^0 - FeSquare(&t, &t) // 2^201 - 2^1 + t.Square(&t) // 2^201 - 2^1 for i := 0; i < 49; i++ { - FeSquare(&t, &t) // 2^250 - 2^50 + t.Square(&t) // 2^250 - 2^50 } - FeMul(&t, &t, &z2_50_0) // 2^250 - 2^0 + t.Mul(&t, &z2_50_0) // 2^250 - 2^0 - FeSquare(&t, &t) // 2^251 - 2^1 - FeSquare(&t, &t) // 2^252 - 2^2 - FeSquare(&t, &t) // 2^253 - 2^3 - FeSquare(&t, &t) // 2^254 - 2^4 - FeSquare(&t, &t) // 2^255 - 2^5 + t.Square(&t) // 2^251 - 2^1 + t.Square(&t) // 2^252 - 2^2 + t.Square(&t) // 2^253 - 2^3 + t.Square(&t) // 2^254 - 2^4 + t.Square(&t) // 2^255 - 2^5 - FeMul(out, &t, &z11) // 2^255 - 21 + v.Mul(&t, &z11) // 2^255 - 21 } -func FeCopy(out, in *FieldElement) { - copy(out[:], in[:]) +func (v *FieldElement) Set(a *FieldElement) { + *v = *a } -func FeFromBytes(v *FieldElement, x *[32]byte) { +func (v *FieldElement) FromBytes(x *[32]byte) { v[0] = uint64(x[0]) v[0] |= uint64(x[1]) << 8 v[0] |= uint64(x[2]) << 16 @@ -241,9 +246,9 @@ func FeFromBytes(v *FieldElement, x *[32]byte) { v[4] |= uint64(x[31]&127) << 44 } -func FeToBytes(r *[32]byte, v *FieldElement) { - var t FieldElement - FeReduce(&t, v) +func (v *FieldElement) ToBytes(r *[32]byte) { + t := *v + t.Reduce() r[0] = byte(t[0] & 0xff) r[1] = byte((t[0] >> 8) & 0xff) @@ -287,7 +292,7 @@ func FeToBytes(r *[32]byte, v *FieldElement) { r[31] = byte((t[4] >> 44)) } -func FeFromBig(h *FieldElement, num *big.Int) { +func (v *FieldElement) FromBig(num *big.Int) { var buf [32]byte offset := 0 @@ -305,12 +310,12 @@ func FeFromBig(h *FieldElement, num *big.Int) { } } - FeFromBytes(h, &buf) + v.FromBytes(&buf) } -func FeToBig(h *FieldElement) *big.Int { +func (v *FieldElement) ToBig() *big.Int { var buf [32]byte - FeToBytes(&buf, h) // does a reduction + v.ToBytes(&buf) // does a reduction numWords := 256 / bits.UintSize words := make([]big.Word, numWords) @@ -333,48 +338,48 @@ func FeToBig(h *FieldElement) *big.Int { return out.SetBits(words) } -// FeEqual returns 1 if a and b are equal, and 0 otherwise. -func FeEqual(a, b *FieldElement) int { - var sa, sb [32]byte - FeToBytes(&sa, a) - FeToBytes(&sb, b) - return subtle.ConstantTimeCompare(sa[:], sb[:]) +// Equal returns 1 if v and u are equal, and 0 otherwise. +func (v *FieldElement) Equal(u *FieldElement) int { + var sa, sv [32]byte + u.ToBytes(&sa) + v.ToBytes(&sv) + return subtle.ConstantTimeCompare(sa[:], sv[:]) } -// FeSelect sets out to v if cond == 1, and to u if cond == 0. -// out, v and u are allowed to overlap. -func FeSelect(out, v, u *FieldElement, cond int) { - b := uint64(cond) * 0xffffffffffffffff - out[0] = (b & v[0]) | (^b & u[0]) - out[1] = (b & v[1]) | (^b & u[1]) - out[2] = (b & v[2]) | (^b & u[2]) - out[3] = (b & v[3]) | (^b & u[3]) - out[4] = (b & v[4]) | (^b & u[4]) +// Select sets v to a if cond == 1, and to b if cond == 0. +// v, a and b are allowed to overlap. +func (v *FieldElement) Select(a, b *FieldElement, cond int) { + m := uint64(cond) * 0xffffffffffffffff + v[0] = (m & a[0]) | (^m & b[0]) + v[1] = (m & a[1]) | (^m & b[1]) + v[2] = (m & a[2]) | (^m & b[2]) + v[3] = (m & a[3]) | (^m & b[3]) + v[4] = (m & a[4]) | (^m & b[4]) } -// FeCondNeg sets u to -u if cond == 1, and to u if cond == 0. -func FeCondNeg(u *FieldElement, cond int) { - var neg FieldElement - FeNeg(&neg, u) +// CondNeg sets v to -v if cond == 1, and to v if cond == 0. +func (v *FieldElement) CondNeg(cond int) { + var t FieldElement + t.Neg(v) b := uint64(cond) * 0xffffffffffffffff - u[0] ^= b & (u[0] ^ neg[0]) - u[1] ^= b & (u[1] ^ neg[1]) - u[2] ^= b & (u[2] ^ neg[2]) - u[3] ^= b & (u[3] ^ neg[3]) - u[4] ^= b & (u[4] ^ neg[4]) + v[0] ^= b & (v[0] ^ t[0]) + v[1] ^= b & (v[1] ^ t[1]) + v[2] ^= b & (v[2] ^ t[2]) + v[3] ^= b & (v[3] ^ t[3]) + v[4] ^= b & (v[4] ^ t[4]) } -// FeIsNegative returns 1 if u is negative, and 0 otherwise. -func FeIsNegative(u *FieldElement) int { +// IsNegative returns 1 if v is negative, and 0 otherwise. +func (v *FieldElement) IsNegative() int { var b [32]byte - FeToBytes(&b, u) + v.ToBytes(&b) return int(b[0] & 1) } -// FeAbs sets out to |u|. out and u are allowed to overlap. -func FeAbs(out, u *FieldElement) { - var neg FieldElement - FeNeg(&neg, u) - FeSelect(out, &neg, u, FeIsNegative(u)) +// Abs sets v to |u|. v and u are allowed to overlap. +func (v *FieldElement) Abs(u *FieldElement) { + var t FieldElement + t.Neg(u) + v.Select(&t, u, u.IsNegative()) } diff --git a/internal/radix51/fe_mul.go b/internal/radix51/fe_mul.go index edc7bd2..64a4493 100644 --- a/internal/radix51/fe_mul.go +++ b/internal/radix51/fe_mul.go @@ -6,8 +6,8 @@ package radix51 -// FeMul sets out = a * b -func FeMul(out, x, y *FieldElement) { +// Mul sets out = a * b. +func (v *FieldElement) Mul(x, y *FieldElement) { var x0, x1, x2, x3, x4 uint64 var y0, y1, y2, y3, y4 uint64 @@ -118,9 +118,9 @@ func FeMul(out, x, y *FieldElement) { r00 += (r40 >> 51) * 19 r40 &= maskLow51Bits - out[0] = r00 - out[1] = r10 - out[2] = r20 - out[3] = r30 - out[4] = r40 + v[0] = r00 + v[1] = r10 + v[2] = r20 + v[3] = r30 + v[4] = r40 } diff --git a/internal/radix51/fe_mul_amd64.go b/internal/radix51/fe_mul_amd64.go index b0d38be..887d1cb 100644 --- a/internal/radix51/fe_mul_amd64.go +++ b/internal/radix51/fe_mul_amd64.go @@ -6,5 +6,10 @@ package radix51 +// Mul sets out = a * b. +func (v *FieldElement) Mul(x, y *FieldElement) { + feMul(v, x, y) +} + // go:noescape -func FeMul(out, a, b *FieldElement) +func feMul(out, a, b *FieldElement) diff --git a/internal/radix51/fe_mul_amd64.s b/internal/radix51/fe_mul_amd64.s index e246262..5a72401 100644 --- a/internal/radix51/fe_mul_amd64.s +++ b/internal/radix51/fe_mul_amd64.s @@ -7,8 +7,8 @@ // +build amd64,!noasm -// func FeMul(outp *uint64, xp *uint64, yp *uint64) -TEXT ·FeMul(SB),$0-24 +// func feMul(outp *uint64, xp *uint64, yp *uint64) +TEXT ·feMul(SB),$0-24 MOVQ outp+0(FP), DI MOVQ xp+8(FP), BX MOVQ yp+16(FP), CX diff --git a/internal/radix51/fe_square.go b/internal/radix51/fe_square.go index ec5bd62..031ec7a 100644 --- a/internal/radix51/fe_square.go +++ b/internal/radix51/fe_square.go @@ -6,8 +6,8 @@ package radix51 -// FeSquare sets out = x*x -func FeSquare(out, x *FieldElement) { +// Square sets v = x * x. +func (v *FieldElement) Square(x *FieldElement) { // Squaring needs only 15 mul instructions. Some inputs are multiplied by 2; // this is combined with multiplication by 19 where possible. The coefficient // reduction after squaring is the same as for multiplication. @@ -90,9 +90,9 @@ func FeSquare(out, x *FieldElement) { r00 += (r40 >> 51) * 19 r40 &= maskLow51Bits - out[0] = r00 - out[1] = r10 - out[2] = r20 - out[3] = r30 - out[4] = r40 + v[0] = r00 + v[1] = r10 + v[2] = r20 + v[3] = r30 + v[4] = r40 } diff --git a/internal/radix51/fe_square_amd64.go b/internal/radix51/fe_square_amd64.go index 8c38356..cdaae53 100644 --- a/internal/radix51/fe_square_amd64.go +++ b/internal/radix51/fe_square_amd64.go @@ -6,5 +6,10 @@ package radix51 +// Square sets v = x*x. +func (v *FieldElement) Square(x *FieldElement) { + feSquare(v, x) +} + // go:noescape -func FeSquare(out, x *FieldElement) +func feSquare(out, x *FieldElement) diff --git a/internal/radix51/fe_square_amd64.s b/internal/radix51/fe_square_amd64.s index fa01331..0103416 100644 --- a/internal/radix51/fe_square_amd64.s +++ b/internal/radix51/fe_square_amd64.s @@ -4,8 +4,8 @@ // +build amd64,!noasm -// func FeSquare(outp *uint64, xp *uint64) -TEXT ·FeSquare(SB),4,$0-16 +// func feSquare(outp *uint64, xp *uint64) +TEXT ·feSquare(SB),4,$0-16 MOVQ outp+0(FP), DI MOVQ xp+8(FP), SI diff --git a/internal/radix51/fe_test.go b/internal/radix51/fe_test.go index 3450871..ba6564b 100644 --- a/internal/radix51/fe_test.go +++ b/internal/radix51/fe_test.go @@ -73,8 +73,8 @@ func TestFeFromBytesRoundTrip(t *testing.T) { in = [32]byte{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32} - FeFromBytes(&fe, &in) - FeToBytes(&out, &fe) + fe.FromBytes(&in) + fe.ToBytes(&out) if !bytes.Equal(in[:], out[:]) { t.Error("Bytes<>FE doesn't roundtrip") @@ -87,8 +87,8 @@ func TestFeFromBytesRoundTrip(t *testing.T) { fe[3] = 0x5e8fca9e0881c fe[4] = 0x5c490f087d796 - FeToBytes(&out, &fe) - FeFromBytes(&r, &out) + fe.ToBytes(&out) + r.FromBytes(&out) for i := 0; i < len(fe); i++ { if r[i] != fe[i] { @@ -104,9 +104,9 @@ func TestSanity(t *testing.T) { // var x2Go, x2sqGo FieldElement x = [5]uint64{1, 1, 1, 1, 1} - FeMul(&x2, &x, &x) + x2.Mul(&x, &x) // FeMulGo(&x2Go, &x, &x) - FeSquare(&x2sq, &x) + x2sq.Square(&x) // FeSquareGo(&x2sqGo, &x) // if !vartimeEqual(x2, x2Go) || !vartimeEqual(x2sq, x2sqGo) || !vartimeEqual(x2, x2sq) { @@ -123,11 +123,11 @@ func TestSanity(t *testing.T) { if err != nil { t.Fatal(err) } - FeFromBytes(&x, &bytes) + x.FromBytes(&bytes) - FeMul(&x2, &x, &x) + x2.Mul(&x, &x) // FeMulGo(&x2Go, &x, &x) - FeSquare(&x2sq, &x) + x2sq.Square(&x) // FeSquareGo(&x2sqGo, &x) // if !vartimeEqual(x2, x2Go) || !vartimeEqual(x2sq, x2sqGo) || !vartimeEqual(x2, x2sq) { @@ -152,13 +152,13 @@ func TestFeEqual(t *testing.T) { var x FieldElement = [5]uint64{1, 1, 1, 1, 1} var y FieldElement = [5]uint64{5, 4, 3, 2, 1} - eq := FeEqual(&x, &x) - if !eq { + eq := x.Equal(&x) + if eq != 1 { t.Errorf("wrong about equality") } - eq = FeEqual(&x, &y) - if eq { + eq = x.Equal(&y) + if eq != 0 { t.Errorf("wrong about inequality") } } @@ -168,9 +168,9 @@ func TestFeInvert(t *testing.T) { var one FieldElement = [5]uint64{1, 0, 0, 0, 0} var xinv, r FieldElement - FeInvert(&xinv, &x) - FeMul(&r, &x, &xinv) - FeReduce(&r, &r) + xinv.Invert(&x) + r.Mul(&x, &xinv) + r.Reduce() if !vartimeEqual(one, r) { t.Errorf("inversion identity failed, got: %x", r) @@ -182,11 +182,11 @@ func TestFeInvert(t *testing.T) { if err != nil { t.Fatal(err) } - FeFromBytes(&x, &bytes) + x.FromBytes(&bytes) - FeInvert(&xinv, &x) - FeMul(&r, &x, &xinv) - FeReduce(&r, &r) + xinv.Invert(&x) + r.Mul(&x, &xinv) + r.Reduce() if !vartimeEqual(one, r) { t.Errorf("random inversion identity failed, got: %x for field element %x", r, x) From 6c2fda803c8fba8709c08e176277a86346cb57c2 Mon Sep 17 00:00:00 2001 From: Filippo Valsorda Date: Sat, 26 Jan 2019 20:29:57 -0500 Subject: [PATCH 20/46] internal/radix51: make all APIs not consider the receiver an input --- ed25519.go | 2 +- internal/radix51/fe.go | 27 ++++++++++----------------- internal/radix51/fe_mul.go | 2 +- internal/radix51/fe_mul_amd64.go | 2 +- internal/radix51/fe_square_amd64.go | 2 +- internal/radix51/fe_test.go | 4 ++-- 6 files changed, 16 insertions(+), 23 deletions(-) diff --git a/ed25519.go b/ed25519.go index c3277f4..b93de42 100644 --- a/ed25519.go +++ b/ed25519.go @@ -74,7 +74,7 @@ func (curve ed25519Curve) IsOnCurve(x, y *big.Int) bool { lh.Neg(&lh) // -x^2 lh.Add(&lh, &y2) // -x^2 + y^2 lh.Sub(&lh, &rh) // -x^2 + y^2 - 1 - dx^2y^2 - lh.Reduce() // mod p + lh.Reduce(&lh) // mod p return lh.Equal(radix51.Zero) == 1 } diff --git a/internal/radix51/fe.go b/internal/radix51/fe.go index d378d02..1249aa8 100644 --- a/internal/radix51/fe.go +++ b/internal/radix51/fe.go @@ -54,7 +54,9 @@ func (v *FieldElement) SetInt(x uint64) { v[4] = 0 } -func (v *FieldElement) Reduce() { +func (v *FieldElement) Reduce(u *FieldElement) { + v.Set(u) + // Lev v = v[0] + v[1]*2^51 + v[2]*2^102 + v[3]*2^153 + v[4]*2^204 // Reduce each limb below 2^51, propagating carries. v[1] += v[0] >> 51 @@ -247,8 +249,8 @@ func (v *FieldElement) FromBytes(x *[32]byte) { } func (v *FieldElement) ToBytes(r *[32]byte) { - t := *v - t.Reduce() + var t FieldElement + t.Reduce(v) r[0] = byte(t[0] & 0xff) r[1] = byte((t[0] >> 8) & 0xff) @@ -357,17 +359,10 @@ func (v *FieldElement) Select(a, b *FieldElement, cond int) { v[4] = (m & a[4]) | (^m & b[4]) } -// CondNeg sets v to -v if cond == 1, and to v if cond == 0. -func (v *FieldElement) CondNeg(cond int) { - var t FieldElement - t.Neg(v) - - b := uint64(cond) * 0xffffffffffffffff - v[0] ^= b & (v[0] ^ t[0]) - v[1] ^= b & (v[1] ^ t[1]) - v[2] ^= b & (v[2] ^ t[2]) - v[3] ^= b & (v[3] ^ t[3]) - v[4] ^= b & (v[4] ^ t[4]) +// CondNeg sets v to -u if cond == 1, and to u if cond == 0. +func (v *FieldElement) CondNeg(u *FieldElement, cond int) { + v.Neg(u) + v.Select(v, u, cond) } // IsNegative returns 1 if v is negative, and 0 otherwise. @@ -379,7 +374,5 @@ func (v *FieldElement) IsNegative() int { // Abs sets v to |u|. v and u are allowed to overlap. func (v *FieldElement) Abs(u *FieldElement) { - var t FieldElement - t.Neg(u) - v.Select(&t, u, u.IsNegative()) + v.CondNeg(u, u.IsNegative()) } diff --git a/internal/radix51/fe_mul.go b/internal/radix51/fe_mul.go index 64a4493..2c28e4f 100644 --- a/internal/radix51/fe_mul.go +++ b/internal/radix51/fe_mul.go @@ -6,7 +6,7 @@ package radix51 -// Mul sets out = a * b. +// Mul sets out = x * y. func (v *FieldElement) Mul(x, y *FieldElement) { var x0, x1, x2, x3, x4 uint64 var y0, y1, y2, y3, y4 uint64 diff --git a/internal/radix51/fe_mul_amd64.go b/internal/radix51/fe_mul_amd64.go index 887d1cb..d629bb5 100644 --- a/internal/radix51/fe_mul_amd64.go +++ b/internal/radix51/fe_mul_amd64.go @@ -6,7 +6,7 @@ package radix51 -// Mul sets out = a * b. +// Mul sets out = x * y. func (v *FieldElement) Mul(x, y *FieldElement) { feMul(v, x, y) } diff --git a/internal/radix51/fe_square_amd64.go b/internal/radix51/fe_square_amd64.go index cdaae53..7d5b9ff 100644 --- a/internal/radix51/fe_square_amd64.go +++ b/internal/radix51/fe_square_amd64.go @@ -6,7 +6,7 @@ package radix51 -// Square sets v = x*x. +// Square sets v = x * x. func (v *FieldElement) Square(x *FieldElement) { feSquare(v, x) } diff --git a/internal/radix51/fe_test.go b/internal/radix51/fe_test.go index ba6564b..d7343c2 100644 --- a/internal/radix51/fe_test.go +++ b/internal/radix51/fe_test.go @@ -170,7 +170,7 @@ func TestFeInvert(t *testing.T) { xinv.Invert(&x) r.Mul(&x, &xinv) - r.Reduce() + r.Reduce(&r) if !vartimeEqual(one, r) { t.Errorf("inversion identity failed, got: %x", r) @@ -186,7 +186,7 @@ func TestFeInvert(t *testing.T) { xinv.Invert(&x) r.Mul(&x, &xinv) - r.Reduce() + r.Reduce(&r) if !vartimeEqual(one, r) { t.Errorf("random inversion identity failed, got: %x for field element %x", r, x) From 2925e841f7fb34c8b3ad19b255c81eb569c8a872 Mon Sep 17 00:00:00 2001 From: Filippo Valsorda Date: Sat, 26 Jan 2019 21:05:03 -0500 Subject: [PATCH 21/46] internal: make all APIs chainable --- ed25519.go | 11 +++--- internal/group/ge.go | 19 +++++----- internal/radix51/fe.go | 54 +++++++++++++++++------------ internal/radix51/fe_mul.go | 3 +- internal/radix51/fe_mul_amd64.go | 3 +- internal/radix51/fe_square.go | 3 +- internal/radix51/fe_square_amd64.go | 3 +- 7 files changed, 53 insertions(+), 43 deletions(-) diff --git a/ed25519.go b/ed25519.go index b93de42..7419f81 100644 --- a/ed25519.go +++ b/ed25519.go @@ -91,12 +91,9 @@ func (curve ed25519Curve) Add(x1, y1, x2, y2 *big.Int) (x, y *big.Int) { // Double returns 2*(x,y). func (curve ed25519Curve) Double(x1, y1 *big.Int) (x, y *big.Int) { - var p group.ProjectiveGroupElement - - p.FromAffine(x1, y1) - + p := new(group.ProjectiveGroupElement).FromAffine(x1, y1) // Use the special-case DoubleZ1 here because we know Z will be 1. - return p.DoubleZ1().ToAffine() + return p.DoubleZ1(p).ToAffine() } // ScalarMult returns k*(Bx,By) where k is a number in big-endian form. @@ -125,10 +122,10 @@ func (curve ed25519Curve) ScalarMult(x1, y1 *big.Int, k []byte) (x, y *big.Int) var b = int32((s[i/8] >> uint(i&7)) & 1) if b == 0 { r1.Add(&r0, &r1) - r0.Double() + r0.Double(&r0) } else { r0.Add(&r0, &r1) - r1.Double() + r1.Double(&r0) } } diff --git a/internal/group/ge.go b/internal/group/ge.go index eb99ce7..8ca4a2d 100644 --- a/internal/group/ge.go +++ b/internal/group/ge.go @@ -38,11 +38,12 @@ type ExtendedGroupElement struct { // described in "Twisted Edwards Curves Revisited", Hisil-Wong-Carter-Dawson // 2008, Section 3.1 (https://eprint.iacr.org/2008/522.pdf) // See also https://hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html#addition-add-2008-hwcd-3 -func (v *ExtendedGroupElement) FromAffine(x, y *big.Int) { +func (v *ExtendedGroupElement) FromAffine(x, y *big.Int) *ExtendedGroupElement { v.X.FromBig(x) v.Y.FromBig(y) v.T.Mul(&v.X, &v.Y) v.Z.One() + return v } // Extended coordinates are XYZT with x = X/Z, y = Y/Z, or the "P3" @@ -130,20 +131,17 @@ func (v *ExtendedGroupElement) Add(p1, p2 *ExtendedGroupElement) *ExtendedGroupE // instead of another point in extended coordinates. I have implemented it // this way to see if more straightforward code is worth the (hopefully small) // performance tradeoff. -func (v *ExtendedGroupElement) Double() *ExtendedGroupElement { +func (v *ExtendedGroupElement) Double(u *ExtendedGroupElement) *ExtendedGroupElement { // TODO: Convert to projective coordinates? Section 4.3 mixed doubling? - // TODO: make a decision about how these APIs work wrt chaining/smashing - // *v = *(v.ToProjective().Double().ToExtended()) - // return v var A, B, C, D, E, F, G, H radix51.FieldElement // A ← X1^2, B ← Y1^2 - A.Square(&v.X) - B.Square(&v.Y) + A.Square(&u.X) + B.Square(&u.Y) // C ← 2*Z1^2 - C.Square(&v.Z) + C.Square(&u.Z) C.Add(&C, &C) // TODO should probably implement FeSquare2 // D ← -1*A @@ -151,7 +149,7 @@ func (v *ExtendedGroupElement) Double() *ExtendedGroupElement { // E ← (X1+Y1)^2 − A − B var t0 radix51.FieldElement - t0.Add(&v.X, &v.Y) + t0.Add(&u.X, &u.Y) t0.Square(&t0) E.Sub(&t0, &A) E.Sub(&E, &B) @@ -174,10 +172,11 @@ type ProjectiveGroupElement struct { X, Y, Z radix51.FieldElement } -func (v *ProjectiveGroupElement) FromAffine(x, y *big.Int) { +func (v *ProjectiveGroupElement) FromAffine(x, y *big.Int) *ProjectiveGroupElement { v.X.FromBig(x) v.Y.FromBig(y) v.Z.Zero() + return v } func (v *ProjectiveGroupElement) ToAffine() (*big.Int, *big.Int) { diff --git a/internal/radix51/fe.go b/internal/radix51/fe.go index 1249aa8..a849544 100644 --- a/internal/radix51/fe.go +++ b/internal/radix51/fe.go @@ -29,32 +29,35 @@ var ( Two = &FieldElement{2, 0, 0, 0, 0} ) -func (v *FieldElement) Zero() { +func (v *FieldElement) Zero() *FieldElement { v[0] = 0 v[1] = 0 v[2] = 0 v[3] = 0 v[4] = 0 + return v } -func (v *FieldElement) One() { +func (v *FieldElement) One() *FieldElement { v[0] = 1 v[1] = 0 v[2] = 0 v[3] = 0 v[4] = 0 + return v } // SetInt sets the receiving FieldElement to the specified small integer. -func (v *FieldElement) SetInt(x uint64) { +func (v *FieldElement) SetInt(x uint64) *FieldElement { v[0] = x v[1] = 0 v[2] = 0 v[3] = 0 v[4] = 0 + return v } -func (v *FieldElement) Reduce(u *FieldElement) { +func (v *FieldElement) Reduce(u *FieldElement) *FieldElement { v.Set(u) // Lev v = v[0] + v[1]*2^51 + v[2]*2^102 + v[3]*2^153 + v[4]*2^204 @@ -92,21 +95,24 @@ func (v *FieldElement) Reduce(u *FieldElement) { v[3] = v[3] & maskLow51Bits // no additional carry v[4] = v[4] & maskLow51Bits + + return v } // Add sets v = a + b. Long sequences of additions without reduction that // let coefficients grow larger than 54 bits would be a problem. Paper // cautions: "do not have such sequences of additions". -func (v *FieldElement) Add(a, b *FieldElement) { +func (v *FieldElement) Add(a, b *FieldElement) *FieldElement { v[0] = a[0] + b[0] v[1] = a[1] + b[1] v[2] = a[2] + b[2] v[3] = a[3] + b[3] v[4] = a[4] + b[4] + return v } // Sub sets v = a - b. -func (v *FieldElement) Sub(a, b *FieldElement) { +func (v *FieldElement) Sub(a, b *FieldElement) *FieldElement { t := *b // Reduce each limb below 2^51, propagating carries. Ensures that results @@ -129,15 +135,17 @@ func (v *FieldElement) Sub(a, b *FieldElement) { v[2] = (a[2] + 0xFFFFFFFFFFFFE) - t[2] v[3] = (a[3] + 0xFFFFFFFFFFFFE) - t[3] v[4] = (a[4] + 0xFFFFFFFFFFFFE) - t[4] + + return v } // Neg sets v = -a. -func (v *FieldElement) Neg(a *FieldElement) { - v.Sub(&FieldElement{}, a) +func (v *FieldElement) Neg(a *FieldElement) *FieldElement { + return v.Sub(Zero, a) } // Invert sets v = 1/z mod p by calculating z^(p-2), p-2 = 2^255 - 21. -func (v *FieldElement) Invert(z *FieldElement) { +func (v *FieldElement) Invert(z *FieldElement) *FieldElement { // Inversion is implemented as exponentiation with exponent p − 2. It uses the // same sequence of 255 squarings and 11 multiplications as [Curve25519]. var z2, z9, z11, z2_5_0, z2_10_0, z2_20_0, z2_50_0, z2_100_0, t FieldElement @@ -198,14 +206,15 @@ func (v *FieldElement) Invert(z *FieldElement) { t.Square(&t) // 2^254 - 2^4 t.Square(&t) // 2^255 - 2^5 - v.Mul(&t, &z11) // 2^255 - 21 + return v.Mul(&t, &z11) // 2^255 - 21 } -func (v *FieldElement) Set(a *FieldElement) { +func (v *FieldElement) Set(a *FieldElement) *FieldElement { *v = *a + return v } -func (v *FieldElement) FromBytes(x *[32]byte) { +func (v *FieldElement) FromBytes(x *[32]byte) *FieldElement { v[0] = uint64(x[0]) v[0] |= uint64(x[1]) << 8 v[0] |= uint64(x[2]) << 16 @@ -246,11 +255,12 @@ func (v *FieldElement) FromBytes(x *[32]byte) { v[4] |= uint64(x[29]) << 28 v[4] |= uint64(x[30]) << 36 v[4] |= uint64(x[31]&127) << 44 + + return v } func (v *FieldElement) ToBytes(r *[32]byte) { - var t FieldElement - t.Reduce(v) + t := new(FieldElement).Reduce(v) r[0] = byte(t[0] & 0xff) r[1] = byte((t[0] >> 8) & 0xff) @@ -294,7 +304,7 @@ func (v *FieldElement) ToBytes(r *[32]byte) { r[31] = byte((t[4] >> 44)) } -func (v *FieldElement) FromBig(num *big.Int) { +func (v *FieldElement) FromBig(num *big.Int) *FieldElement { var buf [32]byte offset := 0 @@ -312,7 +322,7 @@ func (v *FieldElement) FromBig(num *big.Int) { } } - v.FromBytes(&buf) + return v.FromBytes(&buf) } func (v *FieldElement) ToBig() *big.Int { @@ -350,19 +360,19 @@ func (v *FieldElement) Equal(u *FieldElement) int { // Select sets v to a if cond == 1, and to b if cond == 0. // v, a and b are allowed to overlap. -func (v *FieldElement) Select(a, b *FieldElement, cond int) { +func (v *FieldElement) Select(a, b *FieldElement, cond int) *FieldElement { m := uint64(cond) * 0xffffffffffffffff v[0] = (m & a[0]) | (^m & b[0]) v[1] = (m & a[1]) | (^m & b[1]) v[2] = (m & a[2]) | (^m & b[2]) v[3] = (m & a[3]) | (^m & b[3]) v[4] = (m & a[4]) | (^m & b[4]) + return v } // CondNeg sets v to -u if cond == 1, and to u if cond == 0. -func (v *FieldElement) CondNeg(u *FieldElement, cond int) { - v.Neg(u) - v.Select(v, u, cond) +func (v *FieldElement) CondNeg(u *FieldElement, cond int) *FieldElement { + return v.Select(v.Neg(u), u, cond) } // IsNegative returns 1 if v is negative, and 0 otherwise. @@ -373,6 +383,6 @@ func (v *FieldElement) IsNegative() int { } // Abs sets v to |u|. v and u are allowed to overlap. -func (v *FieldElement) Abs(u *FieldElement) { - v.CondNeg(u, u.IsNegative()) +func (v *FieldElement) Abs(u *FieldElement) *FieldElement { + return v.CondNeg(u, u.IsNegative()) } diff --git a/internal/radix51/fe_mul.go b/internal/radix51/fe_mul.go index 2c28e4f..08ffbb4 100644 --- a/internal/radix51/fe_mul.go +++ b/internal/radix51/fe_mul.go @@ -7,7 +7,7 @@ package radix51 // Mul sets out = x * y. -func (v *FieldElement) Mul(x, y *FieldElement) { +func (v *FieldElement) Mul(x, y *FieldElement) *FieldElement { var x0, x1, x2, x3, x4 uint64 var y0, y1, y2, y3, y4 uint64 @@ -123,4 +123,5 @@ func (v *FieldElement) Mul(x, y *FieldElement) { v[2] = r20 v[3] = r30 v[4] = r40 + return v } diff --git a/internal/radix51/fe_mul_amd64.go b/internal/radix51/fe_mul_amd64.go index d629bb5..7ad010f 100644 --- a/internal/radix51/fe_mul_amd64.go +++ b/internal/radix51/fe_mul_amd64.go @@ -7,8 +7,9 @@ package radix51 // Mul sets out = x * y. -func (v *FieldElement) Mul(x, y *FieldElement) { +func (v *FieldElement) Mul(x, y *FieldElement) *FieldElement { feMul(v, x, y) + return v } // go:noescape diff --git a/internal/radix51/fe_square.go b/internal/radix51/fe_square.go index 031ec7a..0a2d4ac 100644 --- a/internal/radix51/fe_square.go +++ b/internal/radix51/fe_square.go @@ -7,7 +7,7 @@ package radix51 // Square sets v = x * x. -func (v *FieldElement) Square(x *FieldElement) { +func (v *FieldElement) Square(x *FieldElement) *FieldElement { // Squaring needs only 15 mul instructions. Some inputs are multiplied by 2; // this is combined with multiplication by 19 where possible. The coefficient // reduction after squaring is the same as for multiplication. @@ -95,4 +95,5 @@ func (v *FieldElement) Square(x *FieldElement) { v[2] = r20 v[3] = r30 v[4] = r40 + return v } diff --git a/internal/radix51/fe_square_amd64.go b/internal/radix51/fe_square_amd64.go index 7d5b9ff..d91d669 100644 --- a/internal/radix51/fe_square_amd64.go +++ b/internal/radix51/fe_square_amd64.go @@ -7,8 +7,9 @@ package radix51 // Square sets v = x * x. -func (v *FieldElement) Square(x *FieldElement) { +func (v *FieldElement) Square(x *FieldElement) *FieldElement { feSquare(v, x) + return v } // go:noescape From 79589420d8668f57ba6aaaeb3259f1d382c8f04c Mon Sep 17 00:00:00 2001 From: Filippo Valsorda Date: Sat, 26 Jan 2019 21:05:26 -0500 Subject: [PATCH 22/46] internal/group: rewrite DoubleZ1 because stack is cheaper than mental state --- internal/group/ge.go | 49 ++++++++++++++++---------------------------- 1 file changed, 18 insertions(+), 31 deletions(-) diff --git a/internal/group/ge.go b/internal/group/ge.go index 8ca4a2d..932d7d6 100644 --- a/internal/group/ge.go +++ b/internal/group/ge.go @@ -1,4 +1,5 @@ // Copyright (c) 2017 George Tankersley. All rights reserved. +// Copyright (c) 2019 The Go Authors. All rights reserved. // Use of this source code is governed by a BSD-style // license that can be found in the LICENSE file. @@ -231,44 +232,30 @@ func (v *ProjectiveGroupElement) Zero() *ProjectiveGroupElement { // This assumption is one reason why this package is internal. For instance, it // will not hold throughout a Montgomery ladder, when we convert to projective // from possibly arbitrary extended coordinates. -func (v *ProjectiveGroupElement) DoubleZ1() *ProjectiveGroupElement { - // TODO This function is inconsistent with the other ones in that it - // returns a copy rather than smashing the receiver. It doesn't matter - // because it is always called on ephemeral intermediate values, but should - // fix. - var p, q ProjectiveGroupElement - var t0, t1 radix51.FieldElement +func (v *ProjectiveGroupElement) DoubleZ1(u *ProjectiveGroupElement) *ProjectiveGroupElement { + var B, C, D, E, F radix51.FieldElement - p = *v + if u.Z.Equal(radix51.Zero) != 1 { + panic("ed25519: DoubleZ1 called with Z != 1") + } - // C = X1^2, D = Y1^2 - t0.Square(&p.X) - t1.Square(&p.Y) - - // B = (X1+Y1)^2 - p.Z.Add(&p.X, &p.Y) // Z is irrelevant but already allocated - q.X.Square(&p.Z) - - // E = a*C where a = -1 - q.Z.Neg(&t0) - - // F = E + D - p.X.Add(&q.Z, &t1) + B.Square(B.Add(&u.X, &u.Y)) // B = (X1+Y1)^2 + C.Square(&u.X) // C = X1^2 + D.Square(&u.Y) // D = Y1^2 + E.Neg(&C) // E = a*C where a = -1 + F.Add(&E, &D) // F = E + D // X3 = (B-C-D)*(F-2) - p.Y.Sub(&q.X, &t0) - p.Y.Sub(&p.Y, &t1) - p.Z.Sub(&p.X, radix51.Two) - q.X.Mul(&p.Y, &p.Z) + v.Y.Sub(v.Y.Sub(&B, &C), &D) + v.X.Mul(&v.Y, v.X.Sub(&F, radix51.Two)) // Y3 = F*(E-D) - p.Y.Sub(&q.Z, &t1) - q.Y.Mul(&p.X, &p.Y) + v.Y.Mul(&F, v.Y.Sub(&E, &D)) // Z3 = F^2 - 2*F - q.Z.Square(&p.X) - q.Z.Sub(&q.Z, &p.X) - q.Z.Sub(&q.Z, &p.X) + v.Z.Square(&F) + v.Z.Sub(&v.Z, &F) + v.Z.Sub(&v.Z, &F) - return &q + return v } From 380ecb67b17c5f2c5a908e88f748720ebe3b666d Mon Sep 17 00:00:00 2001 From: Filippo Valsorda Date: Sat, 26 Jan 2019 21:08:22 -0500 Subject: [PATCH 23/46] internal/group: make conversion APIs caller-allocated --- internal/group/ge.go | 12 ++---------- 1 file changed, 2 insertions(+), 10 deletions(-) diff --git a/internal/group/ge.go b/internal/group/ge.go index 932d7d6..e487bdd 100644 --- a/internal/group/ge.go +++ b/internal/group/ge.go @@ -62,14 +62,10 @@ func (v *ExtendedGroupElement) ToAffine() (*big.Int, *big.Int) { } // Per HWCD, it is safe to move from extended to projective by simply ignoring T. -func (v *ExtendedGroupElement) ToProjective() *ProjectiveGroupElement { - var p ProjectiveGroupElement - +func (v *ExtendedGroupElement) ToProjective(p *ProjectiveGroupElement) { p.X.Set(&v.X) p.Y.Set(&v.Y) p.Z.Set(&v.Z) - - return &p } func (v *ExtendedGroupElement) Zero() *ExtendedGroupElement { @@ -193,15 +189,11 @@ func (v *ProjectiveGroupElement) ToAffine() (*big.Int, *big.Int) { // HWCD Section 3: "Given (X : Y : Z) in [projective coordinates] passing to // [extended coordinates, (X : Y : T : Z)] can be performed in 3M+1S by computing // (XZ, YZ, XY, Z^2)" -func (v *ProjectiveGroupElement) ToExtended() *ExtendedGroupElement { - var r ExtendedGroupElement - +func (v *ProjectiveGroupElement) ToExtended(r *ExtendedGroupElement) { r.X.Mul(&v.X, &v.Z) r.Y.Mul(&v.Y, &v.Z) r.T.Mul(&v.X, &v.Y) r.Z.Square(&v.Z) - - return &r } func (v *ProjectiveGroupElement) Zero() *ProjectiveGroupElement { From 982236110cdc371a3dcca509befe94b688dc941b Mon Sep 17 00:00:00 2001 From: Filippo Valsorda Date: Sat, 26 Jan 2019 22:22:30 -0500 Subject: [PATCH 24/46] internal/radix51: add MinusOne --- internal/radix51/fe.go | 7 ++++--- 1 file changed, 4 insertions(+), 3 deletions(-) diff --git a/internal/radix51/fe.go b/internal/radix51/fe.go index a849544..1320839 100644 --- a/internal/radix51/fe.go +++ b/internal/radix51/fe.go @@ -24,9 +24,10 @@ const ( ) var ( - Zero = &FieldElement{0, 0, 0, 0, 0} - One = &FieldElement{1, 0, 0, 0, 0} - Two = &FieldElement{2, 0, 0, 0, 0} + Zero = &FieldElement{0, 0, 0, 0, 0} + One = &FieldElement{1, 0, 0, 0, 0} + Two = &FieldElement{2, 0, 0, 0, 0} + MinusOne = new(FieldElement).Neg(One) ) func (v *FieldElement) Zero() *FieldElement { From 968377f9c551ce662bf7ab7f288fe8557a9dc3f5 Mon Sep 17 00:00:00 2001 From: George Tankersley Date: Mon, 28 Jan 2019 09:32:56 -0500 Subject: [PATCH 25/46] ed25519: fix ScalarMult and improve test to catch the bug (#1) --- ed25519.go | 2 +- ed25519_test.go | 41 +++++++++++++++++------------------------ 2 files changed, 18 insertions(+), 25 deletions(-) diff --git a/ed25519.go b/ed25519.go index 7419f81..4a2ecfa 100644 --- a/ed25519.go +++ b/ed25519.go @@ -125,7 +125,7 @@ func (curve ed25519Curve) ScalarMult(x1, y1 *big.Int, k []byte) (x, y *big.Int) r0.Double(&r0) } else { r0.Add(&r0, &r1) - r1.Double(&r0) + r1.Double(&r1) } } diff --git a/ed25519_test.go b/ed25519_test.go index 1949df1..26026c9 100644 --- a/ed25519_test.go +++ b/ed25519_test.go @@ -104,35 +104,28 @@ func BenchmarkDouble(b *testing.B) { func TestScalarMult(t *testing.T) { ed := Ed25519() - x, y := ed.Params().Gx, ed.Params().Gy + Bx, By := ed.Params().Gx, ed.Params().Gy + var rX, rY, accX, accY = new(big.Int), new(big.Int), new(big.Int), new(big.Int) - twoX, twoY := ed.ScalarMult(x, y, big.NewInt(2).Bytes()) - xPlusX, yPlusY := ed.Add(x, y, x, y) + for i := 1; i <= 1024; i++ { + rX, rY = ed.ScalarMult(Bx, By, big.NewInt(int64(i)).Bytes()) + if i == 0 && (rX.Cmp(Bx) != 0 || rY.Cmp(By) != 0) { + t.Error("bad ScalarMul") + } + accX.Set(Bx) + accY.Set(By) + for j := 1; j < i; j++ { + accX, accY = ed.Add(accX, accY, Bx, By) + } - if !ed.IsOnCurve(twoX, twoY) { - t.Error("2*B is not on the curve") - } + if !ed.IsOnCurve(rX, rY) || !ed.IsOnCurve(accX, accY) { + t.Error("not on the curve") + } - if twoX.Cmp(xPlusX) != 0 || twoY.Cmp(yPlusY) != 0 { - t.Errorf("2*B != B+B") - } - - // TODO: fuzz like it's going out of style - if !testing.Short() { - buf := make([]byte, 32) - for i := 0; i < 1000; i++ { - _, err := io.ReadFull(rand.Reader, buf) - if err != nil { - t.Fatal(err) - } - randX, randY := ed.ScalarMult(x, y, buf) - - if !ed.IsOnCurve(randX, randY) { - t.Errorf("scalarMult return off-curve point for scalar %x", buf) - } + if rX.Cmp(accX) != 0 || rY.Cmp(accY) != 0 { + t.Errorf("inconsistent ScalarMult: %x", i) } } - } func BenchmarkScalarMult(b *testing.B) { From b9aebb71e82f316a37c9876f89f080a6f0a930fb Mon Sep 17 00:00:00 2001 From: Filippo Valsorda Date: Mon, 28 Jan 2019 09:39:02 -0500 Subject: [PATCH 26/46] internal/group: derive twoD from D --- internal/group/ge.go | 3 +-- 1 file changed, 1 insertion(+), 2 deletions(-) diff --git a/internal/group/ge.go b/internal/group/ge.go index e487bdd..e870840 100644 --- a/internal/group/ge.go +++ b/internal/group/ge.go @@ -76,8 +76,7 @@ func (v *ExtendedGroupElement) Zero() *ExtendedGroupElement { return v } -var twoD = &radix51.FieldElement{1859910466990425, 932731440258426, - 1072319116312658, 1815898335770999, 633789495995903} +var twoD = new(radix51.FieldElement).Add(D, D) // This is the same addition formula everyone uses, "add-2008-hwcd-3". // https://hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html#addition-add-2008-hwcd-3 From 6781d1fcec8e1566095e57a8fa33c34b337db8a6 Mon Sep 17 00:00:00 2001 From: Filippo Valsorda Date: Mon, 28 Jan 2019 09:43:02 -0500 Subject: [PATCH 27/46] internal/group: fix typo --- internal/group/ge.go | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/internal/group/ge.go b/internal/group/ge.go index e870840..a428e76 100644 --- a/internal/group/ge.go +++ b/internal/group/ge.go @@ -141,7 +141,7 @@ func (v *ExtendedGroupElement) Double(u *ExtendedGroupElement) *ExtendedGroupEle C.Add(&C, &C) // TODO should probably implement FeSquare2 // D ← -1*A - D.Neg(&A) // implemented as substraction + D.Neg(&A) // implemented as subtraction // E ← (X1+Y1)^2 − A − B var t0 radix51.FieldElement From 032d7f64e8fedff98bccb78b43cd060719742a00 Mon Sep 17 00:00:00 2001 From: Filippo Valsorda Date: Mon, 28 Jan 2019 09:51:04 -0500 Subject: [PATCH 28/46] internal/group: set Z to 1, not 0 in FromAffine --- internal/group/ge.go | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/internal/group/ge.go b/internal/group/ge.go index a428e76..91fb65f 100644 --- a/internal/group/ge.go +++ b/internal/group/ge.go @@ -171,7 +171,7 @@ type ProjectiveGroupElement struct { func (v *ProjectiveGroupElement) FromAffine(x, y *big.Int) *ProjectiveGroupElement { v.X.FromBig(x) v.Y.FromBig(y) - v.Z.Zero() + v.Z.One() return v } @@ -226,7 +226,7 @@ func (v *ProjectiveGroupElement) Zero() *ProjectiveGroupElement { func (v *ProjectiveGroupElement) DoubleZ1(u *ProjectiveGroupElement) *ProjectiveGroupElement { var B, C, D, E, F radix51.FieldElement - if u.Z.Equal(radix51.Zero) != 1 { + if u.Z.Equal(radix51.One) != 1 { panic("ed25519: DoubleZ1 called with Z != 1") } From 6b0cdf071aa36874d579f0ca718c7b953da29301 Mon Sep 17 00:00:00 2001 From: Filippo Valsorda Date: Mon, 28 Jan 2019 09:54:14 -0500 Subject: [PATCH 29/46] internal/radix51: define a mask64Bits constant --- internal/radix51/fe.go | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/internal/radix51/fe.go b/internal/radix51/fe.go index 1320839..755007c 100644 --- a/internal/radix51/fe.go +++ b/internal/radix51/fe.go @@ -359,10 +359,12 @@ func (v *FieldElement) Equal(u *FieldElement) int { return subtle.ConstantTimeCompare(sa[:], sv[:]) } +const mask64Bits uint64 = (1 << 64) - 1 + // Select sets v to a if cond == 1, and to b if cond == 0. // v, a and b are allowed to overlap. func (v *FieldElement) Select(a, b *FieldElement, cond int) *FieldElement { - m := uint64(cond) * 0xffffffffffffffff + m := uint64(cond) * mask64Bits v[0] = (m & a[0]) | (^m & b[0]) v[1] = (m & a[1]) | (^m & b[1]) v[2] = (m & a[2]) | (^m & b[2]) From 4ac00273fd6d096712f5e6bc0e4e28f2c51b0416 Mon Sep 17 00:00:00 2001 From: George Tankersley Date: Tue, 19 Feb 2019 21:45:46 +0000 Subject: [PATCH 30/46] radix51: use go1.12 intrinsics for 128-bit multiplications --- internal/radix51/fe_mul.go | 50 +++++++++++----------- internal/radix51/fe_square.go | 30 ++++++------- internal/radix51/fe_test.go | 34 ++++----------- internal/radix51/mul_bits.go | 18 ++++++++ internal/radix51/{mul.go => mul_compat.go} | 7 +-- 5 files changed, 70 insertions(+), 69 deletions(-) create mode 100644 internal/radix51/mul_bits.go rename internal/radix51/{mul.go => mul_compat.go} (69%) diff --git a/internal/radix51/fe_mul.go b/internal/radix51/fe_mul.go index 08ffbb4..27e7f5b 100644 --- a/internal/radix51/fe_mul.go +++ b/internal/radix51/fe_mul.go @@ -34,39 +34,39 @@ func (v *FieldElement) Mul(x, y *FieldElement) *FieldElement { x4_19 := x4 * 19 // calculate r0 = x0*y0 + 19*(x1*y4 + x2*y3 + x3*y2 + x4*y1) - r00, r01 := mul64x64(0, 0, x0, y0) - r00, r01 = mul64x64(r00, r01, x1_19, y4) - r00, r01 = mul64x64(r00, r01, x2_19, y3) - r00, r01 = mul64x64(r00, r01, x3_19, y2) - r00, r01 = mul64x64(r00, r01, x4_19, y1) + r00, r01 := madd64(0, 0, x0, y0) + r00, r01 = madd64(r00, r01, x1_19, y4) + r00, r01 = madd64(r00, r01, x2_19, y3) + r00, r01 = madd64(r00, r01, x3_19, y2) + r00, r01 = madd64(r00, r01, x4_19, y1) // calculate r1 = x0*y1 + x1*y0 + 19*(x2*y4 + x3*y3 + x4*y2) - r10, r11 := mul64x64(0, 0, x0, y1) - r10, r11 = mul64x64(r10, r11, x1, y0) - r10, r11 = mul64x64(r10, r11, x2_19, y4) - r10, r11 = mul64x64(r10, r11, x3_19, y3) - r10, r11 = mul64x64(r10, r11, x4_19, y2) + r10, r11 := madd64(0, 0, x0, y1) + r10, r11 = madd64(r10, r11, x1, y0) + r10, r11 = madd64(r10, r11, x2_19, y4) + r10, r11 = madd64(r10, r11, x3_19, y3) + r10, r11 = madd64(r10, r11, x4_19, y2) // calculate r2 = x0*y2 + x1*y1 + x2*y0 + 19*(x3*y4 + x4*y3) - r20, r21 := mul64x64(0, 0, x0, y2) - r20, r21 = mul64x64(r20, r21, x1, y1) - r20, r21 = mul64x64(r20, r21, x2, y0) - r20, r21 = mul64x64(r20, r21, x3_19, y4) - r20, r21 = mul64x64(r20, r21, x4_19, y3) + r20, r21 := madd64(0, 0, x0, y2) + r20, r21 = madd64(r20, r21, x1, y1) + r20, r21 = madd64(r20, r21, x2, y0) + r20, r21 = madd64(r20, r21, x3_19, y4) + r20, r21 = madd64(r20, r21, x4_19, y3) // calculate r3 = x0*y3 + x1*y2 + x2*y1 + x3*y0 + 19*x4*y4 - r30, r31 := mul64x64(0, 0, x0, y3) - r30, r31 = mul64x64(r30, r31, x1, y2) - r30, r31 = mul64x64(r30, r31, x2, y1) - r30, r31 = mul64x64(r30, r31, x3, y0) - r30, r31 = mul64x64(r30, r31, x4_19, y4) + r30, r31 := madd64(0, 0, x0, y3) + r30, r31 = madd64(r30, r31, x1, y2) + r30, r31 = madd64(r30, r31, x2, y1) + r30, r31 = madd64(r30, r31, x3, y0) + r30, r31 = madd64(r30, r31, x4_19, y4) // calculate r4 = x0*y4 + x1*y3 + x2*y2 + x3*y1 + x4*y0 - r40, r41 := mul64x64(0, 0, x0, y4) - r40, r41 = mul64x64(r40, r41, x1, y3) - r40, r41 = mul64x64(r40, r41, x2, y2) - r40, r41 = mul64x64(r40, r41, x3, y1) - r40, r41 = mul64x64(r40, r41, x4, y0) + r40, r41 := madd64(0, 0, x0, y4) + r40, r41 = madd64(r40, r41, x1, y3) + r40, r41 = madd64(r40, r41, x2, y2) + r40, r41 = madd64(r40, r41, x3, y1) + r40, r41 = madd64(r40, r41, x4, y0) // After the multiplication we need to reduce (carry) the 5 coefficients to // obtain a result with coefficients that are at most slightly larger than diff --git a/internal/radix51/fe_square.go b/internal/radix51/fe_square.go index 0a2d4ac..46e01fb 100644 --- a/internal/radix51/fe_square.go +++ b/internal/radix51/fe_square.go @@ -31,29 +31,29 @@ func (v *FieldElement) Square(x *FieldElement) *FieldElement { x4_19 := x4 * 19 // r0 = x0*x0 + x1*38*x4 + x2*38*x3 - r00, r01 := mul64x64(0, 0, x0, x0) - r00, r01 = mul64x64(r00, r01, x1_38, x4) - r00, r01 = mul64x64(r00, r01, x2_38, x3) + r00, r01 := madd64(0, 0, x0, x0) + r00, r01 = madd64(r00, r01, x1_38, x4) + r00, r01 = madd64(r00, r01, x2_38, x3) // r1 = x0*2*x1 + x2*38*x4 + x3*19*x3 - r10, r11 := mul64x64(0, 0, x0_2, x1) - r10, r11 = mul64x64(r10, r11, x2_38, x4) - r10, r11 = mul64x64(r10, r11, x3_19, x3) + r10, r11 := madd64(0, 0, x0_2, x1) + r10, r11 = madd64(r10, r11, x2_38, x4) + r10, r11 = madd64(r10, r11, x3_19, x3) // r2 = x0*2*x2 + x1*x1 + x3*38*x4 - r20, r21 := mul64x64(0, 0, x0_2, x2) - r20, r21 = mul64x64(r20, r21, x1, x1) - r20, r21 = mul64x64(r20, r21, x3_38, x4) + r20, r21 := madd64(0, 0, x0_2, x2) + r20, r21 = madd64(r20, r21, x1, x1) + r20, r21 = madd64(r20, r21, x3_38, x4) // r3 = x0*2*x3 + x1*2*x2 + x4*19*x4 - r30, r31 := mul64x64(0, 0, x0_2, x3) - r30, r31 = mul64x64(r30, r31, x1_2, x2) - r30, r31 = mul64x64(r30, r31, x4_19, x4) + r30, r31 := madd64(0, 0, x0_2, x3) + r30, r31 = madd64(r30, r31, x1_2, x2) + r30, r31 = madd64(r30, r31, x4_19, x4) // r4 = x0*2*x4 + x1*2*x3 + x2*x2 - r40, r41 := mul64x64(0, 0, x0_2, x4) - r40, r41 = mul64x64(r40, r41, x1_2, x3) - r40, r41 = mul64x64(r40, r41, x2, x2) + r40, r41 := madd64(0, 0, x0_2, x4) + r40, r41 = madd64(r40, r41, x1_2, x3) + r40, r41 = madd64(r40, r41, x2, x2) // Same reduction diff --git a/internal/radix51/fe_test.go b/internal/radix51/fe_test.go index d7343c2..2ce6629 100644 --- a/internal/radix51/fe_test.go +++ b/internal/radix51/fe_test.go @@ -9,60 +9,42 @@ import ( "crypto/rand" "io" "testing" - "unsafe" ) func TestMul64to128(t *testing.T) { a := uint64(5) b := uint64(5) - r0, r1 := mul64x64(0, 0, a, b) + r0, r1 := madd64(0, 0, a, b) if r0 != 0x19 || r1 != 0 { t.Errorf("lo-range wide mult failed, got %d + %d*(2**64)", r0, r1) } a = uint64(18014398509481983) // 2^54 - 1 b = uint64(18014398509481983) // 2^54 - 1 - r0, r1 = mul64x64(0, 0, a, b) + r0, r1 = madd64(0, 0, a, b) if r0 != 0xff80000000000001 || r1 != 0xfffffffffff { t.Errorf("hi-range wide mult failed, got %d + %d*(2**64)", r0, r1) } a = uint64(1125899906842661) b = uint64(2097155) - r0, r1 = mul64x64(0, 0, a, b) - r0, r1 = mul64x64(r0, r1, a, b) - r0, r1 = mul64x64(r0, r1, a, b) - r0, r1 = mul64x64(r0, r1, a, b) - r0, r1 = mul64x64(r0, r1, a, b) + r0, r1 = madd64(0, 0, a, b) + r0, r1 = madd64(r0, r1, a, b) + r0, r1 = madd64(r0, r1, a, b) + r0, r1 = madd64(r0, r1, a, b) + r0, r1 = madd64(r0, r1, a, b) if r0 != 16888498990613035 || r1 != 640 { t.Errorf("wrong answer: %d + %d*(2**64)", r0, r1) } } -func BenchmarkWideMultInline(t *testing.B) { - var r0, r1, ol, oh uint64 - a := uint64(18014398509481983) // 2^54 - 1 - b := uint64(18014398509481983) // 2^54 - 1 - - for i := 0; i < t.N; i++ { - t1 := (a>>32)*(b&0xFFFFFFFF) + ((a & 0xFFFFFFFF) * (b & 0xFFFFFFFF) >> 32) - t2 := (a&0xFFFFFFFF)*(b>>32) + (t1 & 0xFFFFFFFF) - ol = (a * b) + r0 - cmp := ol < r0 - oh = r1 + (a>>32)*(b>>32) + t1>>32 + t2>>32 + uint64(*(*byte)(unsafe.Pointer(&cmp))) - - r1 = oh - r0 = ol - } -} - func BenchmarkWideMultCall(t *testing.B) { var r0, r1 uint64 a := uint64(18014398509481983) b := uint64(18014398509481983) for i := 0; i < t.N; i++ { - r0, r1 = mul64x64(r0, r1, a, b) + r0, r1 = madd64(r0, r1, a, b) } } diff --git a/internal/radix51/mul_bits.go b/internal/radix51/mul_bits.go new file mode 100644 index 0000000..0965d03 --- /dev/null +++ b/internal/radix51/mul_bits.go @@ -0,0 +1,18 @@ +// Copyright (c) 2019 George Tankersley. All rights reserved. +// Use of this source code is governed by a BSD-style +// license that can be found in the LICENSE file. + +// +build go1.12 + +package radix51 + +import "math/bits" + +// madd64 multiples two 64-bit numbers and adds them to a split 128-bit accumulator. +func madd64(lo, hi, a, b uint64) (ol uint64, oh uint64) { + oh, ol = bits.Mul64(a, b) + var c uint64 + ol, c = bits.Add64(ol, lo, 0) + oh, _ = bits.Add64(oh, hi, c) + return +} diff --git a/internal/radix51/mul.go b/internal/radix51/mul_compat.go similarity index 69% rename from internal/radix51/mul.go rename to internal/radix51/mul_compat.go index c196d9d..54427a9 100644 --- a/internal/radix51/mul.go +++ b/internal/radix51/mul_compat.go @@ -2,13 +2,14 @@ // Use of this source code is governed by a BSD-style // license that can be found in the LICENSE file. +// +build !go1.12 + package radix51 import "unsafe" -// mul64x64 multiples two 64-bit numbers and adds them to two accumulators. -// This function is written to ensure it inlines. I am so sorry. -func mul64x64(lo, hi, a, b uint64) (ol uint64, oh uint64) { +// madd64 multiples two 64-bit numbers and adds them to a split 128-bit accumulator. +func madd64(lo, hi, a, b uint64) (ol uint64, oh uint64) { t1 := (a>>32)*(b&0xFFFFFFFF) + ((a & 0xFFFFFFFF) * (b & 0xFFFFFFFF) >> 32) t2 := (a&0xFFFFFFFF)*(b>>32) + (t1 & 0xFFFFFFFF) ol = (a * b) + lo From 5758cbf76cde035288e4db709ab84cd6d23394c9 Mon Sep 17 00:00:00 2001 From: Henry de Valence Date: Sat, 2 Mar 2019 17:33:46 -0500 Subject: [PATCH 31/46] internal/radix51: add property-based tests that multiplication distributes over addition Closes #12 --- internal/radix51/fe_test.go | 67 +++++++++++++++++++++++++++++++++++++ 1 file changed, 67 insertions(+) diff --git a/internal/radix51/fe_test.go b/internal/radix51/fe_test.go index 2ce6629..c1812fb 100644 --- a/internal/radix51/fe_test.go +++ b/internal/radix51/fe_test.go @@ -8,9 +8,76 @@ import ( "bytes" "crypto/rand" "io" + mathrand "math/rand" + "reflect" "testing" + "testing/quick" ) +var quickCheckScaleFactor = uint8(3) +var quickCheckConfig = &quick.Config{MaxCount: (1 << (12 + quickCheckScaleFactor))} + +func generateFieldElement(rand *mathrand.Rand) FieldElement { + // Generation strategy: generate random limb values bounded by + // 2**(51+b), where b is a parameter controlling the bit-excess. + b := uint64(0) + mask := (uint64(1) << (51 + b)) - 1 + return FieldElement{ + rand.Uint64() & mask, + rand.Uint64() & mask, + rand.Uint64() & mask, + rand.Uint64() & mask, + rand.Uint64() & mask, + } +} + +func (x FieldElement) Generate(rand *mathrand.Rand, size int) reflect.Value { + return reflect.ValueOf(generateFieldElement(rand)) +} + +func TestFieldElementMulDistributesOverAdd(t *testing.T) { + mulDistributesOverAdd := func(x, y, z FieldElement) bool { + // Compute t1 = (x+y)*z + t1 := new(FieldElement) + t1.Add(&x, &y) + t1.Mul(t1, &z) + + // Compute t2 = x*z + y*z + t2 := new(FieldElement) + t3 := new(FieldElement) + t2.Mul(&x, &z) + t3.Mul(&y, &z) + t2.Add(t2, t3) + + return t1.Equal(t2) == 1 + } + + if err := quick.Check(mulDistributesOverAdd, quickCheckConfig); err != nil { + t.Error(err) + } +} + +func TestMulDistributionFailure1(t *testing.T) { + x := FieldElement{0x592101fd8643a, 0x25a08467381e1, 0x48cb4dcd5dcf5, 0x1074d52744164, 0x91902aac541b} + y := FieldElement{0x165bb67340a7f, 0x52cf7781f4ad6, 0x32534ba21fde4, 0x5b4ba9cbb1736, 0x2e90748c54289} + z := FieldElement{0x25a575665ad1e, 0x124e496e3eeaa, 0x433d2180e2561, 0x221c8be3aa11a, 0x7adc8d0adf806} + + t1 := new(FieldElement) + t1.Add(&x, &y) + t1.Mul(t1, &z) + + // Compute t2 = x*z + y*z + t2 := new(FieldElement) + t3 := new(FieldElement) + t2.Mul(&x, &z) + t3.Mul(&y, &z) + t2.Add(t2, t3) + + if t1.Equal(t2) != 1 { + t.Errorf("t1 = %x should equal t2 = %x", t1, t2) + } +} + func TestMul64to128(t *testing.T) { a := uint64(5) b := uint64(5) From a68796f0119cf3308a542d717659db62ca437233 Mon Sep 17 00:00:00 2001 From: Filippo Valsorda Date: Sat, 2 Mar 2019 18:35:44 -0500 Subject: [PATCH 32/46] internal/radix51: test all combinations of argument and receiver aliasing --- internal/radix51/alias_test.go | 129 +++++++++++++++++++++++++++++++++ internal/radix51/fe_test.go | 5 +- 2 files changed, 132 insertions(+), 2 deletions(-) create mode 100644 internal/radix51/alias_test.go diff --git a/internal/radix51/alias_test.go b/internal/radix51/alias_test.go new file mode 100644 index 0000000..ad3931a --- /dev/null +++ b/internal/radix51/alias_test.go @@ -0,0 +1,129 @@ +// Copyright (c) 2019 The Go Authors. All rights reserved. +// Use of this source code is governed by a BSD-style +// license that can be found in the LICENSE file. + +package radix51 + +import ( + "testing" + "testing/quick" +) + +func checkAliasingOneArg(f func(v, x *FieldElement) *FieldElement) func(v, x FieldElement) bool { + return func(v, x FieldElement) bool { + x1, v1 := x, x + + // Calculate a reference f(x) without aliasing. + if out := f(&v, &x); out != &v { + return false + } + + // Test aliasing the argument and the receiver. + if out := f(&v1, &v1); out != &v1 || v1 != v { + return false + } + + // Ensure the arguments was not modified. + return x == x1 + } +} + +func checkAliasingTwoArgs(f func(v, x, y *FieldElement) *FieldElement) func(v, x, y FieldElement) bool { + return func(v, x, y FieldElement) bool { + x1, y1, v1 := x, y, FieldElement{} + + // Calculate a reference f(x, y) without aliasing. + if out := f(&v, &x, &y); out != &v { + return false + } + + // Test aliasing the first argument and the receiver. + v1 = x + if out := f(&v1, &v1, &y); out != &v1 || v1 != v { + return false + } + // Test aliasing the second argument and the receiver. + v1 = y + if out := f(&v1, &x, &v1); out != &v1 || v1 != v { + return false + } + + // Calculate a reference f(x, x) without aliasing. + if out := f(&v, &x, &x); out != &v { + return false + } + + // Test aliasing the first argument and the receiver. + v1 = x + if out := f(&v1, &v1, &x); out != &v1 || v1 != v { + return false + } + // Test aliasing the second argument and the receiver. + v1 = x + if out := f(&v1, &x, &v1); out != &v1 || v1 != v { + return false + } + // Test aliasing both arguments and the receiver. + v1 = x + if out := f(&v1, &v1, &v1); out != &v1 || v1 != v { + return false + } + + // Ensure the arguments were not modified. + return x == x1 && y == y1 + } +} + +func TestAliasing(t *testing.T) { + type target struct { + name string + oneArgF func(v, x *FieldElement) *FieldElement + twoArgsF func(v, x, y *FieldElement) *FieldElement + } + for _, tt := range []target{ + {name: "Abs", oneArgF: (*FieldElement).Abs}, + {name: "Invert", oneArgF: (*FieldElement).Invert}, + {name: "Neg", oneArgF: (*FieldElement).Neg}, + {name: "Reduce", oneArgF: (*FieldElement).Reduce}, + {name: "Set", oneArgF: (*FieldElement).Set}, + {name: "Square", oneArgF: (*FieldElement).Square}, + { + name: "CondNeg0", + oneArgF: func(v, x *FieldElement) *FieldElement { + return (*FieldElement).CondNeg(v, x, 0) + }, + }, + { + name: "CondNeg1", + oneArgF: func(v, x *FieldElement) *FieldElement { + return (*FieldElement).CondNeg(v, x, 1) + }, + }, + {name: "Mul", twoArgsF: (*FieldElement).Mul}, + {name: "Add", twoArgsF: (*FieldElement).Add}, + {name: "Sub", twoArgsF: (*FieldElement).Sub}, + { + name: "Select0", + twoArgsF: func(v, x, y *FieldElement) *FieldElement { + return (*FieldElement).Select(v, x, y, 0) + }, + }, + { + name: "Select1", + twoArgsF: func(v, x, y *FieldElement) *FieldElement { + return (*FieldElement).Select(v, x, y, 1) + }, + }, + } { + var err error + switch { + case tt.oneArgF != nil: + quick.Check(checkAliasingOneArg(tt.oneArgF), quickCheckConfig) + case tt.twoArgsF != nil: + quick.Check(checkAliasingTwoArgs(tt.twoArgsF), quickCheckConfig) + } + if err != nil { + t.Errorf("%v: %v", tt.name, err) + } + } +} diff --git a/internal/radix51/fe_test.go b/internal/radix51/fe_test.go index c1812fb..b77a50d 100644 --- a/internal/radix51/fe_test.go +++ b/internal/radix51/fe_test.go @@ -14,8 +14,9 @@ import ( "testing/quick" ) -var quickCheckScaleFactor = uint8(3) -var quickCheckConfig = &quick.Config{MaxCount: (1 << (12 + quickCheckScaleFactor))} +// quickCheckConfig will make each quickcheck test run (256 * -quickchecks) +// times. The default value of -quickchecks is 100. +var quickCheckConfig = &quick.Config{MaxCountScale: 1 << 8} func generateFieldElement(rand *mathrand.Rand) FieldElement { // Generation strategy: generate random limb values bounded by From ea3deb459c8c3c7303ec0b530e0b03eae9729716 Mon Sep 17 00:00:00 2001 From: George Tankersley Date: Sun, 3 Mar 2019 02:36:09 +0000 Subject: [PATCH 33/46] radix51: change API of FromBytes and ToBytes to use slices --- internal/radix51/fe.go | 20 +++++++++++++------- internal/radix51/fe_test.go | 12 ++++++------ 2 files changed, 19 insertions(+), 13 deletions(-) diff --git a/internal/radix51/fe.go b/internal/radix51/fe.go index 755007c..1ee3bc6 100644 --- a/internal/radix51/fe.go +++ b/internal/radix51/fe.go @@ -215,7 +215,10 @@ func (v *FieldElement) Set(a *FieldElement) *FieldElement { return v } -func (v *FieldElement) FromBytes(x *[32]byte) *FieldElement { +func (v *FieldElement) FromBytes(x []byte) *FieldElement { + if len(x) != 32 { + panic("invalid input size") + } v[0] = uint64(x[0]) v[0] |= uint64(x[1]) << 8 v[0] |= uint64(x[2]) << 16 @@ -260,7 +263,10 @@ func (v *FieldElement) FromBytes(x *[32]byte) *FieldElement { return v } -func (v *FieldElement) ToBytes(r *[32]byte) { +func (v *FieldElement) ToBytes(r []byte) { + if len(r) != 32 { + panic("invalid input size") + } t := new(FieldElement).Reduce(v) r[0] = byte(t[0] & 0xff) @@ -323,12 +329,12 @@ func (v *FieldElement) FromBig(num *big.Int) *FieldElement { } } - return v.FromBytes(&buf) + return v.FromBytes(buf[:]) } func (v *FieldElement) ToBig() *big.Int { var buf [32]byte - v.ToBytes(&buf) // does a reduction + v.ToBytes(buf[:]) // does a reduction numWords := 256 / bits.UintSize words := make([]big.Word, numWords) @@ -354,8 +360,8 @@ func (v *FieldElement) ToBig() *big.Int { // Equal returns 1 if v and u are equal, and 0 otherwise. func (v *FieldElement) Equal(u *FieldElement) int { var sa, sv [32]byte - u.ToBytes(&sa) - v.ToBytes(&sv) + u.ToBytes(sa[:]) + v.ToBytes(sv[:]) return subtle.ConstantTimeCompare(sa[:], sv[:]) } @@ -381,7 +387,7 @@ func (v *FieldElement) CondNeg(u *FieldElement, cond int) *FieldElement { // IsNegative returns 1 if v is negative, and 0 otherwise. func (v *FieldElement) IsNegative() int { var b [32]byte - v.ToBytes(&b) + v.ToBytes(b[:]) return int(b[0] & 1) } diff --git a/internal/radix51/fe_test.go b/internal/radix51/fe_test.go index b77a50d..1cbbfb7 100644 --- a/internal/radix51/fe_test.go +++ b/internal/radix51/fe_test.go @@ -123,8 +123,8 @@ func TestFeFromBytesRoundTrip(t *testing.T) { in = [32]byte{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32} - fe.FromBytes(&in) - fe.ToBytes(&out) + fe.FromBytes(in[:]) + fe.ToBytes(out[:]) if !bytes.Equal(in[:], out[:]) { t.Error("Bytes<>FE doesn't roundtrip") @@ -137,8 +137,8 @@ func TestFeFromBytesRoundTrip(t *testing.T) { fe[3] = 0x5e8fca9e0881c fe[4] = 0x5c490f087d796 - fe.ToBytes(&out) - r.FromBytes(&out) + fe.ToBytes(out[:]) + r.FromBytes(out[:]) for i := 0; i < len(fe); i++ { if r[i] != fe[i] { @@ -173,7 +173,7 @@ func TestSanity(t *testing.T) { if err != nil { t.Fatal(err) } - x.FromBytes(&bytes) + x.FromBytes(bytes[:]) x2.Mul(&x, &x) // FeMulGo(&x2Go, &x, &x) @@ -232,7 +232,7 @@ func TestFeInvert(t *testing.T) { if err != nil { t.Fatal(err) } - x.FromBytes(&bytes) + x.FromBytes(bytes[:]) xinv.Invert(&x) r.Mul(&x, &xinv) From ec44dec9b876f01f7fe1726afa950f991e456b41 Mon Sep 17 00:00:00 2001 From: Filippo Valsorda Date: Fri, 15 Mar 2019 14:02:08 -0400 Subject: [PATCH 34/46] internal/radix51: actually, uhm, check the result of TestAliasing --- internal/radix51/alias_test.go | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/internal/radix51/alias_test.go b/internal/radix51/alias_test.go index ad3931a..1309cb3 100644 --- a/internal/radix51/alias_test.go +++ b/internal/radix51/alias_test.go @@ -118,9 +118,9 @@ func TestAliasing(t *testing.T) { var err error switch { case tt.oneArgF != nil: - quick.Check(checkAliasingOneArg(tt.oneArgF), quickCheckConfig) + err = quick.Check(checkAliasingOneArg(tt.oneArgF), quickCheckConfig) case tt.twoArgsF != nil: - quick.Check(checkAliasingTwoArgs(tt.twoArgsF), quickCheckConfig) + err = quick.Check(checkAliasingTwoArgs(tt.twoArgsF), quickCheckConfig) } if err != nil { t.Errorf("%v: %v", tt.name, err) From 382beb70629c99f574feac41d366277e2eaacec0 Mon Sep 17 00:00:00 2001 From: George Tankersley Date: Sun, 24 Mar 2019 22:28:21 +0000 Subject: [PATCH 35/46] internal/radix51: fix aliasing bug in CondNeg (#21) --- internal/radix51/fe.go | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/internal/radix51/fe.go b/internal/radix51/fe.go index 1ee3bc6..c07b9b3 100644 --- a/internal/radix51/fe.go +++ b/internal/radix51/fe.go @@ -381,7 +381,8 @@ func (v *FieldElement) Select(a, b *FieldElement, cond int) *FieldElement { // CondNeg sets v to -u if cond == 1, and to u if cond == 0. func (v *FieldElement) CondNeg(u *FieldElement, cond int) *FieldElement { - return v.Select(v.Neg(u), u, cond) + tmp := new(FieldElement).Neg(u) + return v.Select(tmp, u, cond) } // IsNegative returns 1 if v is negative, and 0 otherwise. From 2a0f4ba3cc9a80857d3d4fd35d112b2e817e7401 Mon Sep 17 00:00:00 2001 From: Filippo Valsorda Date: Sat, 2 Mar 2019 22:48:44 -0500 Subject: [PATCH 36/46] internal/radix51: replace ToBytes with AppendBytes --- internal/radix51/fe.go | 33 ++++++++++++++++++++++++--------- internal/radix51/fe_test.go | 4 ++-- 2 files changed, 26 insertions(+), 11 deletions(-) diff --git a/internal/radix51/fe.go b/internal/radix51/fe.go index c07b9b3..43515e2 100644 --- a/internal/radix51/fe.go +++ b/internal/radix51/fe.go @@ -263,10 +263,10 @@ func (v *FieldElement) FromBytes(x []byte) *FieldElement { return v } -func (v *FieldElement) ToBytes(r []byte) { - if len(r) != 32 { - panic("invalid input size") - } +// AppendBytes appends a 32 bytes little-endian encoding of v to b. +func (v *FieldElement) AppendBytes(b []byte) []byte { + res, r := sliceForAppend(b, 32) + t := new(FieldElement).Reduce(v) r[0] = byte(t[0] & 0xff) @@ -309,6 +309,22 @@ func (v *FieldElement) ToBytes(r []byte) { r[29] = byte((t[4] >> 28) & 0xff) r[30] = byte((t[4] >> 36) & 0xff) r[31] = byte((t[4] >> 44)) + + return res +} + +// sliceForAppend extends the input slice by n bytes. head is the full extended +// slice, while tail is the appended part. If the original slice has sufficient +// capacity no allocation is performed. +func sliceForAppend(in []byte, n int) (head, tail []byte) { + if total := len(in) + n; cap(in) >= total { + head = in[:total] + } else { + head = make([]byte, total) + copy(head, in) + } + tail = head[len(in):] + return } func (v *FieldElement) FromBig(num *big.Int) *FieldElement { @@ -333,8 +349,7 @@ func (v *FieldElement) FromBig(num *big.Int) *FieldElement { } func (v *FieldElement) ToBig() *big.Int { - var buf [32]byte - v.ToBytes(buf[:]) // does a reduction + buf := v.AppendBytes(nil) numWords := 256 / bits.UintSize words := make([]big.Word, numWords) @@ -360,8 +375,8 @@ func (v *FieldElement) ToBig() *big.Int { // Equal returns 1 if v and u are equal, and 0 otherwise. func (v *FieldElement) Equal(u *FieldElement) int { var sa, sv [32]byte - u.ToBytes(sa[:]) - v.ToBytes(sv[:]) + u.AppendBytes(sa[:0]) + v.AppendBytes(sv[:0]) return subtle.ConstantTimeCompare(sa[:], sv[:]) } @@ -388,7 +403,7 @@ func (v *FieldElement) CondNeg(u *FieldElement, cond int) *FieldElement { // IsNegative returns 1 if v is negative, and 0 otherwise. func (v *FieldElement) IsNegative() int { var b [32]byte - v.ToBytes(b[:]) + v.AppendBytes(b[:0]) return int(b[0] & 1) } diff --git a/internal/radix51/fe_test.go b/internal/radix51/fe_test.go index 1cbbfb7..2366573 100644 --- a/internal/radix51/fe_test.go +++ b/internal/radix51/fe_test.go @@ -124,7 +124,7 @@ func TestFeFromBytesRoundTrip(t *testing.T) { 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32} fe.FromBytes(in[:]) - fe.ToBytes(out[:]) + fe.AppendBytes(out[:0]) if !bytes.Equal(in[:], out[:]) { t.Error("Bytes<>FE doesn't roundtrip") @@ -137,7 +137,7 @@ func TestFeFromBytesRoundTrip(t *testing.T) { fe[3] = 0x5e8fca9e0881c fe[4] = 0x5c490f087d796 - fe.ToBytes(out[:]) + fe.AppendBytes(out[:0]) r.FromBytes(out[:]) for i := 0; i < len(fe); i++ { From 22cdf749a2586c1cc6d90e67551a1e266b8e9783 Mon Sep 17 00:00:00 2001 From: Filippo Valsorda Date: Sat, 2 Mar 2019 22:50:13 -0500 Subject: [PATCH 37/46] internal/radix51: refactor ToBig and FromBig --- internal/radix51/fe.go | 47 ++++++++++++++++++------------------------ 1 file changed, 20 insertions(+), 27 deletions(-) diff --git a/internal/radix51/fe.go b/internal/radix51/fe.go index 43515e2..ec4639f 100644 --- a/internal/radix51/fe.go +++ b/internal/radix51/fe.go @@ -327,49 +327,42 @@ func sliceForAppend(in []byte, n int) (head, tail []byte) { return } -func (v *FieldElement) FromBig(num *big.Int) *FieldElement { - var buf [32]byte +// FromBig sets v = n and returns v. The bit length of n must not exceed 256. +func (v *FieldElement) FromBig(n *big.Int) *FieldElement { + if n.BitLen() > 32*8 { + panic("ed25519: invalid field element input size") + } - offset := 0 - words := num.Bits() - numWords := len(words) - - for n := 0; n < numWords; n++ { - word := words[n] - for i := 0; i < bits.UintSize/8; i++ { - if offset >= len(buf) { + buf := make([]byte, 0, 32) + for _, word := range n.Bits() { + for i := 0; i < bits.UintSize; i += 8 { + if len(buf) >= cap(buf) { break } - buf[offset] = byte(word >> uint((i << 3))) - offset++ + buf = append(buf, byte(word)) + word >>= 8 } } - return v.FromBytes(buf[:]) + return v.FromBytes(buf[:32]) } +// ToBig returns v as a big.Int. func (v *FieldElement) ToBig() *big.Int { buf := v.AppendBytes(nil) - numWords := 256 / bits.UintSize - words := make([]big.Word, numWords) - - offset := 0 - byteSize := uint(bits.UintSize >> 3) - for n := 0; n < numWords; n++ { - word := uint(0) - for i := uint(0); i < byteSize; i++ { - if offset >= len(buf) { + words := make([]big.Word, 32*8/bits.UintSize) + for n := range words { + for i := 0; i < bits.UintSize; i += 8 { + if len(buf) == 0 { break } - word |= uint(buf[offset]) << (i << 3) - offset++ + words[n] |= big.Word(buf[0]) << big.Word(i) + buf = buf[1:] } - words[n] = big.Word(word) } - out := new(big.Int) - return out.SetBits(words) + return new(big.Int).SetBits(words) } // Equal returns 1 if v and u are equal, and 0 otherwise. From 010995eaa9f84d1a86411da534c9d032fd8f44b8 Mon Sep 17 00:00:00 2001 From: Filippo Valsorda Date: Sat, 2 Mar 2019 18:54:11 -0500 Subject: [PATCH 38/46] internal/radix51: remove unused (and a bit broken) SetInt It would return an invalid element if the input was greater than 2^54 - 1. --- internal/radix51/fe.go | 10 ---------- 1 file changed, 10 deletions(-) diff --git a/internal/radix51/fe.go b/internal/radix51/fe.go index ec4639f..f7eb759 100644 --- a/internal/radix51/fe.go +++ b/internal/radix51/fe.go @@ -48,16 +48,6 @@ func (v *FieldElement) One() *FieldElement { return v } -// SetInt sets the receiving FieldElement to the specified small integer. -func (v *FieldElement) SetInt(x uint64) *FieldElement { - v[0] = x - v[1] = 0 - v[2] = 0 - v[3] = 0 - v[4] = 0 - return v -} - func (v *FieldElement) Reduce(u *FieldElement) *FieldElement { v.Set(u) From d23de5461ed262f4a54d5f3805975ea99bb4ed0c Mon Sep 17 00:00:00 2001 From: Filippo Valsorda Date: Sat, 2 Mar 2019 22:55:27 -0500 Subject: [PATCH 39/46] internal/radix51: add docs and some light readability refactors --- internal/radix51/fe.go | 41 ++++++++++++++++++----------- internal/radix51/fe_mul.go | 2 +- internal/radix51/fe_mul_amd64.go | 2 +- internal/radix51/fe_mul_amd64.s | 8 +++--- internal/radix51/fe_square.go | 2 +- internal/radix51/fe_square_amd64.go | 2 +- internal/radix51/fe_square_amd64.s | 6 ++--- 7 files changed, 36 insertions(+), 27 deletions(-) diff --git a/internal/radix51/fe.go b/internal/radix51/fe.go index f7eb759..ef06d66 100644 --- a/internal/radix51/fe.go +++ b/internal/radix51/fe.go @@ -3,8 +3,11 @@ // Use of this source code is governed by a BSD-style // license that can be found in the LICENSE file. -// Field arithmetic in radix 2^51 representation. This code is a port of the -// public domain amd64-51-30k version of ed25519 from SUPERCOP. +// GF(2^255-19) field arithmetic in radix 2^51 representation. This code is a +// port of the public domain amd64-51-30k version of ed25519 from SUPERCOP. +// +// The interface works similarly to math/big.Int, and all arguments and +// receivers are allowed to alias. package radix51 import ( @@ -15,13 +18,12 @@ import ( // FieldElement represents an element of the field GF(2^255-19). An element t // represents the integer t[0] + t[1]*2^51 + t[2]*2^102 + t[3]*2^153 + -// t[4]*2^204. The zero value is a valid zero element. +// t[4]*2^204. Each limb must not exceed 2^54 - 1 to be valid. +// +// The zero value is a valid zero element. type FieldElement [5]uint64 -const ( - // The vaule 2^51-1, used in carry propagation - maskLow51Bits = uint64(1)<<51 - 1 -) +const maskLow51Bits uint64 = (1 << 51) - 1 var ( Zero = &FieldElement{0, 0, 0, 0, 0} @@ -30,6 +32,7 @@ var ( MinusOne = new(FieldElement).Neg(One) ) +// Zero sets v = 0 and returns v. func (v *FieldElement) Zero() *FieldElement { v[0] = 0 v[1] = 0 @@ -39,6 +42,7 @@ func (v *FieldElement) Zero() *FieldElement { return v } +// One sets v = 1 and returns v. func (v *FieldElement) One() *FieldElement { v[0] = 1 v[1] = 0 @@ -48,6 +52,7 @@ func (v *FieldElement) One() *FieldElement { return v } +// Reduce reduces v modulo 2^255 - 19 and returns it. func (v *FieldElement) Reduce(u *FieldElement) *FieldElement { v.Set(u) @@ -90,9 +95,11 @@ func (v *FieldElement) Reduce(u *FieldElement) *FieldElement { return v } -// Add sets v = a + b. Long sequences of additions without reduction that -// let coefficients grow larger than 54 bits would be a problem. Paper -// cautions: "do not have such sequences of additions". +// Add sets v = a + b and returns v. +// +// Long sequences of additions without reduction that let coefficients grow +// larger than 54 bits would be a problem. Paper cautions: "do not have such +// sequences of additions". func (v *FieldElement) Add(a, b *FieldElement) *FieldElement { v[0] = a[0] + b[0] v[1] = a[1] + b[1] @@ -102,7 +109,7 @@ func (v *FieldElement) Add(a, b *FieldElement) *FieldElement { return v } -// Sub sets v = a - b. +// Sub sets v = a - b and returns v. func (v *FieldElement) Sub(a, b *FieldElement) *FieldElement { t := *b @@ -130,12 +137,12 @@ func (v *FieldElement) Sub(a, b *FieldElement) *FieldElement { return v } -// Neg sets v = -a. +// Neg sets v = -a and returns v. func (v *FieldElement) Neg(a *FieldElement) *FieldElement { return v.Sub(Zero, a) } -// Invert sets v = 1/z mod p by calculating z^(p-2), p-2 = 2^255 - 21. +// Invert sets v = 1/z mod p and returns v. func (v *FieldElement) Invert(z *FieldElement) *FieldElement { // Inversion is implemented as exponentiation with exponent p − 2. It uses the // same sequence of 255 squarings and 11 multiplications as [Curve25519]. @@ -200,15 +207,18 @@ func (v *FieldElement) Invert(z *FieldElement) *FieldElement { return v.Mul(&t, &z11) // 2^255 - 21 } +// Set sets v = a and returns v. func (v *FieldElement) Set(a *FieldElement) *FieldElement { *v = *a return v } +// FromBytes sets v to x, which must be a 32 bytes little-endian encoding. func (v *FieldElement) FromBytes(x []byte) *FieldElement { if len(x) != 32 { - panic("invalid input size") + panic("ed25519: invalid field element input size") } + v[0] = uint64(x[0]) v[0] |= uint64(x[1]) << 8 v[0] |= uint64(x[2]) << 16 @@ -366,7 +376,6 @@ func (v *FieldElement) Equal(u *FieldElement) int { const mask64Bits uint64 = (1 << 64) - 1 // Select sets v to a if cond == 1, and to b if cond == 0. -// v, a and b are allowed to overlap. func (v *FieldElement) Select(a, b *FieldElement, cond int) *FieldElement { m := uint64(cond) * mask64Bits v[0] = (m & a[0]) | (^m & b[0]) @@ -390,7 +399,7 @@ func (v *FieldElement) IsNegative() int { return int(b[0] & 1) } -// Abs sets v to |u|. v and u are allowed to overlap. +// Abs sets v to |u| and returns v. func (v *FieldElement) Abs(u *FieldElement) *FieldElement { return v.CondNeg(u, u.IsNegative()) } diff --git a/internal/radix51/fe_mul.go b/internal/radix51/fe_mul.go index 27e7f5b..3ea249f 100644 --- a/internal/radix51/fe_mul.go +++ b/internal/radix51/fe_mul.go @@ -6,7 +6,7 @@ package radix51 -// Mul sets out = x * y. +// Mul sets v = x * y and returns v. func (v *FieldElement) Mul(x, y *FieldElement) *FieldElement { var x0, x1, x2, x3, x4 uint64 var y0, y1, y2, y3, y4 uint64 diff --git a/internal/radix51/fe_mul_amd64.go b/internal/radix51/fe_mul_amd64.go index 7ad010f..eeadefa 100644 --- a/internal/radix51/fe_mul_amd64.go +++ b/internal/radix51/fe_mul_amd64.go @@ -6,7 +6,7 @@ package radix51 -// Mul sets out = x * y. +// Mul sets v = x * y and returns v. func (v *FieldElement) Mul(x, y *FieldElement) *FieldElement { feMul(v, x, y) return v diff --git a/internal/radix51/fe_mul_amd64.s b/internal/radix51/fe_mul_amd64.s index 5a72401..a4cb3ad 100644 --- a/internal/radix51/fe_mul_amd64.s +++ b/internal/radix51/fe_mul_amd64.s @@ -7,11 +7,11 @@ // +build amd64,!noasm -// func feMul(outp *uint64, xp *uint64, yp *uint64) +// func feMul(out, a, b *FieldElement) TEXT ·feMul(SB),$0-24 - MOVQ outp+0(FP), DI - MOVQ xp+8(FP), BX - MOVQ yp+16(FP), CX + MOVQ out+0(FP), DI + MOVQ a+8(FP), BX + MOVQ b+16(FP), CX // Calculate r0 MOVQ 0(BX), AX // rax <-- x0 diff --git a/internal/radix51/fe_square.go b/internal/radix51/fe_square.go index 46e01fb..eb9a565 100644 --- a/internal/radix51/fe_square.go +++ b/internal/radix51/fe_square.go @@ -6,7 +6,7 @@ package radix51 -// Square sets v = x * x. +// Square sets v = x * x and returns v. func (v *FieldElement) Square(x *FieldElement) *FieldElement { // Squaring needs only 15 mul instructions. Some inputs are multiplied by 2; // this is combined with multiplication by 19 where possible. The coefficient diff --git a/internal/radix51/fe_square_amd64.go b/internal/radix51/fe_square_amd64.go index d91d669..fc08056 100644 --- a/internal/radix51/fe_square_amd64.go +++ b/internal/radix51/fe_square_amd64.go @@ -6,7 +6,7 @@ package radix51 -// Square sets v = x * x. +// Square sets v = x * x and returns v. func (v *FieldElement) Square(x *FieldElement) *FieldElement { feSquare(v, x) return v diff --git a/internal/radix51/fe_square_amd64.s b/internal/radix51/fe_square_amd64.s index 0103416..953c7bb 100644 --- a/internal/radix51/fe_square_amd64.s +++ b/internal/radix51/fe_square_amd64.s @@ -4,10 +4,10 @@ // +build amd64,!noasm -// func feSquare(outp *uint64, xp *uint64) +// func feSquare(out, x *FieldElement) TEXT ·feSquare(SB),4,$0-16 - MOVQ outp+0(FP), DI - MOVQ xp+8(FP), SI + MOVQ out+0(FP), DI + MOVQ x+8(FP), SI // r0 = x0*x0 + x1*38*x4 + x2*38*x3 MOVQ 0(SI), AX From c9d2135504c4317faa08cfeb36ebf1e604d5f5c8 Mon Sep 17 00:00:00 2001 From: Filippo Valsorda Date: Sun, 3 Mar 2019 04:17:18 -0500 Subject: [PATCH 40/46] internal/radix51: rewrite FromBytes and AppendBytes with encoding/binary --- internal/radix51/fe.go | 111 ++++++++++-------------------------- internal/radix51/fe_test.go | 87 +++++++++++++++++++++------- 2 files changed, 97 insertions(+), 101 deletions(-) diff --git a/internal/radix51/fe.go b/internal/radix51/fe.go index ef06d66..f6a6522 100644 --- a/internal/radix51/fe.go +++ b/internal/radix51/fe.go @@ -12,6 +12,7 @@ package radix51 import ( "crypto/subtle" + "encoding/binary" "math/big" "math/bits" ) @@ -214,101 +215,51 @@ func (v *FieldElement) Set(a *FieldElement) *FieldElement { } // FromBytes sets v to x, which must be a 32 bytes little-endian encoding. +// +// Consistently with RFC 7748, the most significant bit (the high bit of the +// last byte) is ignored, and non-canonical values (2^255-19 through 2^255-1) +// are accepted. func (v *FieldElement) FromBytes(x []byte) *FieldElement { if len(x) != 32 { panic("ed25519: invalid field element input size") } - v[0] = uint64(x[0]) - v[0] |= uint64(x[1]) << 8 - v[0] |= uint64(x[2]) << 16 - v[0] |= uint64(x[3]) << 24 - v[0] |= uint64(x[4]) << 32 - v[0] |= uint64(x[5]) << 40 - v[0] |= uint64(x[6]&7) << 48 + // Provide headroom for the slight binary.LittleEndian.Uint64 overread. (We + // read 64 bits at an offset of 200, but then take only 4+51 into account.) + var buf [33]byte + copy(buf[:], x) - v[1] = uint64(x[6]) >> 3 - v[1] |= uint64(x[7]) << 5 - v[1] |= uint64(x[8]) << 13 - v[1] |= uint64(x[9]) << 21 - v[1] |= uint64(x[10]) << 29 - v[1] |= uint64(x[11]) << 37 - v[1] |= uint64(x[12]&63) << 45 - - v[2] = uint64(x[12]) >> 6 - v[2] |= uint64(x[13]) << 2 - v[2] |= uint64(x[14]) << 10 - v[2] |= uint64(x[15]) << 18 - v[2] |= uint64(x[16]) << 26 - v[2] |= uint64(x[17]) << 34 - v[2] |= uint64(x[18]) << 42 - v[2] |= uint64(x[19]&1) << 50 - - v[3] = uint64(x[19]) >> 1 - v[3] |= uint64(x[20]) << 7 - v[3] |= uint64(x[21]) << 15 - v[3] |= uint64(x[22]) << 23 - v[3] |= uint64(x[23]) << 31 - v[3] |= uint64(x[24]) << 39 - v[3] |= uint64(x[25]&15) << 47 - - v[4] = uint64(x[25]) >> 4 - v[4] |= uint64(x[26]) << 4 - v[4] |= uint64(x[27]) << 12 - v[4] |= uint64(x[28]) << 20 - v[4] |= uint64(x[29]) << 28 - v[4] |= uint64(x[30]) << 36 - v[4] |= uint64(x[31]&127) << 44 + for i := range v { + bitsOffset := i * 51 + v[i] = binary.LittleEndian.Uint64(buf[bitsOffset/8:]) + v[i] >>= uint(bitsOffset % 8) + v[i] &= maskLow51Bits + } return v } // AppendBytes appends a 32 bytes little-endian encoding of v to b. func (v *FieldElement) AppendBytes(b []byte) []byte { - res, r := sliceForAppend(b, 32) - t := new(FieldElement).Reduce(v) - r[0] = byte(t[0] & 0xff) - r[1] = byte((t[0] >> 8) & 0xff) - r[2] = byte((t[0] >> 16) & 0xff) - r[3] = byte((t[0] >> 24) & 0xff) - r[4] = byte((t[0] >> 32) & 0xff) - r[5] = byte((t[0] >> 40) & 0xff) - r[6] = byte((t[0] >> 48)) + res, out := sliceForAppend(b, 32) + for i := range out { + out[i] = 0 + } - r[6] ^= byte((t[1] << 3) & 0xf8) - r[7] = byte((t[1] >> 5) & 0xff) - r[8] = byte((t[1] >> 13) & 0xff) - r[9] = byte((t[1] >> 21) & 0xff) - r[10] = byte((t[1] >> 29) & 0xff) - r[11] = byte((t[1] >> 37) & 0xff) - r[12] = byte((t[1] >> 45)) - - r[12] ^= byte((t[2] << 6) & 0xc0) - r[13] = byte((t[2] >> 2) & 0xff) - r[14] = byte((t[2] >> 10) & 0xff) - r[15] = byte((t[2] >> 18) & 0xff) - r[16] = byte((t[2] >> 26) & 0xff) - r[17] = byte((t[2] >> 34) & 0xff) - r[18] = byte((t[2] >> 42) & 0xff) - r[19] = byte((t[2] >> 50)) - - r[19] ^= byte((t[3] << 1) & 0xfe) - r[20] = byte((t[3] >> 7) & 0xff) - r[21] = byte((t[3] >> 15) & 0xff) - r[22] = byte((t[3] >> 23) & 0xff) - r[23] = byte((t[3] >> 31) & 0xff) - r[24] = byte((t[3] >> 39) & 0xff) - r[25] = byte((t[3] >> 47)) - - r[25] ^= byte((t[4] << 4) & 0xf0) - r[26] = byte((t[4] >> 4) & 0xff) - r[27] = byte((t[4] >> 12) & 0xff) - r[28] = byte((t[4] >> 20) & 0xff) - r[29] = byte((t[4] >> 28) & 0xff) - r[30] = byte((t[4] >> 36) & 0xff) - r[31] = byte((t[4] >> 44)) + var buf [8]byte + for i := range t { + bitsOffset := i * 51 + binary.LittleEndian.PutUint64(buf[:], t[i]<= len(out) { + break + } + out[off] |= b + } + } return res } diff --git a/internal/radix51/fe_test.go b/internal/radix51/fe_test.go index 2366573..413cce7 100644 --- a/internal/radix51/fe_test.go +++ b/internal/radix51/fe_test.go @@ -1,4 +1,5 @@ // Copyright (c) 2017 George Tankersley. All rights reserved. +// Copyright (c) 2019 The Go Authors. All rights reserved. // Use of this source code is governed by a BSD-style // license that can be found in the LICENSE file. @@ -8,6 +9,7 @@ import ( "bytes" "crypto/rand" "io" + "math/big" mathrand "math/rand" "reflect" "testing" @@ -21,7 +23,8 @@ var quickCheckConfig = &quick.Config{MaxCountScale: 1 << 8} func generateFieldElement(rand *mathrand.Rand) FieldElement { // Generation strategy: generate random limb values bounded by // 2**(51+b), where b is a parameter controlling the bit-excess. - b := uint64(0) + // TODO: randomly decide to set the limbs to "weird" values. + b := uint64(0) // TODO: set this higher once we know the bounds. mask := (uint64(1) << (51 + b)) - 1 return FieldElement{ rand.Uint64() & mask, @@ -116,35 +119,77 @@ func BenchmarkWideMultCall(t *testing.B) { } } -func TestFeFromBytesRoundTrip(t *testing.T) { - var in, out [32]byte - var fe, r FieldElement +func TestFromBytesRoundTrip(t *testing.T) { + f1 := func(in, out [32]byte, fe FieldElement) bool { + fe.FromBytes(in[:]) + fe.AppendBytes(out[:0]) - in = [32]byte{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, - 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32} + // Mask the most significant bit as it's ignored by FromBytes. (Now + // instead of earlier so we check the masking in FromBytes is working.) + in[len(in)-1] &= (1 << 7) - 1 - fe.FromBytes(in[:]) - fe.AppendBytes(out[:0]) + // TODO: values in the range [2^255-19, 2^255-1] will still fail the + // comparison as they will have been reduced in the round-trip, but the + // current quickcheck generation strategy will never hit them, which is + // not good. We should have a weird generator that aims for edge cases, + // and we'll know it works when this test breaks. - if !bytes.Equal(in[:], out[:]) { - t.Error("Bytes<>FE doesn't roundtrip") + return bytes.Equal(in[:], out[:]) + } + if err := quick.Check(f1, nil); err != nil { + t.Errorf("failed bytes->FE->bytes round-trip: %v", err) } - // Random field element - fe[0] = 0x4e645be9215a2 - fe[1] = 0x4e9654922df12 - fe[2] = 0x5829e468b0205 - fe[3] = 0x5e8fca9e0881c - fe[4] = 0x5c490f087d796 + f2 := func(fe, r FieldElement, out [32]byte) bool { + fe.AppendBytes(out[:0]) + r.FromBytes(out[:]) - fe.AppendBytes(out[:0]) - r.FromBytes(out[:]) + // Intentionally not using Equal not to go through AppendBytes again. + // Calling Reduce because both Generate and FromBytes can produce + // non-canonical representations. + fe.Reduce(&fe) + r.Reduce(&r) + return fe == r + } + if err := quick.Check(f2, nil); err != nil { + t.Errorf("failed FE->bytes->FE round-trip: %v", err) + } +} - for i := 0; i < len(fe); i++ { - if r[i] != fe[i] { - t.Error("FE<>Bytes doesn't roundtrip") +func swapEndianness(buf []byte) []byte { + for i := 0; i < len(buf)/2; i++ { + buf[i], buf[len(buf)-i-1] = buf[len(buf)-i-1], buf[i] + } + return buf +} + +func TestBytesBigEquivalence(t *testing.T) { + f1 := func(in, out [32]byte, fe, fe1 FieldElement) bool { + fe.FromBytes(in[:]) + + in[len(in)-1] &= (1 << 7) - 1 // mask the most significant bit + b := new(big.Int).SetBytes(swapEndianness(in[:])) + fe1.FromBig(b) + + if fe != fe1 { + return false } + + fe.AppendBytes(out[:0]) + buf := make([]byte, 32) // pad with zeroes + copy(buf, swapEndianness(fe1.ToBig().Bytes())) + + return bytes.Equal(out[:], buf) } + if err := quick.Check(f1, nil); err != nil { + t.Error(err) + } +} + +func TestFromBytesRoundTripEdgeCases(t *testing.T) { + // TODO: values close to 0, close to 2^255-19, between 2^255-19 and 2^255-1, + // and between 2^255 and 2^256-1. Test both the documented FromBytes + // behavior, and that AppendBytes reduces them. } // Tests self-consistency between FeMul and FeSquare. From 1e528602b8012ede6413fd9203f750b9c9848ac9 Mon Sep 17 00:00:00 2001 From: Filippo Valsorda Date: Sat, 30 Mar 2019 19:10:39 -0400 Subject: [PATCH 41/46] internal/radix51: rename AppendBytes to Bytes --- internal/radix51/fe.go | 12 ++++++------ internal/radix51/fe_test.go | 10 +++++----- 2 files changed, 11 insertions(+), 11 deletions(-) diff --git a/internal/radix51/fe.go b/internal/radix51/fe.go index f6a6522..1c09245 100644 --- a/internal/radix51/fe.go +++ b/internal/radix51/fe.go @@ -239,8 +239,8 @@ func (v *FieldElement) FromBytes(x []byte) *FieldElement { return v } -// AppendBytes appends a 32 bytes little-endian encoding of v to b. -func (v *FieldElement) AppendBytes(b []byte) []byte { +// Bytes appends a 32 bytes little-endian encoding of v to b. +func (v *FieldElement) Bytes(b []byte) []byte { t := new(FieldElement).Reduce(v) res, out := sliceForAppend(b, 32) @@ -300,7 +300,7 @@ func (v *FieldElement) FromBig(n *big.Int) *FieldElement { // ToBig returns v as a big.Int. func (v *FieldElement) ToBig() *big.Int { - buf := v.AppendBytes(nil) + buf := v.Bytes(nil) words := make([]big.Word, 32*8/bits.UintSize) for n := range words { @@ -319,8 +319,8 @@ func (v *FieldElement) ToBig() *big.Int { // Equal returns 1 if v and u are equal, and 0 otherwise. func (v *FieldElement) Equal(u *FieldElement) int { var sa, sv [32]byte - u.AppendBytes(sa[:0]) - v.AppendBytes(sv[:0]) + u.Bytes(sa[:0]) + v.Bytes(sv[:0]) return subtle.ConstantTimeCompare(sa[:], sv[:]) } @@ -346,7 +346,7 @@ func (v *FieldElement) CondNeg(u *FieldElement, cond int) *FieldElement { // IsNegative returns 1 if v is negative, and 0 otherwise. func (v *FieldElement) IsNegative() int { var b [32]byte - v.AppendBytes(b[:0]) + v.Bytes(b[:0]) return int(b[0] & 1) } diff --git a/internal/radix51/fe_test.go b/internal/radix51/fe_test.go index 413cce7..17cfaeb 100644 --- a/internal/radix51/fe_test.go +++ b/internal/radix51/fe_test.go @@ -122,7 +122,7 @@ func BenchmarkWideMultCall(t *testing.B) { func TestFromBytesRoundTrip(t *testing.T) { f1 := func(in, out [32]byte, fe FieldElement) bool { fe.FromBytes(in[:]) - fe.AppendBytes(out[:0]) + fe.Bytes(out[:0]) // Mask the most significant bit as it's ignored by FromBytes. (Now // instead of earlier so we check the masking in FromBytes is working.) @@ -141,10 +141,10 @@ func TestFromBytesRoundTrip(t *testing.T) { } f2 := func(fe, r FieldElement, out [32]byte) bool { - fe.AppendBytes(out[:0]) + fe.Bytes(out[:0]) r.FromBytes(out[:]) - // Intentionally not using Equal not to go through AppendBytes again. + // Intentionally not using Equal not to go through Bytes again. // Calling Reduce because both Generate and FromBytes can produce // non-canonical representations. fe.Reduce(&fe) @@ -175,7 +175,7 @@ func TestBytesBigEquivalence(t *testing.T) { return false } - fe.AppendBytes(out[:0]) + fe.Bytes(out[:0]) buf := make([]byte, 32) // pad with zeroes copy(buf, swapEndianness(fe1.ToBig().Bytes())) @@ -189,7 +189,7 @@ func TestBytesBigEquivalence(t *testing.T) { func TestFromBytesRoundTripEdgeCases(t *testing.T) { // TODO: values close to 0, close to 2^255-19, between 2^255-19 and 2^255-1, // and between 2^255 and 2^256-1. Test both the documented FromBytes - // behavior, and that AppendBytes reduces them. + // behavior, and that Bytes reduces them. } // Tests self-consistency between FeMul and FeSquare. From 71368c02ff42e48aaed23a2c34763246a7d11918 Mon Sep 17 00:00:00 2001 From: George Tankersley Date: Sun, 31 Mar 2019 01:13:56 +0000 Subject: [PATCH 42/46] internal/radix51: test field encoding roundtrip with fixed vectors --- internal/radix51/fe_test.go | 22 ++++++++++++++++++++++ 1 file changed, 22 insertions(+) diff --git a/internal/radix51/fe_test.go b/internal/radix51/fe_test.go index 17cfaeb..3b52e38 100644 --- a/internal/radix51/fe_test.go +++ b/internal/radix51/fe_test.go @@ -154,6 +154,28 @@ func TestFromBytesRoundTrip(t *testing.T) { if err := quick.Check(f2, nil); err != nil { t.Errorf("failed FE->bytes->FE round-trip: %v", err) } + + // Check some fixed vectors from dalek + type feRTTest struct { + fe FieldElement + b []byte + } + var tests = []feRTTest{ + { + fe: FieldElement([5]uint64{358744748052810, 1691584618240980, 977650209285361, 1429865912637724, 560044844278676}), + b: []byte{74, 209, 69, 197, 70, 70, 161, 222, 56, 226, 229, 19, 112, 60, 25, 92, 187, 74, 222, 56, 50, 153, 51, 233, 40, 74, 57, 6, 160, 185, 213, 31}, + }, + { + fe: FieldElement([5]uint64{84926274344903, 473620666599931, 365590438845504, 1028470286882429, 2146499180330972}), + b: []byte{199, 23, 106, 112, 61, 77, 216, 79, 186, 60, 11, 118, 13, 16, 103, 15, 42, 32, 83, 250, 44, 57, 204, 198, 78, 199, 253, 119, 146, 172, 3, 122}, + }, + } + + for _, tt := range tests { + if !bytes.Equal(tt.fe.Bytes(nil), tt.b) || new(FieldElement).FromBytes(tt.b).Equal(&tt.fe) != 1 { + t.Errorf("Failed fixed roundtrip: %v", tt) + } + } } func swapEndianness(buf []byte) []byte { From 09d91297791d98f1954820efafa06f9f465b6ec4 Mon Sep 17 00:00:00 2001 From: Dimitris Apostolou Date: Sun, 31 Mar 2019 04:29:01 +0300 Subject: [PATCH 43/46] all: fix typos --- ed25519_test.go | 2 +- internal/group/ge.go | 2 +- internal/radix51/mul_compat.go | 2 +- 3 files changed, 3 insertions(+), 3 deletions(-) diff --git a/ed25519_test.go b/ed25519_test.go index 26026c9..88ed88a 100644 --- a/ed25519_test.go +++ b/ed25519_test.go @@ -337,7 +337,7 @@ func BenchmarkScalarMult(b *testing.B) { // } // // Test vector generated by instrumenting x/crypto/ed25519 GenerateKey(). These -// // are raw values. The edwards code interprets them as little-endian, so they +// // are raw values. The Edwards code interprets them as little-endian, so they // // need to be reversed before use with big.Int. // var genKeyTest = struct { // seed, expanded, public string diff --git a/internal/group/ge.go b/internal/group/ge.go index 91fb65f..1edbad1 100644 --- a/internal/group/ge.go +++ b/internal/group/ge.go @@ -3,7 +3,7 @@ // Use of this source code is governed by a BSD-style // license that can be found in the LICENSE file. -// Package group mplements group logic for the Ed25519 curve. +// Package group implements group logic for the Ed25519 curve. package group import ( diff --git a/internal/radix51/mul_compat.go b/internal/radix51/mul_compat.go index 54427a9..3ffb93b 100644 --- a/internal/radix51/mul_compat.go +++ b/internal/radix51/mul_compat.go @@ -8,7 +8,7 @@ package radix51 import "unsafe" -// madd64 multiples two 64-bit numbers and adds them to a split 128-bit accumulator. +// madd64 multiplies two 64-bit numbers and adds them to a split 128-bit accumulator. func madd64(lo, hi, a, b uint64) (ol uint64, oh uint64) { t1 := (a>>32)*(b&0xFFFFFFFF) + ((a & 0xFFFFFFFF) * (b & 0xFFFFFFFF) >> 32) t2 := (a&0xFFFFFFFF)*(b>>32) + (t1 & 0xFFFFFFFF) From ce6d218ef30e9eabca34f660e2e26199dd08d7f6 Mon Sep 17 00:00:00 2001 From: Filippo Valsorda Date: Sat, 30 Mar 2019 21:56:34 -0400 Subject: [PATCH 44/46] internal/radix51: actually apply go:noescape --- internal/radix51/fe_mul_amd64.go | 2 +- internal/radix51/fe_square_amd64.go | 2 +- 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/internal/radix51/fe_mul_amd64.go b/internal/radix51/fe_mul_amd64.go index eeadefa..53b761c 100644 --- a/internal/radix51/fe_mul_amd64.go +++ b/internal/radix51/fe_mul_amd64.go @@ -12,5 +12,5 @@ func (v *FieldElement) Mul(x, y *FieldElement) *FieldElement { return v } -// go:noescape +//go:noescape func feMul(out, a, b *FieldElement) diff --git a/internal/radix51/fe_square_amd64.go b/internal/radix51/fe_square_amd64.go index fc08056..afa1295 100644 --- a/internal/radix51/fe_square_amd64.go +++ b/internal/radix51/fe_square_amd64.go @@ -12,5 +12,5 @@ func (v *FieldElement) Square(x *FieldElement) *FieldElement { return v } -// go:noescape +//go:noescape func feSquare(out, x *FieldElement) From 620415daa4f07457fa06e1030f05d6a1a8dc4366 Mon Sep 17 00:00:00 2001 From: Filippo Valsorda Date: Sat, 30 Mar 2019 22:01:30 -0400 Subject: [PATCH 45/46] internal/radix51: make reduction an invariant and unexport Reduce Now every operation returns a light-reduced value, so the reduction is an invariant, and there's no need to ever explicitly call Reduce. Safety! --- ed25519.go | 1 - internal/radix51/alias_test.go | 2 +- internal/radix51/fe.go | 76 ++++++++++++++-------------------- internal/radix51/fe_mul.go | 43 ++++++------------- internal/radix51/fe_square.go | 31 ++++---------- internal/radix51/fe_test.go | 10 ++--- 6 files changed, 56 insertions(+), 107 deletions(-) diff --git a/ed25519.go b/ed25519.go index 4a2ecfa..820b77e 100644 --- a/ed25519.go +++ b/ed25519.go @@ -74,7 +74,6 @@ func (curve ed25519Curve) IsOnCurve(x, y *big.Int) bool { lh.Neg(&lh) // -x^2 lh.Add(&lh, &y2) // -x^2 + y^2 lh.Sub(&lh, &rh) // -x^2 + y^2 - 1 - dx^2y^2 - lh.Reduce(&lh) // mod p return lh.Equal(radix51.Zero) == 1 } diff --git a/internal/radix51/alias_test.go b/internal/radix51/alias_test.go index 1309cb3..0ac0521 100644 --- a/internal/radix51/alias_test.go +++ b/internal/radix51/alias_test.go @@ -84,7 +84,7 @@ func TestAliasing(t *testing.T) { {name: "Abs", oneArgF: (*FieldElement).Abs}, {name: "Invert", oneArgF: (*FieldElement).Invert}, {name: "Neg", oneArgF: (*FieldElement).Neg}, - {name: "Reduce", oneArgF: (*FieldElement).Reduce}, + {name: "reduce", oneArgF: (*FieldElement).reduce}, {name: "Set", oneArgF: (*FieldElement).Set}, {name: "Square", oneArgF: (*FieldElement).Square}, { diff --git a/internal/radix51/fe.go b/internal/radix51/fe.go index 1c09245..00b0d65 100644 --- a/internal/radix51/fe.go +++ b/internal/radix51/fe.go @@ -35,51 +35,53 @@ var ( // Zero sets v = 0 and returns v. func (v *FieldElement) Zero() *FieldElement { - v[0] = 0 - v[1] = 0 - v[2] = 0 - v[3] = 0 - v[4] = 0 + *v = *Zero return v } // One sets v = 1 and returns v. func (v *FieldElement) One() *FieldElement { - v[0] = 1 - v[1] = 0 - v[2] = 0 - v[3] = 0 - v[4] = 0 + *v = *One return v } -// Reduce reduces v modulo 2^255 - 19 and returns it. -func (v *FieldElement) Reduce(u *FieldElement) *FieldElement { - v.Set(u) - - // Lev v = v[0] + v[1]*2^51 + v[2]*2^102 + v[3]*2^153 + v[4]*2^204 - // Reduce each limb below 2^51, propagating carries. +// lightReduce brings the limbs below 52, 51, 51, 51, 51 bits. It is split in +// two because the inliner works actively against us. The two functions MUST be +// called one after the other. +func (v *FieldElement) lightReduce1() *FieldElement { v[1] += v[0] >> 51 v[0] = v[0] & maskLow51Bits v[2] += v[1] >> 51 v[1] = v[1] & maskLow51Bits v[3] += v[2] >> 51 + return v +} +func (v *FieldElement) lightReduce2() *FieldElement { v[2] = v[2] & maskLow51Bits v[4] += v[3] >> 51 v[3] = v[3] & maskLow51Bits v[0] += (v[4] >> 51) * 19 v[4] = v[4] & maskLow51Bits + return v +} - // We now hate a field element v < 2^255, but need v <= 2^255-19 - // TODO Document why this works. It's the elaborate comment about r = h-pq etc etc. +// reduce reduces v modulo 2^255 - 19 and returns it. +func (v *FieldElement) reduce(u *FieldElement) *FieldElement { + v.Set(u).lightReduce1().lightReduce2() - // Get the carry bit + // After the light reduction we now have a field element representation + // v < 2^255 + 2^13 * 19, but need v < 2^255 - 19. + + // If v >= 2^255 - 19, then v + 19 >= 2^255, which would overflow 2^255 - 1, + // generating a carry. That is, c will be 0 if v < 2^255 - 19, and 1 otherwise. c := (v[0] + 19) >> 51 c = (v[1] + c) >> 51 c = (v[2] + c) >> 51 c = (v[3] + c) >> 51 c = (v[4] + c) >> 51 + // If v < 2^255 - 19 and c = 0, this will be a no-op. Otherwise, it's + // effectively applying the reduction identity to the carry. v[0] += 19 * c v[1] += v[0] >> 51 @@ -107,35 +109,19 @@ func (v *FieldElement) Add(a, b *FieldElement) *FieldElement { v[2] = a[2] + b[2] v[3] = a[3] + b[3] v[4] = a[4] + b[4] - return v + return v.lightReduce1().lightReduce2() } // Sub sets v = a - b and returns v. func (v *FieldElement) Sub(a, b *FieldElement) *FieldElement { - t := *b - - // Reduce each limb below 2^51, propagating carries. Ensures that results - // fit within the limbs. This would not be required for reduced input. - t[1] += t[0] >> 51 - t[0] = t[0] & maskLow51Bits - t[2] += t[1] >> 51 - t[1] = t[1] & maskLow51Bits - t[3] += t[2] >> 51 - t[2] = t[2] & maskLow51Bits - t[4] += t[3] >> 51 - t[3] = t[3] & maskLow51Bits - t[0] += (t[4] >> 51) * 19 - t[4] = t[4] & maskLow51Bits - - // This is slightly more complicated. Because we use unsigned coefficients, we - // first add a multiple of p and then subtract. - v[0] = (a[0] + 0xFFFFFFFFFFFDA) - t[0] - v[1] = (a[1] + 0xFFFFFFFFFFFFE) - t[1] - v[2] = (a[2] + 0xFFFFFFFFFFFFE) - t[2] - v[3] = (a[3] + 0xFFFFFFFFFFFFE) - t[3] - v[4] = (a[4] + 0xFFFFFFFFFFFFE) - t[4] - - return v + // We first add 2 * p, to guarantee the subtraction won't underflow, and + // then subtract b (which can be up to 2^255 + 2^13 * 19). + v[0] = (a[0] + 0xFFFFFFFFFFFDA) - b[0] + v[1] = (a[1] + 0xFFFFFFFFFFFFE) - b[1] + v[2] = (a[2] + 0xFFFFFFFFFFFFE) - b[2] + v[3] = (a[3] + 0xFFFFFFFFFFFFE) - b[3] + v[4] = (a[4] + 0xFFFFFFFFFFFFE) - b[4] + return v.lightReduce1().lightReduce2() } // Neg sets v = -a and returns v. @@ -241,7 +227,7 @@ func (v *FieldElement) FromBytes(x []byte) *FieldElement { // Bytes appends a 32 bytes little-endian encoding of v to b. func (v *FieldElement) Bytes(b []byte) []byte { - t := new(FieldElement).Reduce(v) + t := new(FieldElement).reduce(v) res, out := sliceForAppend(b, 32) for i := range out { diff --git a/internal/radix51/fe_mul.go b/internal/radix51/fe_mul.go index 3ea249f..c5bb94e 100644 --- a/internal/radix51/fe_mul.go +++ b/internal/radix51/fe_mul.go @@ -8,20 +8,17 @@ package radix51 // Mul sets v = x * y and returns v. func (v *FieldElement) Mul(x, y *FieldElement) *FieldElement { - var x0, x1, x2, x3, x4 uint64 - var y0, y1, y2, y3, y4 uint64 + x0 := x[0] + x1 := x[1] + x2 := x[2] + x3 := x[3] + x4 := x[4] - x0 = x[0] - x1 = x[1] - x2 = x[2] - x3 = x[3] - x4 = x[4] - - y0 = y[0] - y1 = y[1] - y2 = y[2] - y3 = y[3] - y4 = y[4] + y0 := y[0] + y1 := y[1] + y2 := y[2] + y3 := y[3] + y4 := y[4] // Reduction can be carried out simultaneously to multiplication. For // example, we do not compute a coefficient r_5 . Whenever the result of a @@ -106,22 +103,6 @@ func (v *FieldElement) Mul(x, y *FieldElement) *FieldElement { // r_0 to r_1 , from r_1 to r_2 , from r_2 to r_3 , from r_3 to r_4 , and // finally from r_4 to r_0 . Each of these carries is done as one copy, one // right shift by 51, one logical and with 2^51 − 1, and one addition. - - r10 += r00 >> 51 - r00 &= maskLow51Bits - r20 += r10 >> 51 - r10 &= maskLow51Bits - r30 += r20 >> 51 - r20 &= maskLow51Bits - r40 += r30 >> 51 - r30 &= maskLow51Bits - r00 += (r40 >> 51) * 19 - r40 &= maskLow51Bits - - v[0] = r00 - v[1] = r10 - v[2] = r20 - v[3] = r30 - v[4] = r40 - return v + *v = FieldElement{r00, r10, r20, r30, r40} + return v.lightReduce1().lightReduce2() } diff --git a/internal/radix51/fe_square.go b/internal/radix51/fe_square.go index eb9a565..c04c72a 100644 --- a/internal/radix51/fe_square.go +++ b/internal/radix51/fe_square.go @@ -12,13 +12,11 @@ func (v *FieldElement) Square(x *FieldElement) *FieldElement { // this is combined with multiplication by 19 where possible. The coefficient // reduction after squaring is the same as for multiplication. - var x0, x1, x2, x3, x4 uint64 - - x0 = x[0] - x1 = x[1] - x2 = x[2] - x3 = x[3] - x4 = x[4] + x0 := x[0] + x1 := x[1] + x2 := x[2] + x3 := x[3] + x4 := x[4] x0_2 := x0 << 1 x1_2 := x1 << 1 @@ -79,21 +77,6 @@ func (v *FieldElement) Square(x *FieldElement) *FieldElement { r41 *= 19 r00 += r41 - r10 += r00 >> 51 - r00 &= maskLow51Bits - r20 += r10 >> 51 - r10 &= maskLow51Bits - r30 += r20 >> 51 - r20 &= maskLow51Bits - r40 += r30 >> 51 - r30 &= maskLow51Bits - r00 += (r40 >> 51) * 19 - r40 &= maskLow51Bits - - v[0] = r00 - v[1] = r10 - v[2] = r20 - v[3] = r30 - v[4] = r40 - return v + *v = FieldElement{r00, r10, r20, r30, r40} + return v.lightReduce1().lightReduce2() } diff --git a/internal/radix51/fe_test.go b/internal/radix51/fe_test.go index 3b52e38..6b9424c 100644 --- a/internal/radix51/fe_test.go +++ b/internal/radix51/fe_test.go @@ -145,10 +145,10 @@ func TestFromBytesRoundTrip(t *testing.T) { r.FromBytes(out[:]) // Intentionally not using Equal not to go through Bytes again. - // Calling Reduce because both Generate and FromBytes can produce + // Calling reduce because both Generate and FromBytes can produce // non-canonical representations. - fe.Reduce(&fe) - r.Reduce(&r) + fe.reduce(&fe) + r.reduce(&r) return fe == r } if err := quick.Check(f2, nil); err != nil { @@ -287,7 +287,7 @@ func TestFeInvert(t *testing.T) { xinv.Invert(&x) r.Mul(&x, &xinv) - r.Reduce(&r) + r.reduce(&r) if !vartimeEqual(one, r) { t.Errorf("inversion identity failed, got: %x", r) @@ -303,7 +303,7 @@ func TestFeInvert(t *testing.T) { xinv.Invert(&x) r.Mul(&x, &xinv) - r.Reduce(&r) + r.reduce(&r) if !vartimeEqual(one, r) { t.Errorf("random inversion identity failed, got: %x for field element %x", r, x) From c37f1d835957022192acbdc7be0f546f2c001a3e Mon Sep 17 00:00:00 2001 From: Filippo Valsorda Date: Sat, 30 Mar 2019 22:05:36 -0400 Subject: [PATCH 46/46] internal/radix51: minor tests cleanup --- internal/radix51/alias_test.go | 4 +-- internal/radix51/fe_test.go | 48 +++++++--------------------------- 2 files changed, 11 insertions(+), 41 deletions(-) diff --git a/internal/radix51/alias_test.go b/internal/radix51/alias_test.go index 0ac0521..54089b9 100644 --- a/internal/radix51/alias_test.go +++ b/internal/radix51/alias_test.go @@ -118,9 +118,9 @@ func TestAliasing(t *testing.T) { var err error switch { case tt.oneArgF != nil: - err = quick.Check(checkAliasingOneArg(tt.oneArgF), quickCheckConfig) + err = quick.Check(checkAliasingOneArg(tt.oneArgF), &quick.Config{MaxCountScale: 1 << 8}) case tt.twoArgsF != nil: - err = quick.Check(checkAliasingTwoArgs(tt.twoArgsF), quickCheckConfig) + err = quick.Check(checkAliasingTwoArgs(tt.twoArgsF), &quick.Config{MaxCountScale: 1 << 8}) } if err != nil { t.Errorf("%v: %v", tt.name, err) diff --git a/internal/radix51/fe_test.go b/internal/radix51/fe_test.go index 6b9424c..5c83d22 100644 --- a/internal/radix51/fe_test.go +++ b/internal/radix51/fe_test.go @@ -16,9 +16,9 @@ import ( "testing/quick" ) -// quickCheckConfig will make each quickcheck test run (256 * -quickchecks) +// quickCheckConfig will make each quickcheck test run (1024 * -quickchecks) // times. The default value of -quickchecks is 100. -var quickCheckConfig = &quick.Config{MaxCountScale: 1 << 8} +var quickCheckConfig = &quick.Config{MaxCountScale: 1 << 10} func generateFieldElement(rand *mathrand.Rand) FieldElement { // Generation strategy: generate random limb values bounded by @@ -39,7 +39,7 @@ func (x FieldElement) Generate(rand *mathrand.Rand, size int) reflect.Value { return reflect.ValueOf(generateFieldElement(rand)) } -func TestFieldElementMulDistributesOverAdd(t *testing.T) { +func TestMulDistributesOverAdd(t *testing.T) { mulDistributesOverAdd := func(x, y, z FieldElement) bool { // Compute t1 = (x+y)*z t1 := new(FieldElement) @@ -61,27 +61,6 @@ func TestFieldElementMulDistributesOverAdd(t *testing.T) { } } -func TestMulDistributionFailure1(t *testing.T) { - x := FieldElement{0x592101fd8643a, 0x25a08467381e1, 0x48cb4dcd5dcf5, 0x1074d52744164, 0x91902aac541b} - y := FieldElement{0x165bb67340a7f, 0x52cf7781f4ad6, 0x32534ba21fde4, 0x5b4ba9cbb1736, 0x2e90748c54289} - z := FieldElement{0x25a575665ad1e, 0x124e496e3eeaa, 0x433d2180e2561, 0x221c8be3aa11a, 0x7adc8d0adf806} - - t1 := new(FieldElement) - t1.Add(&x, &y) - t1.Mul(t1, &z) - - // Compute t2 = x*z + y*z - t2 := new(FieldElement) - t3 := new(FieldElement) - t2.Mul(&x, &z) - t3.Mul(&y, &z) - t2.Add(t2, t3) - - if t1.Equal(t2) != 1 { - t.Errorf("t1 = %x should equal t2 = %x", t1, t2) - } -} - func TestMul64to128(t *testing.T) { a := uint64(5) b := uint64(5) @@ -230,7 +209,7 @@ func TestSanity(t *testing.T) { // t.Fatalf("all ones failed\nmul.s: %d\nmul.g: %d\nsqr.s: %d\nsqr.g: %d\n", x2, x2Go, x2sq, x2sqGo) // } - if !vartimeEqual(x2, x2sq) { + if x2 != x2sq { t.Fatalf("all ones failed\nmul: %x\nsqr: %x\n", x2, x2sq) } @@ -251,21 +230,12 @@ func TestSanity(t *testing.T) { // t.Fatalf("random field element failed\nfe: %x\n\nmul.s: %x\nmul.g: %x\nsqr.s: %x\nsqr.g: %x\n", x, x2, x2Go, x2sq, x2sqGo) // } - if !vartimeEqual(x2, x2sq) { + if x2 != x2sq { t.Fatalf("all ones failed\nmul: %x\nsqr: %x\n", x2, x2sq) } } -func vartimeEqual(x, y FieldElement) bool { - for i := 0; i < 5; i++ { - if x[i] != y[i] { - return false - } - } - return true -} - -func TestFeEqual(t *testing.T) { +func TestEqual(t *testing.T) { var x FieldElement = [5]uint64{1, 1, 1, 1, 1} var y FieldElement = [5]uint64{5, 4, 3, 2, 1} @@ -280,7 +250,7 @@ func TestFeEqual(t *testing.T) { } } -func TestFeInvert(t *testing.T) { +func TestInvert(t *testing.T) { var x FieldElement = [5]uint64{1, 1, 1, 1, 1} var one FieldElement = [5]uint64{1, 0, 0, 0, 0} var xinv, r FieldElement @@ -289,7 +259,7 @@ func TestFeInvert(t *testing.T) { r.Mul(&x, &xinv) r.reduce(&r) - if !vartimeEqual(one, r) { + if one != r { t.Errorf("inversion identity failed, got: %x", r) } @@ -305,7 +275,7 @@ func TestFeInvert(t *testing.T) { r.Mul(&x, &xinv) r.reduce(&r) - if !vartimeEqual(one, r) { + if one != r { t.Errorf("random inversion identity failed, got: %x for field element %x", r, x) } }