Merge remote-tracking branch 'origin/maint-0.2.4' into maint-0.2.5
This commit is contained in:
Nick Mathewson
commit
5c200d9be2
|
|
@ -0,0 +1,12 @@
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|||
o Major bugfixes:
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- Fix a bug in the bounds-checking in the 32-bit curve25519-donna
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implementation that caused incorrect results on 32-bit
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implementations when certain malformed inputs were used along with
|
||||
a small class of private ntor keys. This bug does not currently
|
||||
appear to allow an attacker to learn private keys or impersonate a
|
||||
Tor server, but it could provide a means to distinguish 32-bit Tor
|
||||
implementations from 64-bit Tor implementations. Fixes bug 12694;
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||||
bugfix on 0.2.4.8-alpha. Bug found by Robert Ransom; fix from
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||||
Adam Langley.
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@ -43,8 +43,7 @@
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*
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* This is, almost, a clean room reimplementation from the curve25519 paper. It
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* uses many of the tricks described therein. Only the crecip function is taken
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* from the sample implementation.
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*/
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* from the sample implementation. */
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#include "orconfig.h"
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@ -61,25 +60,23 @@ typedef int64_t limb;
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* significant first. The value of the field element is:
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* x[0] + 2^26·x[1] + x^51·x[2] + 2^102·x[3] + ...
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*
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* i.e. the limbs are 26, 25, 26, 25, ... bits wide.
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*/
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* i.e. the limbs are 26, 25, 26, 25, ... bits wide. */
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/* Sum two numbers: output += in */
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static void fsum(limb *output, const limb *in) {
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unsigned i;
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for (i = 0; i < 10; i += 2) {
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output[0+i] = (output[0+i] + in[0+i]);
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output[1+i] = (output[1+i] + in[1+i]);
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output[0+i] = output[0+i] + in[0+i];
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output[1+i] = output[1+i] + in[1+i];
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}
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}
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/* Find the difference of two numbers: output = in - output
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* (note the order of the arguments!)
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*/
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* (note the order of the arguments!). */
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static void fdifference(limb *output, const limb *in) {
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unsigned i;
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for (i = 0; i < 10; ++i) {
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output[i] = (in[i] - output[i]);
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output[i] = in[i] - output[i];
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}
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}
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@ -95,7 +92,8 @@ static void fscalar_product(limb *output, const limb *in, const limb scalar) {
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*
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* output must be distinct to both inputs. The inputs are reduced coefficient
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* form, the output is not.
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*/
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*
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* output[x] <= 14 * the largest product of the input limbs. */
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static void fproduct(limb *output, const limb *in2, const limb *in) {
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output[0] = ((limb) ((s32) in2[0])) * ((s32) in[0]);
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output[1] = ((limb) ((s32) in2[0])) * ((s32) in[1]) +
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@ -199,9 +197,15 @@ static void fproduct(limb *output, const limb *in2, const limb *in) {
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output[18] = 2 * ((limb) ((s32) in2[9])) * ((s32) in[9]);
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}
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/* Reduce a long form to a short form by taking the input mod 2^255 - 19. */
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/* Reduce a long form to a short form by taking the input mod 2^255 - 19.
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*
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* On entry: |output[i]| < 14*2^54
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* On exit: |output[0..8]| < 280*2^54 */
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static void freduce_degree(limb *output) {
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/* Each of these shifts and adds ends up multiplying the value by 19. */
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/* Each of these shifts and adds ends up multiplying the value by 19.
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*
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* For output[0..8], the absolute entry value is < 14*2^54 and we add, at
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* most, 19*14*2^54 thus, on exit, |output[0..8]| < 280*2^54. */
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output[8] += output[18] << 4;
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output[8] += output[18] << 1;
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output[8] += output[18];
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@ -235,11 +239,13 @@ static void freduce_degree(limb *output) {
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#error "This code only works on a two's complement system"
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#endif
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/* return v / 2^26, using only shifts and adds. */
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/* return v / 2^26, using only shifts and adds.
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*
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* On entry: v can take any value. */
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static inline limb
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div_by_2_26(const limb v)
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{
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/* High word of v; no shift needed*/
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/* High word of v; no shift needed. */
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const uint32_t highword = (uint32_t) (((uint64_t) v) >> 32);
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/* Set to all 1s if v was negative; else set to 0s. */
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const int32_t sign = ((int32_t) highword) >> 31;
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|
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@ -249,7 +255,9 @@ div_by_2_26(const limb v)
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return (v + roundoff) >> 26;
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}
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|
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/* return v / (2^25), using only shifts and adds. */
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/* return v / (2^25), using only shifts and adds.
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*
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* On entry: v can take any value. */
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static inline limb
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div_by_2_25(const limb v)
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{
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|
|
@ -263,17 +271,21 @@ div_by_2_25(const limb v)
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return (v + roundoff) >> 25;
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}
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|
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#if 0
|
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/* return v / (2^25), using only shifts and adds.
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*
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||||
* On entry: v can take any value. */
|
||||
static inline s32
|
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div_s32_by_2_25(const s32 v)
|
||||
{
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const s32 roundoff = ((uint32_t)(v >> 31)) >> 7;
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return (v + roundoff) >> 25;
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}
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#endif
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|
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/* Reduce all coefficients of the short form input so that |x| < 2^26.
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*
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* On entry: |output[i]| < 2^62
|
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*/
|
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* On entry: |output[i]| < 280*2^54 */
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static void freduce_coefficients(limb *output) {
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unsigned i;
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|
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|
|
@ -281,56 +293,65 @@ static void freduce_coefficients(limb *output) {
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|
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for (i = 0; i < 10; i += 2) {
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limb over = div_by_2_26(output[i]);
|
||||
/* The entry condition (that |output[i]| < 280*2^54) means that over is, at
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* most, 280*2^28 in the first iteration of this loop. This is added to the
|
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* next limb and we can approximate the resulting bound of that limb by
|
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* 281*2^54. */
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output[i] -= over << 26;
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output[i+1] += over;
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|
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/* For the first iteration, |output[i+1]| < 281*2^54, thus |over| <
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||||
* 281*2^29. When this is added to the next limb, the resulting bound can
|
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* be approximated as 281*2^54.
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*
|
||||
* For subsequent iterations of the loop, 281*2^54 remains a conservative
|
||||
* bound and no overflow occurs. */
|
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over = div_by_2_25(output[i+1]);
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output[i+1] -= over << 25;
|
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output[i+2] += over;
|
||||
}
|
||||
/* Now |output[10]| < 2 ^ 38 and all other coefficients are reduced. */
|
||||
/* Now |output[10]| < 281*2^29 and all other coefficients are reduced. */
|
||||
output[0] += output[10] << 4;
|
||||
output[0] += output[10] << 1;
|
||||
output[0] += output[10];
|
||||
|
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output[10] = 0;
|
||||
|
||||
/* Now output[1..9] are reduced, and |output[0]| < 2^26 + 19 * 2^38
|
||||
* So |over| will be no more than 77825 */
|
||||
/* Now output[1..9] are reduced, and |output[0]| < 2^26 + 19*281*2^29
|
||||
* So |over| will be no more than 2^16. */
|
||||
{
|
||||
limb over = div_by_2_26(output[0]);
|
||||
output[0] -= over << 26;
|
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output[1] += over;
|
||||
}
|
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|
||||
/* Now output[0,2..9] are reduced, and |output[1]| < 2^25 + 77825
|
||||
* So |over| will be no more than 1. */
|
||||
{
|
||||
/* output[1] fits in 32 bits, so we can use div_s32_by_2_25 here. */
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s32 over32 = div_s32_by_2_25((s32) output[1]);
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output[1] -= over32 << 25;
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output[2] += over32;
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}
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/* Finally, output[0,1,3..9] are reduced, and output[2] is "nearly reduced":
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* we have |output[2]| <= 2^26. This is good enough for all of our math,
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* but it will require an extra freduce_coefficients before fcontract. */
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/* Now output[0,2..9] are reduced, and |output[1]| < 2^25 + 2^16 < 2^26. The
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* bound on |output[1]| is sufficient to meet our needs. */
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}
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/* A helpful wrapper around fproduct: output = in * in2.
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*
|
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* output must be distinct to both inputs. The output is reduced degree and
|
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* reduced coefficient.
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*/
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* On entry: |in[i]| < 2^27 and |in2[i]| < 2^27.
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*
|
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* output must be distinct to both inputs. The output is reduced degree
|
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* (indeed, one need only provide storage for 10 limbs) and |output[i]| < 2^26. */
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static void
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fmul(limb *output, const limb *in, const limb *in2) {
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limb t[19];
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fproduct(t, in, in2);
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/* |t[i]| < 14*2^54 */
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freduce_degree(t);
|
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freduce_coefficients(t);
|
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/* |t[i]| < 2^26 */
|
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memcpy(output, t, sizeof(limb) * 10);
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}
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|
||||
/* Square a number: output = in**2
|
||||
*
|
||||
* output must be distinct from the input. The inputs are reduced coefficient
|
||||
* form, the output is not.
|
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*
|
||||
* output[x] <= 14 * the largest product of the input limbs. */
|
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static void fsquare_inner(limb *output, const limb *in) {
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output[0] = ((limb) ((s32) in[0])) * ((s32) in[0]);
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output[1] = 2 * ((limb) ((s32) in[0])) * ((s32) in[1]);
|
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|
|
@ -389,12 +410,23 @@ static void fsquare_inner(limb *output, const limb *in) {
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output[18] = 2 * ((limb) ((s32) in[9])) * ((s32) in[9]);
|
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}
|
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|
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/* fsquare sets output = in^2.
|
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*
|
||||
* On entry: The |in| argument is in reduced coefficients form and |in[i]| <
|
||||
* 2^27.
|
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*
|
||||
* On exit: The |output| argument is in reduced coefficients form (indeed, one
|
||||
* need only provide storage for 10 limbs) and |out[i]| < 2^26. */
|
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static void
|
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fsquare(limb *output, const limb *in) {
|
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limb t[19];
|
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fsquare_inner(t, in);
|
||||
/* |t[i]| < 14*2^54 because the largest product of two limbs will be <
|
||||
* 2^(27+27) and fsquare_inner adds together, at most, 14 of those
|
||||
* products. */
|
||||
freduce_degree(t);
|
||||
freduce_coefficients(t);
|
||||
/* |t[i]| < 2^26 */
|
||||
memcpy(output, t, sizeof(limb) * 10);
|
||||
}
|
||||
|
||||
|
|
@ -423,60 +455,143 @@ fexpand(limb *output, const u8 *input) {
|
|||
#error "This code only works when >> does sign-extension on negative numbers"
|
||||
#endif
|
||||
|
||||
/* s32_eq returns 0xffffffff iff a == b and zero otherwise. */
|
||||
static s32 s32_eq(s32 a, s32 b) {
|
||||
a = ~(a ^ b);
|
||||
a &= a << 16;
|
||||
a &= a << 8;
|
||||
a &= a << 4;
|
||||
a &= a << 2;
|
||||
a &= a << 1;
|
||||
return a >> 31;
|
||||
}
|
||||
|
||||
/* s32_gte returns 0xffffffff if a >= b and zero otherwise, where a and b are
|
||||
* both non-negative. */
|
||||
static s32 s32_gte(s32 a, s32 b) {
|
||||
a -= b;
|
||||
/* a >= 0 iff a >= b. */
|
||||
return ~(a >> 31);
|
||||
}
|
||||
|
||||
/* Take a fully reduced polynomial form number and contract it into a
|
||||
* little-endian, 32-byte array
|
||||
*/
|
||||
* little-endian, 32-byte array.
|
||||
*
|
||||
* On entry: |input_limbs[i]| < 2^26 */
|
||||
static void
|
||||
fcontract(u8 *output, limb *input) {
|
||||
fcontract(u8 *output, limb *input_limbs) {
|
||||
int i;
|
||||
int j;
|
||||
s32 input[10];
|
||||
s32 mask;
|
||||
|
||||
/* |input_limbs[i]| < 2^26, so it's valid to convert to an s32. */
|
||||
for (i = 0; i < 10; i++) {
|
||||
input[i] = (s32) input_limbs[i];
|
||||
}
|
||||
|
||||
for (j = 0; j < 2; ++j) {
|
||||
for (i = 0; i < 9; ++i) {
|
||||
if ((i & 1) == 1) {
|
||||
/* This calculation is a time-invariant way to make input[i] positive
|
||||
by borrowing from the next-larger limb.
|
||||
*/
|
||||
const s32 mask = (s32)(input[i]) >> 31;
|
||||
const s32 carry = -(((s32)(input[i]) & mask) >> 25);
|
||||
input[i] = (s32)(input[i]) + (carry << 25);
|
||||
input[i+1] = (s32)(input[i+1]) - carry;
|
||||
/* This calculation is a time-invariant way to make input[i]
|
||||
* non-negative by borrowing from the next-larger limb. */
|
||||
const s32 mask = input[i] >> 31;
|
||||
const s32 carry = -((input[i] & mask) >> 25);
|
||||
input[i] = input[i] + (carry << 25);
|
||||
input[i+1] = input[i+1] - carry;
|
||||
} else {
|
||||
const s32 mask = (s32)(input[i]) >> 31;
|
||||
const s32 carry = -(((s32)(input[i]) & mask) >> 26);
|
||||
input[i] = (s32)(input[i]) + (carry << 26);
|
||||
input[i+1] = (s32)(input[i+1]) - carry;
|
||||
const s32 mask = input[i] >> 31;
|
||||
const s32 carry = -((input[i] & mask) >> 26);
|
||||
input[i] = input[i] + (carry << 26);
|
||||
input[i+1] = input[i+1] - carry;
|
||||
}
|
||||
}
|
||||
|
||||
/* There's no greater limb for input[9] to borrow from, but we can multiply
|
||||
* by 19 and borrow from input[0], which is valid mod 2^255-19. */
|
||||
{
|
||||
const s32 mask = (s32)(input[9]) >> 31;
|
||||
const s32 carry = -(((s32)(input[9]) & mask) >> 25);
|
||||
input[9] = (s32)(input[9]) + (carry << 25);
|
||||
input[0] = (s32)(input[0]) - (carry * 19);
|
||||
const s32 mask = input[9] >> 31;
|
||||
const s32 carry = -((input[9] & mask) >> 25);
|
||||
input[9] = input[9] + (carry << 25);
|
||||
input[0] = input[0] - (carry * 19);
|
||||
}
|
||||
|
||||
/* After the first iteration, input[1..9] are non-negative and fit within
|
||||
* 25 or 26 bits, depending on position. However, input[0] may be
|
||||
* negative. */
|
||||
}
|
||||
|
||||
/* The first borrow-propagation pass above ended with every limb
|
||||
except (possibly) input[0] non-negative.
|
||||
|
||||
Since each input limb except input[0] is decreased by at most 1
|
||||
by a borrow-propagation pass, the second borrow-propagation pass
|
||||
could only have wrapped around to decrease input[0] again if the
|
||||
first pass left input[0] negative *and* input[1] through input[9]
|
||||
were all zero. In that case, input[1] is now 2^25 - 1, and this
|
||||
last borrow-propagation step will leave input[1] non-negative.
|
||||
*/
|
||||
If input[0] was negative after the first pass, then it was because of a
|
||||
carry from input[9]. On entry, input[9] < 2^26 so the carry was, at most,
|
||||
one, since (2**26-1) >> 25 = 1. Thus input[0] >= -19.
|
||||
|
||||
In the second pass, each limb is decreased by at most one. Thus the second
|
||||
borrow-propagation pass could only have wrapped around to decrease
|
||||
input[0] again if the first pass left input[0] negative *and* input[1]
|
||||
through input[9] were all zero. In that case, input[1] is now 2^25 - 1,
|
||||
and this last borrow-propagation step will leave input[1] non-negative. */
|
||||
{
|
||||
const s32 mask = (s32)(input[0]) >> 31;
|
||||
const s32 carry = -(((s32)(input[0]) & mask) >> 26);
|
||||
input[0] = (s32)(input[0]) + (carry << 26);
|
||||
input[1] = (s32)(input[1]) - carry;
|
||||
const s32 mask = input[0] >> 31;
|
||||
const s32 carry = -((input[0] & mask) >> 26);
|
||||
input[0] = input[0] + (carry << 26);
|
||||
input[1] = input[1] - carry;
|
||||
}
|
||||
|
||||
/* Both passes through the above loop, plus the last 0-to-1 step, are
|
||||
necessary: if input[9] is -1 and input[0] through input[8] are 0,
|
||||
negative values will remain in the array until the end.
|
||||
*/
|
||||
/* All input[i] are now non-negative. However, there might be values between
|
||||
* 2^25 and 2^26 in a limb which is, nominally, 25 bits wide. */
|
||||
for (j = 0; j < 2; j++) {
|
||||
for (i = 0; i < 9; i++) {
|
||||
if ((i & 1) == 1) {
|
||||
const s32 carry = input[i] >> 25;
|
||||
input[i] &= 0x1ffffff;
|
||||
input[i+1] += carry;
|
||||
} else {
|
||||
const s32 carry = input[i] >> 26;
|
||||
input[i] &= 0x3ffffff;
|
||||
input[i+1] += carry;
|
||||
}
|
||||
}
|
||||
|
||||
{
|
||||
const s32 carry = input[9] >> 25;
|
||||
input[9] &= 0x1ffffff;
|
||||
input[0] += 19*carry;
|
||||
}
|
||||
}
|
||||
|
||||
/* If the first carry-chain pass, just above, ended up with a carry from
|
||||
* input[9], and that caused input[0] to be out-of-bounds, then input[0] was
|
||||
* < 2^26 + 2*19, because the carry was, at most, two.
|
||||
*
|
||||
* If the second pass carried from input[9] again then input[0] is < 2*19 and
|
||||
* the input[9] -> input[0] carry didn't push input[0] out of bounds. */
|
||||
|
||||
/* It still remains the case that input might be between 2^255-19 and 2^255.
|
||||
* In this case, input[1..9] must take their maximum value and input[0] must
|
||||
* be >= (2^255-19) & 0x3ffffff, which is 0x3ffffed. */
|
||||
mask = s32_gte(input[0], 0x3ffffed);
|
||||
for (i = 1; i < 10; i++) {
|
||||
if ((i & 1) == 1) {
|
||||
mask &= s32_eq(input[i], 0x1ffffff);
|
||||
} else {
|
||||
mask &= s32_eq(input[i], 0x3ffffff);
|
||||
}
|
||||
}
|
||||
|
||||
/* mask is either 0xffffffff (if input >= 2^255-19) and zero otherwise. Thus
|
||||
* this conditionally subtracts 2^255-19. */
|
||||
input[0] -= mask & 0x3ffffed;
|
||||
|
||||
for (i = 1; i < 10; i++) {
|
||||
if ((i & 1) == 1) {
|
||||
input[i] -= mask & 0x1ffffff;
|
||||
} else {
|
||||
input[i] -= mask & 0x3ffffff;
|
||||
}
|
||||
}
|
||||
|
||||
input[1] <<= 2;
|
||||
input[2] <<= 3;
|
||||
|
|
@ -514,7 +629,9 @@ fcontract(u8 *output, limb *input) {
|
|||
* x z: short form, destroyed
|
||||
* xprime zprime: short form, destroyed
|
||||
* qmqp: short form, preserved
|
||||
*/
|
||||
*
|
||||
* On entry and exit, the absolute value of the limbs of all inputs and outputs
|
||||
* are < 2^26. */
|
||||
static void fmonty(limb *x2, limb *z2, /* output 2Q */
|
||||
limb *x3, limb *z3, /* output Q + Q' */
|
||||
limb *x, limb *z, /* input Q */
|
||||
|
|
@ -525,43 +642,69 @@ static void fmonty(limb *x2, limb *z2, /* output 2Q */
|
|||
|
||||
memcpy(origx, x, 10 * sizeof(limb));
|
||||
fsum(x, z);
|
||||
fdifference(z, origx); // does x - z
|
||||
/* |x[i]| < 2^27 */
|
||||
fdifference(z, origx); /* does x - z */
|
||||
/* |z[i]| < 2^27 */
|
||||
|
||||
memcpy(origxprime, xprime, sizeof(limb) * 10);
|
||||
fsum(xprime, zprime);
|
||||
/* |xprime[i]| < 2^27 */
|
||||
fdifference(zprime, origxprime);
|
||||
/* |zprime[i]| < 2^27 */
|
||||
fproduct(xxprime, xprime, z);
|
||||
/* |xxprime[i]| < 14*2^54: the largest product of two limbs will be <
|
||||
* 2^(27+27) and fproduct adds together, at most, 14 of those products.
|
||||
* (Approximating that to 2^58 doesn't work out.) */
|
||||
fproduct(zzprime, x, zprime);
|
||||
/* |zzprime[i]| < 14*2^54 */
|
||||
freduce_degree(xxprime);
|
||||
freduce_coefficients(xxprime);
|
||||
/* |xxprime[i]| < 2^26 */
|
||||
freduce_degree(zzprime);
|
||||
freduce_coefficients(zzprime);
|
||||
/* |zzprime[i]| < 2^26 */
|
||||
memcpy(origxprime, xxprime, sizeof(limb) * 10);
|
||||
fsum(xxprime, zzprime);
|
||||
/* |xxprime[i]| < 2^27 */
|
||||
fdifference(zzprime, origxprime);
|
||||
/* |zzprime[i]| < 2^27 */
|
||||
fsquare(xxxprime, xxprime);
|
||||
/* |xxxprime[i]| < 2^26 */
|
||||
fsquare(zzzprime, zzprime);
|
||||
/* |zzzprime[i]| < 2^26 */
|
||||
fproduct(zzprime, zzzprime, qmqp);
|
||||
/* |zzprime[i]| < 14*2^52 */
|
||||
freduce_degree(zzprime);
|
||||
freduce_coefficients(zzprime);
|
||||
/* |zzprime[i]| < 2^26 */
|
||||
memcpy(x3, xxxprime, sizeof(limb) * 10);
|
||||
memcpy(z3, zzprime, sizeof(limb) * 10);
|
||||
|
||||
fsquare(xx, x);
|
||||
/* |xx[i]| < 2^26 */
|
||||
fsquare(zz, z);
|
||||
/* |zz[i]| < 2^26 */
|
||||
fproduct(x2, xx, zz);
|
||||
/* |x2[i]| < 14*2^52 */
|
||||
freduce_degree(x2);
|
||||
freduce_coefficients(x2);
|
||||
/* |x2[i]| < 2^26 */
|
||||
fdifference(zz, xx); // does zz = xx - zz
|
||||
/* |zz[i]| < 2^27 */
|
||||
memset(zzz + 10, 0, sizeof(limb) * 9);
|
||||
fscalar_product(zzz, zz, 121665);
|
||||
/* |zzz[i]| < 2^(27+17) */
|
||||
/* No need to call freduce_degree here:
|
||||
fscalar_product doesn't increase the degree of its input. */
|
||||
freduce_coefficients(zzz);
|
||||
/* |zzz[i]| < 2^26 */
|
||||
fsum(zzz, xx);
|
||||
/* |zzz[i]| < 2^27 */
|
||||
fproduct(z2, zz, zzz);
|
||||
/* |z2[i]| < 14*2^(26+27) */
|
||||
freduce_degree(z2);
|
||||
freduce_coefficients(z2);
|
||||
/* |z2|i| < 2^26 */
|
||||
}
|
||||
|
||||
/* Conditionally swap two reduced-form limb arrays if 'iswap' is 1, but leave
|
||||
|
|
@ -572,8 +715,7 @@ static void fmonty(limb *x2, limb *z2, /* output 2Q */
|
|||
* wrong results. Also, the two limb arrays must be in reduced-coefficient,
|
||||
* reduced-degree form: the values in a[10..19] or b[10..19] aren't swapped,
|
||||
* and all all values in a[0..9],b[0..9] must have magnitude less than
|
||||
* INT32_MAX.
|
||||
*/
|
||||
* INT32_MAX. */
|
||||
static void
|
||||
swap_conditional(limb a[19], limb b[19], limb iswap) {
|
||||
unsigned i;
|
||||
|
|
@ -590,8 +732,7 @@ swap_conditional(limb a[19], limb b[19], limb iswap) {
|
|||
*
|
||||
* resultx/resultz: the x coordinate of the resulting curve point (short form)
|
||||
* n: a little endian, 32-byte number
|
||||
* q: a point of the curve (short form)
|
||||
*/
|
||||
* q: a point of the curve (short form) */
|
||||
static void
|
||||
cmult(limb *resultx, limb *resultz, const u8 *n, const limb *q) {
|
||||
limb a[19] = {0}, b[19] = {1}, c[19] = {1}, d[19] = {0};
|
||||
|
|
@ -709,7 +850,7 @@ crecip(limb *out, const limb *z) {
|
|||
/* 2^255 - 21 */ fmul(out,t1,z11);
|
||||
}
|
||||
|
||||
int curve25519_donna(u8 *, const u8 *, const u8 *);
|
||||
int curve25519_donna(u8 *mypublic, const u8 *secret, const u8 *basepoint);
|
||||
|
||||
int
|
||||
curve25519_donna(u8 *mypublic, const u8 *secret, const u8 *basepoint) {
|
||||
|
|
@ -726,7 +867,6 @@ curve25519_donna(u8 *mypublic, const u8 *secret, const u8 *basepoint) {
|
|||
cmult(x, z, e, bp);
|
||||
crecip(zmone, z);
|
||||
fmul(z, x, zmone);
|
||||
freduce_coefficients(z);
|
||||
fcontract(mypublic, z);
|
||||
return 0;
|
||||
}
|
||||
|
|
|
|||
Loading…
Reference in New Issue