From bf95da4be17834c9dad28f75613c2dcf6e9bcc8d Mon Sep 17 00:00:00 2001 From: Sarah Jamie Lewis Date: Tue, 10 Aug 2021 21:25:26 -0700 Subject: [PATCH] Fix push --- header.html | 8 ++ index.html | 9 ++ index.md | 2 + obfuscated_apples.html | 131 +++++++++++-------- obfuscated_apples.png | Bin 0 -> 2299 bytes posts/obfuscated_apples.md | 249 +++++++++++++++++++++++-------------- ssb | 3 +- styles.css | 39 +++++- 8 files changed, 289 insertions(+), 152 deletions(-) create mode 100644 obfuscated_apples.png diff --git a/header.html b/header.html index 7b454cf..0598090 100644 --- a/header.html +++ b/header.html @@ -5,6 +5,14 @@ $TITLE | pseudorandom + + + + + + + + diff --git a/index.html b/index.html index 828f4be..a5aa83d 100644 --- a/index.html +++ b/index.html @@ -5,6 +5,14 @@ Welcome! | pseudorandom + + + + + + + + @@ -30,6 +38,7 @@

Welcome!

+

A site by Sarah Jamie Lewis

diff --git a/index.md b/index.md index 03bc49b..70a19d3 100644 --- a/index.md +++ b/index.md @@ -1,2 +1,4 @@ # Welcome! +A site by Sarah Jamie Lewis + diff --git a/obfuscated_apples.html b/obfuscated_apples.html index fd17be0..199d1a2 100644 --- a/obfuscated_apples.html +++ b/obfuscated_apples.html @@ -5,6 +5,14 @@ Obfuscated Apples | pseudorandom + + + + + + + + @@ -30,88 +38,103 @@

Obfuscated Apples

-Generating noise in a way which is indistinguishable from real signal is a ridiculously hard problem. Obfuscation does not hide signal, it only adds noise. +

Generating noise in a way which is indistinguishable from real signal is a ridiculously hard problem. Obfuscation does not hide signal, it only adds noise.

if you take anything away from this article please let it be this fact.

-

Sadly, most people operate under the assumption that adding noise to a system is all that it takes to make the signal unrecoverable. This logic is very clearly in operation in Apple’s new proposal of on-device scanning technical summary which,among other things, proposes generating synthetic matches to hide the true number of real matches in the system.

-I want to take this opportunity to break down how this kind of obfuscation can be defeated even when not considering the fact that it is Apple themselves who are charged with generating and maintaining the safety parameters of the system. +

Sadly, most people operate under the assumption that adding noise to a system is all that it takes to make the signal unrecoverable. This logic is very clearly in operation in Apple’s new proposal for on-device scanning technical summary which, among other things, proposes generating synthetic matches to hide the true number of real matches in the system.

+I want to take this opportunity to break down how this kind of obfuscation can be defeated even when not considering the fact that it is Apple themselves who are charged with generating and maintaining the safety parameters of the system.

i.e. even if we treat the people who design and build this system as honest adversaries.

Sketching a Basic Scheme

-

For the sake of clarity I will omit the technical details of the private set intersection protocol, and the threshold scheme, and we will operate under the assumption that both are cryptographically secure. We will also, for now, assume that the database of images to compare is incorruptible.

-

At the heart of system is a (mostly) black box that contains a perceptual hash function that analyzes an images and spits out a hash, this hash is then compared against a database of known hashes and if a match is found the system reports true and otherwise reports false.

-

Throughout this article I will use match when talking about both true and false positives.

-

Without question, the server learns a match occurred on the phone during the PSI protocol.

+

For the sake of clarity I will omit the technical details of the private set intersection protocol, and the threshold scheme, and we will operate under the assumption that both are cryptographically secure. We will also assume that the database of images to compare is incorruptible This is clearly not the case..

+

At the heart of system is a (mostly) black box that contains a perceptual hash function that analyzes an image and spits out a hash, this hash is then compared against a database of known hashes and if a match is found the system reports true and otherwise reports false. As we will see later on, perceptual hashes are not cryptographic hashes.

+

Throughout this article I will use the term match when talking about both true and false positives, though I will mostly assume any matches are false positives.

+

According to documentation provided by Apple, the server learns any matches occurred on the phone during the PSI protocol.

“The output of PSI protocol on the server reveals whether there is a match or not” - Apple Technical Summary
-

When a certain threshold of real matches is reached, the server gains the ability to decrypt all real matches, a human reviews them, and a determination is made.

-As presented the system above has one major flaw -

-(besides the gross nature of co-opting a personal device as a surveillance system) -

-

: the server learns how many matches the device has reported prior to being able to decrypt those matches. This is obviously very important metadata in the context of the system and as such needs to be protected.

-

To do so Apple rely on the invocation of “Synthetic Vouchers” with the following property:

-
    -
  • “The probability that a device uploads a synthetic voucher instead of a real voucher for an image is calibrated to ensure the total number of synthetics is of the same order of magnitude as the threshold”
    • -
-

Additional Metadata

+

When a certain threshold of matches are reached, the server gains the ability to decrypt all data associated , a human reviews that data, and a determination is made.

+

As presented the system above has one major flaw (besides the gross nature of co-opting a personal device as a surveillance system): the server learns how many matches the device has reported prior to being able to decrypt those matches.

+

This is obviously very important metadata in the context of the system and as such needs to be protected - if it is not then Apple, or someone who can compel Apple to release the data, can identify potential targets based on this metadata. As we shall soon discuss targeting people in way would be highly irrational if your goal was to actually hunt people doing harm, but people are not rational actors.

+

To protect this data Apple relies on the invocation of so-called “Synthetic Vouchers” with the following property:

+
+“The probability that a device uploads a synthetic voucher instead of a real voucher for an image is calibrated to ensure the total number of synthetics is of the same order of magnitude as the threshold” +
+

Deriving Probabilities

So, that is it right? Problem averted? Let’s not be too hasty…

-

From what we know so far there are a few interest parameters in this system, a threshold t, a probability of generating a synthetic match P(synthetic) and the probability of a false positive match P(falsepositive).

+

From what we know so far there are a few interesting parameters in this system that Apple must determine values for.

+

There is the threshold t of matches necessary to decrypt the data, there is the probability of a device generating a synthetic match P(synthetic) and there is the probability of a false positive match P(falsepositive).

We also know that Apple has constructed these parameters such that the probability of an account being flagged for human review (i.e. when number of matches M > t is P(flag) = 1e − 12 or one in one trillion.

“The threshold is selected to provide an extremely low (1 in 1 trillion) probability of incorrectly flagging a given account.” - Apple Technical Summary
-

From these parameters we can start to derive some additional metadata that exists within the system:

-

Matches over Threshold without Decryption

-

One of the most obvious sources of distinguishing information is built explicitly into the design. The threshold scheme as proposed requires t real matches in order to decrypt the inner envelopes containing the matches images.

-

Taking the number of observed matches, o, the system is able to derive a value t′ = o/t or, the number of observed matches divided by the threshold.

-

Every time a new match is observed and t′ is over the threshold the system learns additional information regarding the makeup of previous matches.

-

For example, if we set the t = 10, then after the system observes 10 matches (made up of a number of an unknown number of real or synthetic matches) then every additional match until decryption can be used to derive information about the previous matches. If the next match to arrive does not allow decryption then the server can derive that there are at least 2 synthetic matches in the bucket. This confidence increases with every observation that does not result in decryption.

-

Further we know that synthetic matches happen with probability P(synthetic) and as such the rate at which observed matches in combination with the above allows us to define a probability P(match) given that the observation did not result in decryption, and the known probability of a device generating a synthetic value P(synthetic).

-


$$P(\texttt{match}| \texttt{observation}) = \frac{P(\texttt{match}|\texttt{real}) \times P(\texttt{observation})}{P(\texttt{synthetic})}$$

-

We know that all matches will result in an observation and so…

-


$$P(\texttt{match}| \texttt{observation}) = \frac{1 \times P(\texttt{observation})}{P(\texttt{synthetic})}$$

-

Or more simply:

-


$$P(\texttt{match}| \texttt{observation}) = \frac{P(\texttt{observation})}{P(\texttt{synthetic})}$$

-

Finally, we can state the the probability of an observation (say over the period of a day) is the probability of a real match on any given day plus the probability of a synthetic match on any given day:

-


$$P(\texttt{match}| \texttt{observation}) = \frac{P(\texttt{synthetic})+P(\texttt{match})}{P(\texttt{synthetic})}$$

-

Given that the probability of a synthetic match is defined by Apple, the only unknown in the system is the probability of a match.

-While this may start as an unknown -

-setting aside the fact that Apple already has enough data to derive this themselves -

-

, Apple quickly generates a large amount of data relating to when new observations are made. Since people are different in the ways these take and store photos and live in different parts of the world, the exact probability of them triggering a check is dependent on them.

-

It is also obvious at this point that if P(synthetic) is generated naively (naively as in not account specific) then any additional information known about a particular account e.g. the number of photos they are likely to store over a period of time is enough for anyone who knows P(synthetic) to derive P(match). We will come back to this point shortly

-

The secrecy then is highly dependent on both Apple never deriving P(observation) for themselves and on Apple generating a distinct P(synthetic) for each account. Or rather, the privacy of one of the most sensitive aspects of this system requires Apple both collecting no information on accounts, and also on Apple knowing enough about accounts to derive the parameters necessary to keep the information private.

-
-

It is actually worse than that through. Remember that Apple has stated that the actual probability of observing M > t false positive matches is 1 in a trillion.

-

We can work backwards from that number to derive P(falsepositive):

-


$$P(\texttt{flag}) = \sum_{\substack{x = t}}^T {T \choose x} \cdot P(\texttt{falsepositive})^x \cdot P(\texttt{falsepositive})^{100 - x} \approx 1\mathrm{e}^{-12}$$

-

In order to finalize this we only need to make educated guesses about 2 parameters: the threshold value t and the total number of photos checked per year T. Apple throws out the number t = 10 in their technical summary, which seems like a good place to start.

-

Assuming an average account generates 3-4 pictures a day to be checked then T ≈ 1278 over a year. Plugging in those numbers, and we get P(falsepositive) ≈ 0.00035 or 1 in 2858. yikes.

+

We can actually work backwards from that number to derive P(falsepositive):

+


$$P(\texttt{flag}) = \sum_{\substack{x = t}}^T {T \choose x} \cdot P(\texttt{falsepositive})^x \cdot P(\texttt{falsepositive})^{T - x} \approx 1\mathrm{e}^{-12}$$

+

In order to finalize this we only need to make educated guesses about 2 parameters: the threshold value, t, and the total number of photos checked per year, T. Apple throws out the number t = 10 in their technical summary, which seems like a good place to start.

+

Assuming an average account generates 3-4 pictures a day to be checked then T ≈ 1278 over a year. Plugging in those numbers, and we get P(falsepositive) ≈ 0.00035 or 1 in 2858.

+

Does that number have any relation to reality? There is evidence to suggest Adversarial Detection Avoidance Attacks: Evaluating the robustness of perceptual hashing-based client-side scanning. Shubham Jain, Ana-Maria Cretu, Yves-Alexandre de Montjoye that the false acceptance rate for common perceptual hashing algorithms is between 0.001-0.01 for a database size of 500K.

+

That makes our guesstimate of 0.00035 an order of magnitude smaller than the most generous empirical estimate. We will be generous and assume Apple broke some new ground with NeuralHash and 0.00035 represents a major improvement in perceptual hashing false acceptance rates.

Given that we can go back and calculate the probability of observing, P(match), a match each day…

-


$$P(\texttt{match}) = 1 - (( 1 - 0.00035)^3.5) \approx 0.001225 \approx \frac{1}{816}$$

+


$$P(\texttt{match}) = 1 - (( 1 - {0.00035})^{3.5}) \approx {0.001225} \approx \frac{1}{{816}}$$

Or, a match once on average every 816 days for a person that only stores 3-4 photos per day.

Not everybody is every person though, if we applied the same P(falsepositive) to a new parent who takes upwards of 50 photos per day, then their P(match) is:

-


$$P(\texttt{match}) = 1 - (( 1 - 0.00035)^50) \approx 0.01735 \approx \frac{1}{57}$$

+


$$P(\texttt{match}) = 1 - (( 1 - {0.00035})^{50}) \approx {0.01735} \approx \frac{1}{{57}}$$

Or, a match on average every 57 days.

At this point I feel compelled to point out that these are average match probabilities. For the prolific photo taking parent who takes 18250 photos a year, the probability that they actually exceed the threshold in false matches is 6% assuming t is 10.

It is also worth mentioning that even though we ballparked t and T there are explicit constraints on what their values can be. If Apple generates a single t for all accounts, then T needs to be an approximation on the average number of photos an account stores per year. If Apple generates a different t value for every account, then it has enough information already to derive P(observation) and break its own obfuscation.

-

Using what we now know we can revisit the server side operations show how the observer can calculate the probability of a real match given the probability of any q observation and the probability of a synthetic match.

+

Using what we now know we can assess the server side operations show how the observer can calculate the probability of a real match given the probability of any q observation and the probability of a synthetic match.

+

The Probability of Synthetic Matches

+

Before we go any further, we should address one particular complication. Apple appears to suggest that images are probabilistically replaced with synthetic matches:

+
+“the device occasionally produces synthetic vouchers for images as opposed to ones corresponding to their image” +
+

Given that, the rate of real matches v.s. synthetic matches isn’t independent. Actual matches might be replaced by synthetic matches.

+


$$P(\texttt{observation}) = (P(\texttt{match}) \cdot (1 - P(\texttt{synthetic}))) + ((1-P(\texttt{match})) \cdot P(\texttt{synthetic})) \vphantom{+ \frac{1}{1}}$$

+

Or, to put it another way, the probability of a match being reported as a match is dependent on the probability it isn’t reported as a synthetic. Either way, P(observation|match) = 1

+

Further we can actually make a guess at the value of P(synthetic) under the assumption that it is calculated globally. Since Apple have stated that P(synthetic) is dependent on t and is designed such that it generates synthetic matches in the same order or magnitude as t we can derive P(synthetic) such that each device generates t synthetic matches a year on average.

+

Using our numbers from earlier we can place P(synthetic) ≈ 0.01 which would mean that over the course of a year, an average account storing 3-4 messages a day would have ~70% chance of generating 10 or more synthetic vouchers.

+

The exact value doesn’t really matter for our purposes. Any order of magnitude greater than 0.01 results in too many synthetic matches, and any order of magnitude smaller results in too few.

+

Calculating Synthetic Probabilities

+

Given what we know about the probabilities in this system we can now piece together a server-side attribution attack to break the privacy provided by synthetic matches.

+


$$P(\texttt{match}| \texttt{observation}) = \frac{P(\texttt{observation}|\texttt{match}) \times P(\texttt{match})}{P(\texttt{observation})}$$

+

We know that all matches will result in an observation and so…

+


$$P(\texttt{match}| \texttt{observation}) = \frac{1 \times P(\texttt{match})}{P(\texttt{observation})}$$

+

Or more simply:

+


$$P(\texttt{match}| \texttt{observation}) = \frac{P(\texttt{match})}{P(\texttt{observation})}$$

+

Given that the probability of a synthetic match is defined by Apple, the only unknown in the system is the probability of a match.

+

We can now plug in our values from earlier. Remember, Apple has stated that the actual probability of observing t false positive matches, P(flag), is 1 in a trillion and as such we have been able to derive approximate probabilities for false positives.

For an “average” account that stores 3-4 photos per day we know that P(match) =  ≈ 0.001225, allowing Apple, who defines P() to calculate:

-


$$P(\texttt{match}| \texttt{observation}) = \frac{P(\texttt{synthetic}) + 0.001225}{P(\texttt{synthetic})}$$

-

For our prolific “parent” account that stores 50 photos per day we know that P(observation) ≈ 0.01735, allowing Apple, who defines P() to calculate:

-


$$P(\texttt{match}| \texttt{observation}) = \frac{P(\texttt{synthetic}) + 0.01735}{P(\texttt{synthetic})}$$

-This clarifies our observation from earlier, if Apple define a global P(synthetic) then different accounts will naturally have different server-side distributions of observations, and these can be used to tighten the estimates of true matches. +


$$P(\texttt{match}| \texttt{observation}) = \frac{(0.001225 \cdot 0.99)}{(0.001225 \cdot 0.99) + (0.998775 \cdot 0.01))} \approx 0.11 $$

+

Given that we are aiming for ~10 synthetics over the course of a year, an 11% probability of any observation being a real match seems about the right level of indistinguishably.

+

But, what about our prolific “parent” account that stores 50 photos per day? We know that P(match) ≈ 0.01735, allowing Apple, who defines P(synthetic) to calculate:

+


$$P(\texttt{match}| \texttt{observation}) = \frac{(0.01735 \cdot 0.99)}{(0.01735 \cdot 0.99) + (0.98265 \cdot 0.01))} \approx 0.63 $$

+

That is a 63% probability that any reported match is a real match and not a synthetic one!

+If Apple define a global P(synthetic) then different accounts will naturally have different server-side distributions of observations, and these can be used to tighten the estimates of true matches.

And, again, if Apple can define P(synthetic) on a per-account basis then they have more information to use when tightening these estimates

+

The secrecy of this metadata is then is paradoxically dependent on both Apple never deriving P(observation) for themselves and on Apple generating a distinct P(synthetic) for each account. Or rather, the privacy of one of the most sensitive aspects of this system requires Apple both collecting no information on accounts, and also on Apple knowing enough about accounts to derive the parameters necessary to keep the information private.

+

It is actually much worse than that though.

+

While priors may start as an unknown (setting aside the fact that Apple already has enough data to derive this themselves), Apple quickly generates a large amount of data relating to when new observations are made. Since people are different in the ways these take and store photos and live in different parts of the world, the exact probability of them triggering a check is dependent on them.

+

There are also additional discriminating events in the system itself.

+

Matches over Threshold without Decryption

+

One of the most obvious sources of discriminating information is built explicitly into the design. The threshold scheme as proposed requires t real matches in order to decrypt the inner envelopes containing the matches images.

+

As such. every time a new match is observed and t′ is over the threshold the system learns additional information regarding the makeup of previous matches.

+

To illustrate, if we set the t = 10, then after the system observes 10 matches (made up of a number of an unknown number of real or synthetic matches) then every additional match until decryption can be used to derive information about the previous matches. If the next match to arrive does not allow decryption then the server can derive that there are at least 2 synthetic matches in the bucket. This confidence increases with every observation that does not result in decryption.

+

Combining that with our bayesian estimates fom earlier we can see how an adversarial observer could update their estimates of P(synthetic) based on the total number of observations, and the fact that no decryption has taken place.

+
+

What does this all mean?

In this analysis we have deliberately left out other information that Apple, or someone who can compel Apple, may use to tighten these estimates e.g. derived from public social media feeds.

+

In reality, an adversary will have far greater access to auxiliary data about the environment and target sets than simply raw probabilities.

+

In that kind of environment, no amount of server-defined obfuscation is enough to protect the metadata that the server holds.

+

In this case that metadata is a rather controversial number i.e. the number of possible matches to illegal material detected on the device.

+

That is interesting metadata to countless entities including the law enforcement and intelligence agencies of multiple jurisdictions and states.

+

Even if we strictly limit the type of material that Apple is searching for, the high likelihood of false positive events combined with the ease at which Apple can likely distinguish true matching events from synthetic events (as worked through above) should concern any potential subject of the system.

+

Innocence is no defense against judgements made using derived metadata.

diff --git a/obfuscated_apples.png b/obfuscated_apples.png new file mode 100644 index 0000000000000000000000000000000000000000..2db9946998bfb7d995dffa8840a4e8b03de283c0 GIT binary patch literal 2299 zcmeAS@N?(olHy`uVBq!ia0y~yU{+vYU|ht(1{9H>zGxDVVkvg=4B-HR8jh3>1_q8z zo-U3d6?5KRH)K4ZAaLM75c~UhogEy^bP0 Hl+XkKrw&$0 literal 0 HcmV?d00001 diff --git a/posts/obfuscated_apples.md b/posts/obfuscated_apples.md index fab7974..ba81905 100644 --- a/posts/obfuscated_apples.md +++ b/posts/obfuscated_apples.md @@ -1,136 +1,94 @@ # Obfuscated Apples -Generating noise in a way which is indistinguishable from real signal is a ridiculously hard problem. Obfuscation does -not hide signal, it only adds noise.

if you take anything away from this article please let +Generating noise in a way which is indistinguishable from real signal is a ridiculously hard problem. Obfuscation does not hide signal, it only adds noise. + +

if you take anything away from this article please let it be this fact.

Sadly, most people operate under the assumption that adding noise to a system is all that it takes to make the -signal unrecoverable. This logic is very clearly in operation in Apple's new proposal of on-device scanning -technical summary which,among other things, proposes generating *synthetic* matches to hide the true number of *real* matches in the system. +signal unrecoverable. This logic is very clearly in operation in Apple's new proposal for on-device scanning +technical summary which, among other things, proposes generating *synthetic* matches to hide the true number of *real* matches in the system. I want to take this opportunity to break down how this kind of obfuscation can be defeated even when not considering -the fact that it is Apple themselves who are charged with generating and maintaining the safety parameters of the system. +the fact that it is **Apple themselves who are charged with generating and maintaining the safety parameters of the system**.

i.e. even if we treat the people who design and build this system as honest adversaries.

## Sketching a Basic Scheme For the sake of clarity I will omit the technical details of the private set intersection protocol, and the threshold -scheme, and we will operate under the assumption that both are cryptographically secure. We will also, for now, -assume that the database of images to compare is incorruptible. +scheme, and we will operate under the assumption that both are cryptographically secure. We will also +assume that the database of images to compare is *incorruptible* This is clearly not the +case.. -At the heart of system is a (mostly) black box that contains a perceptual hash function that analyzes an images and +At the heart of system is a (mostly) black box that contains a **perceptual** hash function that analyzes an image and spits out a hash, this hash is then compared against a database of known hashes and if a match is found the system -reports `true` and otherwise reports `false`. +reports `true` and otherwise reports `false`. As we will see later on, perceptual hashes +are **not** cryptographic hashes. -Throughout this article I will use **match** when talking about both true and false positives. +Throughout this article I will use the term **match** when talking about both true and false positives, though I will +mostly assume any matches are false positives. -Without question, the server learns a match occurred on the phone during the PSI protocol. +According to documentation provided by Apple, the server learns any matches occurred on the phone during the PSI protocol.
"The output of PSI protocol on the server reveals whether there is a match or not" - Apple Technical Summary
-When a certain threshold of real matches is reached, the server gains the ability to decrypt all real matches, a human -reviews them, and a determination is made. +When a certain threshold of matches are reached, the server gains the ability to decrypt all data associated , a human +reviews that data, and a determination is made. -As presented the system above has one major flaw

(besides the gross nature of co-opting a personal device -as a surveillance system)

: the server learns how many matches the device has reported prior to being able to -decrypt those matches. This is obviously very important metadata in the context of the system and as such needs to be -protected. +As presented the system above has one major flaw (besides the gross nature of co-opting a personal device +as a surveillance system): the server learns how many matches the device has reported prior to being able to +decrypt those matches. -To do so Apple rely on the invocation of "Synthetic Vouchers" with the following property: +This is obviously very important metadata in the context of the system and as such needs to be +protected - if it is not then Apple, or someone who can compel Apple to release the data, can identify potential +targets based on this metadata. As we shall soon discuss targeting people in way would be highly +irrational if your goal was to actually hunt people doing harm, but people are not rational actors. -* "The probability that a device uploads a synthetic voucher instead of a real voucher for an image is calibrated to ensure the total number of synthetics is of the same order of magnitude as the threshold" +To protect this data Apple relies on the invocation of so-called "Synthetic Vouchers" with the following property: -## Additional Metadata +
"The probability that a device uploads a synthetic voucher instead of a real voucher for an image is calibrated to ensure the total number of synthetics is of the same order of magnitude as the threshold"
+ +## Deriving Probabilities So, that is it right? Problem averted? Let's not be too hasty... -From what we know so far there are a few interest parameters in this system, a threshold $t$, a probability of generating -a synthetic match $P(\texttt{synthetic})$ and the probability of a false positive match $P(\texttt{falsepositive})$. +From what we know so far there are a few interesting parameters in this system that Apple must determine +values for. + +There is the threshold $t$ of matches necessary to decrypt the data, there is the probability of a device generating +a synthetic match $P(\texttt{synthetic})$ and there is the probability of a false positive match $P(\texttt{falsepositive})$. We also know that Apple has constructed these parameters such that the probability of an account being flagged for human review (i.e. when number of matches $M > t$ is $P(flag) = 1\mathrm{e}^{-12}$ or one in one trillion.
"The threshold is selected to provide an extremely low (1 in 1 trillion) probability of incorrectly flagging a given account." - Apple Technical Summary
-From these parameters we can start to derive some additional metadata that exists within the system: +We can actually work backwards from that number to derive $P(\texttt{falsepositive})$: -### Matches over Threshold without Decryption +$$P(\texttt{flag}) = \sum_{\substack{x = t}}^T {T \choose x} \cdot P(\texttt{falsepositive})^x \cdot P(\texttt{falsepositive})^{T - x} \approx 1\mathrm{e}^{-12}$$ -One of the most obvious sources of distinguishing information is built explicitly into the design. The threshold -scheme as proposed requires $t$ real matches in order to decrypt the inner envelopes containing the matches images. - -Taking the number of observed matches, $o$, the system is able to derive a value $t' = o / t$ or, the number of -observed matches divided by the threshold. - -Every time a new match is observed and $t'$ is over the threshold the system learns additional information regarding -the makeup of previous matches. - -For example, if we set the $t = 10$, then after the system observes $10$ matches (made up of a number of an unknown number of real -or synthetic matches) then every additional match until decryption can be used to derive information about the previous -matches. If the next match to arrive does not allow decryption then the server can derive that there are at least 2 synthetic -matches in the bucket. This confidence increases with every observation that does not result in decryption. - -Further we know that synthetic matches happen with probability $P(\texttt{synthetic})$ and as such the **rate** at which -observed matches in combination with the above allows us to define a probability $P(\texttt{match})$ given that -the observation did not result in decryption, and the known probability of a device generating a synthetic value $P(\texttt{synthetic})$. - -$$P(\texttt{match}| \texttt{observation}) = \frac{P(\texttt{match}|\texttt{real}) \times P(\texttt{observation})}{P(\texttt{synthetic})}$$ - -We know that all matches will result in an observation and so... - -$$P(\texttt{match}| \texttt{observation}) = \frac{1 \times P(\texttt{observation})}{P(\texttt{synthetic})}$$ - -Or more simply: - -$$P(\texttt{match}| \texttt{observation}) = \frac{P(\texttt{observation})}{P(\texttt{synthetic})}$$ - -Finally, we can state the the probability of an observation (say over the period of a day) is the probability of a -real match on any given day plus the probability of a synthetic match on any given day: - -$$P(\texttt{match}| \texttt{observation}) = \frac{P(\texttt{synthetic})+P(\texttt{match})}{P(\texttt{synthetic})}$$ - -Given that the probability of a synthetic match is defined by Apple, the only unknown in the system is the probability -of a match. - -While this may start as an unknown

setting aside the fact that Apple already has enough data to derive this themselves

, Apple quickly generates a large amount of data relating to when new observations -are made. Since people are different in the ways these take and store photos and live in different parts of the world, -the exact probability of them triggering a check is dependent on them. - -It is also obvious at this point that if $P(\texttt{synthetic})$ is generated *naively* (*naively* as in -not account specific) then *any* additional information known about a particular account e.g. the number of photos -they are likely to store over a period of time is enough for **anyone** who knows $P(\texttt{synthetic})$ to derive -$P(\texttt{match})$. We will come back to this point shortly - -The secrecy then is highly dependent on both Apple never deriving $P(\texttt{observation})$ for themselves *and* on Apple -generating a distinct $P(\texttt{synthetic})$ for each account. Or rather, the privacy of one of the most sensitive aspects -of this system requires Apple both collecting no information on accounts, and also on Apple knowing enough about accounts -to derive the parameters necessary to keep the information private. - -
- -It is actually worse than that through. Remember that Apple has stated that the actual probability of observing $M > t$ -false positive matches is 1 in a trillion. - -We can work backwards from that number to derive $P(\texttt{falsepositive})$: - -$$P(\texttt{flag}) = \sum_{\substack{x = t}}^T {T \choose x} \cdot P(\texttt{falsepositive})^x \cdot P(\texttt{falsepositive})^{100 - x} \approx 1\mathrm{e}^{-12}$$ - -In order to finalize this we only need to make educated guesses about 2 parameters: the threshold value $t$ and the total -number of photos checked per year $T$. Apple throws out the number $t = 10$ in their technical summary, which seems +In order to finalize this we only need to make educated guesses about 2 parameters: the threshold value, $t$, and the total +number of photos checked per year, $T$. Apple throws out the number $t = 10$ in their technical summary, which seems like a good place to start. -Assuming an average account generates 3-4 pictures a day to be checked then $T \approx 1278$ over a year. Plugging in those -numbers, and we get $P(\texttt{falsepositive}) \approx 0.00035$ or 1 in 2858. yikes. +Assuming an average account generates 3-4 pictures a day to be checked then $T \approx {1278}$ over a year. Plugging in those +numbers, and we get $P(\texttt{falsepositive}) \approx 0.00035$ or **1 in 2858**. + +Does that number have any relation to reality? There is evidence to suggest Adversarial Detection Avoidance Attacks: Evaluating the robustness of perceptual hashing-based client-side scanning. Shubham Jain, Ana-Maria Cretu, Yves-Alexandre de Montjoye that the false acceptance rate for +common perceptual hashing algorithms is between 0.001-0.01 for a database size of 500K. + +That makes our guesstimate of 0.00035 an order of magnitude smaller than the most generous empirical estimate. We will +be generous and assume Apple broke some new ground with NeuralHash and 0.00035 represents a major improvement in perceptual hashing false acceptance rates. Given that we can go back and calculate the probability of observing, $P(\texttt{match})$, a match each day... -$$P(\texttt{match}) = 1 - (( 1 - 0.00035)^3.5) \approx 0.001225 \approx \frac{1}{816}$$ +$$P(\texttt{match}) = 1 - (( 1 - {0.00035})^{3.5}) \approx {0.001225} \approx \frac{1}{{816}}$$ Or, a match once on average every 816 days for a person that only stores 3-4 photos per day. Not everybody is every person though, if we applied the same $P(\texttt{falsepositive})$ to a new parent who takes upwards of 50 photos per day, then their $P(\texttt{match})$ is: -$$P(\texttt{match}) = 1 - (( 1 - 0.00035)^50) \approx 0.01735 \approx \frac{1}{57}$$ +$$P(\texttt{match}) = 1 - (( 1 - {0.00035})^{50}) \approx {0.01735} \approx \frac{1}{{57}}$$ Or, a match on average every 57 days. @@ -145,26 +103,127 @@ enough information already to derive $P(\texttt{observation})$ and break its own
-Using what we now know we can revisit the server side operations show how the observer can calculate the probability of a real +Using what we now know we can assess the server side operations show how the observer can calculate the probability of a real match given the probability of any q observation and the probability of a synthetic match. +### The Probability of Synthetic Matches + +Before we go any further, we should address one particular complication. Apple appears to suggest that +images are probabilistically replaced with synthetic matches: + +
"the device occasionally produces synthetic vouchers for images as opposed to ones corresponding to their image"
+ +Given that, the rate of real matches v.s. synthetic matches isn't independent. **Actual matches might be replaced +by synthetic matches**. + +$$P(\texttt{observation}) = (P(\texttt{match}) \cdot (1 - P(\texttt{synthetic}))) + ((1-P(\texttt{match})) \cdot P(\texttt{synthetic})) \vphantom{+ \frac{1}{1}}$$ + +Or, to put it another way, the probability of a match being reported as a match is dependent on the probability +it isn't reported as a synthetic. Either way, $P(\texttt{observation}|\texttt{match}) = 1$ + +Further we can actually make a guess at the value of $P(\texttt{synthetic})$ under the assumption that it is calculated +globally. Since Apple have stated that $P(\texttt{synthetic})$ is dependent on $t$ and is designed such that it generates +synthetic matches in the same order or magnitude as $t$ we can derive $P(\texttt{synthetic})$ such that each device +generates $t$ synthetic matches a year on average. + +Using our numbers from earlier we can place $P(\texttt{synthetic}) \approx {0.01}$ which would mean that over the course +of a year, an average account storing 3-4 messages a day would have ~70% chance of generating 10 or more synthetic +vouchers. + +The exact value doesn't really matter for our purposes. Any order of magnitude greater than ${0.01}$ results in too many synthetic +matches, and any order of magnitude smaller results in too few. + +### Calculating Synthetic Probabilities + +Given what we know about the probabilities in this system we can now piece together a server-side attribution attack +to break the privacy provided by synthetic matches. + +$$P(\texttt{match}| \texttt{observation}) = \frac{P(\texttt{observation}|\texttt{match}) \times P(\texttt{match})}{P(\texttt{observation})}$$ + +We know that all matches will result in an observation and so... + +$$P(\texttt{match}| \texttt{observation}) = \frac{1 \times P(\texttt{match})}{P(\texttt{observation})}$$ + +Or more simply: + +$$P(\texttt{match}| \texttt{observation}) = \frac{P(\texttt{match})}{P(\texttt{observation})}$$ + +Given that the probability of a synthetic match is defined by Apple, the only unknown in the system is the probability +of a match. + +We can now plug in our values from earlier. Remember, **Apple has stated** that the actual probability of observing $t$ +false positive matches, $P(\texttt{flag})$, is **1 in a trillion** and as such we have been able to derive approximate +probabilities for false positives. + For an "average" account that stores 3-4 photos per day we know that $P(\texttt{match}) = \approx 0.001225$, allowing Apple, who defines P(\texttt{synthetic}) to calculate: -$$P(\texttt{match}| \texttt{observation}) = \frac{P(\texttt{synthetic}) + 0.001225}{P(\texttt{synthetic})}$$ +$$P(\texttt{match}| \texttt{observation}) = \frac{(0.001225 \cdot 0.99)}{(0.001225 \cdot 0.99) + (0.998775 \cdot 0.01))} \approx 0.11 $$ -For our prolific "parent" account that stores 50 photos per day we know that $P(\texttt{observation}) \approx 0.01735$, allowing -Apple, who defines P(\texttt{synthetic}) to calculate: +Given that we are aiming for ~10 synthetics over the course of a year, an 11% probability of any observation being a +real match seems about the right level of indistinguishably. -$$P(\texttt{match}| \texttt{observation}) = \frac{P(\texttt{synthetic}) + 0.01735}{P(\texttt{synthetic})}$$ +But, what about our prolific "parent" account that stores 50 photos per day? We know that $P(\texttt{match}) \approx 0.01735$, allowing Apple, who defines $P(\texttt{synthetic})$ to calculate: -This clarifies our observation from earlier, if Apple define a global $P(\texttt{synthetic})$ then different accounts will +$$P(\texttt{match}| \texttt{observation}) = \frac{(0.01735 \cdot 0.99)}{(0.01735 \cdot 0.99) + (0.98265 \cdot 0.01))} \approx 0.63 $$ + +That is a 63% probability that any reported match is a real match and not a synthetic one! + +If Apple define a global $P(\texttt{synthetic})$ then different accounts will naturally have different server-side distributions of observations, and these can be used to tighten the estimates of true matches.

And, again, if Apple can define $P(\texttt{synthetic})$ on a per-account basis then they have **more** information to use when tightening these estimates

+The secrecy of this metadata is then is **paradoxically dependent** on both Apple never deriving $P(\texttt{observation})$ for themselves *and* on Apple generating a distinct $P(\texttt{synthetic})$ for each account. Or rather, the privacy of one of the most sensitive aspects of this system requires Apple both collecting no information on accounts, and also on Apple knowing enough about accounts to derive the parameters necessary to keep the information private. +
+It is actually much worse than that though. + +While priors may start as an unknown (setting aside the fact that Apple already has enough data to derive this themselves), Apple quickly generates a large amount of data relating to when new observations +are made. Since people are different in the ways these take and store photos and live in different parts of the world, +the exact probability of them triggering a check is dependent on them. + +There are also additional discriminating events in the system itself. + +### Matches over Threshold without Decryption + +One of the most obvious sources of discriminating information is built explicitly into the design. The threshold +scheme as proposed requires $t$ real matches in order to decrypt the inner envelopes containing the matches images. + +As such. every time a new match is observed and $t'$ is over the threshold the system learns additional information regarding +the makeup of previous matches. + +To illustrate, if we set the $t = 10$, then after the system observes $10$ matches (made up of a number of an unknown number of real or synthetic matches) then every additional match until decryption can be used to derive information about the previous +matches. If the next match to arrive does not allow decryption then the server can derive that there are at least 2 synthetic +matches in the bucket. This confidence increases with every observation that does not result in decryption. + +Combining that with our bayesian estimates fom earlier we can see how an adversarial observer could update their +estimates of $P(\texttt{synthetic})$ based on the total number of observations, and the fact that no decryption has taken +place. + +
+ +### What does this all mean? + In this analysis we have deliberately left out other information that Apple, or someone who can compel Apple, may use to tighten these estimates e.g. derived from public social media feeds. +In reality, an adversary will have far greater access to auxiliary data about the environment and target sets than simply +raw probabilities. + +In that kind of environment, no amount of server-defined obfuscation is enough to protect the metadata that the server +holds. + +In this case that metadata is a rather controversial number i.e. **the number of possible matches to illegal material detected +on the device**. + +That is interesting metadata to countless entities including the law enforcement and intelligence agencies of multiple +jurisdictions and states. + +Even if we strictly limit the type of material that Apple is searching for, the high likelihood of false positive events +combined with the ease at which Apple can likely distinguish true matching events from synthetic events +(as worked through above) should concern any potential subject of the system. + +**Innocence is no defense against judgements made using derived metadata**. + diff --git a/ssb b/ssb index 2bbfe29..00cc44d 100755 --- a/ssb +++ b/ssb @@ -86,8 +86,9 @@ function make_html_files file_base=`basename $md_file .md` output_file="$OUTPUT_DIR/$file_base.html" post_title=`grep -m 1 "^# .*" $md_file | cut -c 3-` + description=`head -3 $md_file | tail -1` append_posts_list $posts | cat $md_file - | $MD_RENDERER > $output_file - cat $HEADER_PATH $output_file $FOOTER_PATH | sed -e "s/\$TITLE/$post_title/g" | tee $output_file + cat $HEADER_PATH $output_file $FOOTER_PATH | sed -e "s/\$TITLE/$post_title/g" | sed -e "s/\$LINK/$file_base/g" | sed -e "s/\$DESCRIPTION/$description/g" | tee $output_file done } diff --git a/styles.css b/styles.css index a5c8493..b5ca7cf 100644 --- a/styles.css +++ b/styles.css @@ -13,10 +13,12 @@ body { padding: 1.5em; } + + article { margin-left: 0px; margin-right: auto; - max-width: 60%; + max-width: 60%; } .sidenote { @@ -30,6 +32,39 @@ article { transform: translateY(-100%); } +@media only screen and (max-width: 796px) { + + + body { + font-family: Spectral; + background: #111; + color: #eee; + width:95%; + max-width: 100%; + padding:0.5em; + margin: 0px 0px; + } + + article { + max-width: 100%; + } + + .sidenote { + float:none; + transform: translateY(0%); + color: #999; + font-style: italic; + margin-right: 0; + font-size: 0.7em; + } + + .display .katex { + font-size: 0.7em; + overflow: scroll; + } + +} + header { font-size: 1.1em; } footer { border-top: 0.25em solid #eee; margin-top: 1.5em; text-align: right; } nav { margin-bottom: 1.5em; padding-bottom: 0.5em; } @@ -74,7 +109,7 @@ td { } blockquote { padding: 0 1em; - color: #646464; + color: #ddd; border-left: 0.25em solid #bfbfbf; } pre {