# 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 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 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 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 SummaryWhen 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 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 SummaryWe can actually work backwards from that number to derive $P(\texttt{falsepositive})$: $$P(\texttt{flag}) = \sum_{\substack{x = t}}^T {T \choose x} \cdot P(\texttt{falsepositive})^x \cdot P(1-\texttt{falsepositive})^{T - x} \approx 1\mathrm{e}^{-12}$$ 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 $T \approx {1278}$ 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... $$P(\texttt{match}) = 1 - (( 1 - {0.00035})^{3.5}) \approx {0.001225} \approx \frac{1}{{816}}$$ 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: $$P(\texttt{match}) = 1 - (( 1 - {0.00035})^{50}) \approx {0.01735} \approx \frac{1}{{57}}$$ 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 $P(\texttt{observation})$ and break its own obfuscation.
"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: $$P(\texttt{match}| \texttt{observation}) = \frac{(0.001225 \cdot 0.99)}{(0.001225 \cdot 0.99) + (0.998775 \cdot 0.01))} \approx 0.11 $$ 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 $P(\texttt{match}) \approx 0.01735$, allowing Apple, who defines $P(\texttt{synthetic})$ 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 $P(\texttt{synthetic})$ 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.