Zero Knowledge Proofs: An illustrated primer, Part 2

This post is the second in a two-part series on zero-knowledge proofs. Click here t2380271980_b2a66bd47d_zo read Part 1.

In this post I’m going to continue the short, (relatively) non-technical overview of zero knowledge proofs that I started a couple of years ago. Yes, that was a very long time! If you didn’t catch the first post, now would be an excellent time to go read it.

Before we go much further, a bit of a warning. While this series is still intended as a high-level overview, at a certain point it’s necessary to dig a bit deeper into some specific algorithms. So you should expect this post to get a bit wonkier than the last.

A quick recap, and a bit more on Zero Knowledge(ness)

First, a brief refresher.

In the last post we defined a zero knowledge proof as an interaction between two computer programs (or Turing machines) — respectively called a Prover and a Verifier — where the Prover works to convince the Verifier that some mathematical statement is true. We also covered a specific example: a clever protocol by Goldreich, Micali and Wigderson that allows us to prove, in zero knowledge, that a graph possesses a three-coloring.

In the course of that discussion, we described three critical properties that any zero knowledge proof must satisfy:

  • Completeness: If the Prover is honest, then she will eventually convince the Verifier.
  • Soundness: The Prover can only convince the Verifier if the statement is true.
  • Zero-knowledge(ness): The Verifier learns no information beyond the fact that the statement is true.

The real challenge turns out to be finding a way to formally define the last property. How do you state that a Verifier learns nothing beyond the truth of a statement?

In case you didn’t read the previous post — the answer to this question came from Goldwasser, Micali and Rackoff, and it’s very cool. What they argued is that a protocol can be proven zero knowledge if for every possible Verifier, you can demonstrate the existence of an algorithm called a ‘Simulator’, and show that this algorithm has some very special properties.

From a purely mechanical perspective, the Simulator is like a special kind of Prover. However, unlike a real Prover — which starts with some special knowledge that allows it to prove the truth of a statement — the Simulator gets no special knowledge at all.* Nonetheless, the Simulator (or Simulators) must be able to ‘fool’ every Verifier into believing that the statement is true, while producing a transcript that’s statistically identical top (or indistinguishable from) the output of a real Prover.

The logic here flows pretty cleanly: since Simulator has no ‘knowledge’ to extract in the first place, then clearly a Verifier can’t obtain any meaningful amount of information after interacting with it. Moreover, if the transcript of the interaction is distributed identically to a real protocol run with a normal Prover, then the Verifier can’t do better against the real prover than it can do against the Simulator. (If the Verifier could do better, then that would imply that the distributions were not statistically identical.) Ergo, the Verifier can’t extract useful information from the real protocol run.

This is incredibly wonky, and worse, it seems contradictory! We’re asking that a protocol be both sound — meaning that a bogus Prover can’t trick some Verifier into accepting a statement unless it has special knowledge allowing it to prove the statement — but we’re also asking for the existence of an algorithm (the simulator) that can literally cheat. Clearly both properties can’t hold at the same time.

The solution to this problem is that both properties don’t hold at the same time.

To build our simulator, we’re allowed to do things to the Verifier that would never happen in the real world. The example that I gave in the previous post was to use a ‘time machine’ — that is, our ‘Simulator’ can rewind the Verifier program’s execution in order to ‘fool’ it. Thus, in a world where we can wind the Verifier back in time, it’s easy to show that a Simulator exists. In the real world, of course it doesn’t. This ‘trick’ gets us around the contradiction.

As a last reminder, to illustrate all of these ideas, we covered one of the first general zero knowledge proofs, devised by Goldreich, Micali and Wigderson (GMW). That protocol allowed us to prove, in zero knowledge, that a graph supports a three-coloring. Of course, proving three colorings isn’t terribly interesting. The real significance of the GMW result is theoretical. Since graph three coloring is known to be in the complexity class NP-complete, the GMW protocol can be used to prove any statement in the class NP. And that’s quite powerful.

Let me elaborate slightly on what that means:

  1. If there exists any decision problem (that is, a problem with a yes/no answer) whose witness (solution) can be verified in polynomial time, then:
  2. We can prove that said solution exists by (1) translating the problem into an instance of the graph three-coloring problem, and (2) running the GMW protocol.*

This amazing result gives us interactive zero knowledge proofs for every statement in NP. The only problem is that it’s almost totally unusable.

From theory into practice

If you’re of a practical mindset, you’re probably shaking your head at all this talk of ZK proofs. That’s because actually using this approach would be an insanely expensive and stupid thing to do. Most likely you’d first represent your input problem as a boolean circuit where the circuit is satisfied if and only if you know the correct input. Then you’d have to translate your circuit into a graph, resulting in some further blowup. Finally you’d need to run the GMW protocol, which is damned expensive all by itself.

So in practice nobody does this. It’s really considered a ‘feasibility’ result. Once you show that something is possible, the next step is to make it efficient.

But we do use zero knowledge proofs, almost every day. In this post I’m going to spend some time talking about the more practical ZK proofs that we actually use. To do that I just need give just a tiny bit of extra background.

Proofs vs. Proofs of Knowledge

Before we go on, there’s one more concept we need to cover. Specifically, we need to discuss what precisely we’re proving when we conduct a zero knowledge proof.

Let me explain. At a high level, there are two kinds of statement you might want to prove in zero knowledge. Roughly speaking, these break up as follows.

Statements about “facts”. For example, I might wish to prove that “a specific graph has a three coloring” or “some number N is in the set of composite numbers“. Each of these is a statement about some intrinsic property of the universe.

Statements about my personal knowledge. Alternatively, I might wish to prove that I know some piece information. Examples of this kind of statement include: “I know a three coloring for this graph”, or “I know the factorization of N”. These go beyond merely proving that a fact is true, and actually rely on what the Prover knows.

It’s important to recognize that there’s a big difference between these two kinds of statements! For example, it may be possible to prove that a number N is composite even if you don’t know the full factorization. So merely proving the first statement is not equivalent to proving the second one.

The second class of proof is known as a “proof of knowledge”. It turns out to be extremely useful for proving a variety of statements that we use in real life. In this post, we’ll mostly be focusing on this kind of proof.

The Schnorr identification protocol

Now that we’ve covered some of the required background, it’s helpful to move on to a specific and very useful proof of knowledge that was invented by Claus-Peter Schnorr in the 1980s. At first glance, the Schnorr protocol may seem a bit odd, but in fact it’s the basis of many of our modern signature schemes today.

Schnorr wasn’t really concerned with digital signatures, however. His concern was with identification. Specifically, let’s imagine that Alice has published her public key to the world, and later on wants to prove that she knows the secret key corresponding to that public key. This is the exact problem that we encounter in real-world protocols such as public-key SSH, so it turns out to be well-motivated.

Schnorr began with the assumption that the public key would be of a very specific format. Specifically, let p be some prime number, and let g be a generator of a cyclic group of prime-order q. To generate a keypair, Alice would first pick a random integer a between 1 and q, and then compute the keypair as:

PK_{A} = g^a~mod~p, SK_{A} = a

(If you’ve been around the block a time or two, you’ll probably notice that this is the same type of key used for Diffie-Hellman and the DSA signing algorithm. That’s not a coincidence, and it makes this protocol very useful.)

Alice keeps her secret key to herself, but she’s free to publish her public key to the world. Later on, when she wants to prove knowledge of her secret key, she conducts the following simple interactive protocol with Bob:

There’s a lot going on in here, so let’s take a minute to unpack things.

First off, we should ask ourselves if the protocol is complete. This is usually the easiest property to verify: if Alice performs the protocol honestly, should Bob be satisfied at the end of it? In this case, completeness is pretty easy to see just by doing a bit of substitution:

Proving soundness

The harder property is soundness. Mainly because we don’t yet have a good definition of what it means for a proof of knowledge to be sound. Remember that what we want to show is the following:

If Alice successfully convinces Bob, then she must know the secret key a.

It’s easy to look at the equations above and try to convince yourself that Alice’s only way to cheat the protocol is to know a. But that’s hardly a proof.

When it comes to demonstrating the soundness of a proof of knowledge, we have a really nice formal approach. Just as with the Simulator we discussed above, we need to demonstrate the existence of a special algorithm. This algorithm is called a knowledge extractor, and it does exactly what it claims to. A knowledge extractor (or just ‘Extractor’ for short) is a special type of Verifier that interacts with a Prover, and — if the Prover succeeds in completing the proof — the Extractor should be able to extract the Prover’s original secret.

And this answers our question above. To prove soundness for a proof of knowledge, we must show that an Extractor exists for every possible Prover.

Of course this again seems totally contradictory to the purpose of a zero knowledge protocol — where we’re not supposed to be able to learn secrets from a Prover. Fortunately we’ve already resolved this conundrum once for the case of the Simulator. Here again, we take the same approach. The Extractor is not required to exist during a normal run of the protocol. We simply show that it exists if we’re allowed to take special liberties with the Prover — in this case, we’ll use ‘rewinding’ to wind back the Prover’s execution and allow us to extract secrets.

The extractor for the Schnorr protocol is extremely clever — and it’s also pretty simple. Let’s illustrate it in terms of a protocol diagram. Alice (the Prover) is on the left, and the Extractor is on the right:

The key observation here is that by rewinding Alice’s execution, the Extractor can ‘trick’ Alice into making two different proof transcripts using the same k. This shouldn’t normally happen in a real protocol run, where Alice specifically picks a new k for each execution of the protocol.

If the Extractor can trick Alice into doing this, then he can solve the following simple equation to recover Alice’s secret:

It’s worth taking a moment right now to note that this also implies a serious vulnerability in bad implementations of the Schnorr protocol. If you ever accidentally use the same k for two different runs of the protocol, an attacker may be able to recover your secret key! This can happen if you use a bad random number generator.

Indeed, those with a bit more experience will notice that this is similar to a real attack on systems (with bad random number generators) that implement ECDSA or DSA signatures! This is also not a coincidence. The (EC)DSA signature family is based on Schnorr. Ironically, the developers of DSA managed to retain this vulnerability of the Schorr family of protocols while at the same time ditching the security proof that makes Schnorr so nice.

Proving zero-knowledge(ness) against an honest Verifier

Having demonstrated that Schnorr signatures are complete and sound, it remains only to prove that they’re ‘zero knowledge’. Remember that to do this, normally we require a Simulator that can interact with any possible Verifier and produce a ‘simulated’ transcript of the proof, even if the Simulator doesn’t know the secret it’s proving it knows.

The standard Schnorr protocol does not have such a Simulator, for reasons we’ll get into in a second. Instead, to make the proof work we need to make a special assumption. Specifically, the Verifier needs to be ‘honest’. That is, we need to make the special assumption that it will run its part of the protocol correctly — namely, that it will pick its challenge “c” using only its random number generator, and will not choose this value based on any input we provide it. As long as it does this, we can construct a Simulator.

Here’s how the Simulator works.

Let’s say we are trying to prove knowledge of a secret a for some public key g^a~mod~pbut we don’t actually know the value aOur Simulator assumes that the Verifier will choose some value c as its challenge, and moreover, it knows that the honest Verifier will choose the value c only based on its random number generator — and not based on any inputs the Prover has provided.

  1. First, output some initial g^{k_1} as the Prover’s first message, and find out what challenge c the Verifier chooses.
  2. Rewind the Verifier, and pick a random integer z in the range \{0,\dots,q-1\}.
  3. Compute g^{k_2} = g^z * g^{a (-c)} and output g^{k_2} as the Prover’s new initial message.
  4. When the Verifier challenges on c again, output z.
Notice that the transcript g^k, c, z will verify correctly as a perfectly valid, well-distributed proof of knowledge of the value a. The Verifier will accept this output as a valid proof of knowledge of a, even though the Simulator does not know a in the first place!
What this proves is that if we can rewind a Verifier, then (just as in the first post in this series) we can always trick the Verifier into believing we have knowledge of a value, even when we don’t. And since the statistical distribution of our protocol is identical to the real protocol, this means that our protocol must be zero knowledge — against an honest Verifier.

From interactive to non-interactive

So far we’ve shown how to use the Schnorr protocol to interactively prove knowledge of a secret key a that corresponds to a public key g^{a}. This is an incredibly useful protocol, but it only works if our Verifier is online and willing to interact with us.

An obvious question is whether we can make this protocol work without interaction. Specifically, can I make a proof that I can send you without you even being online. Such a proof is called a non-interactive zero knowledge proof (NIZK). Turning Schnorr into a non-interactive proof seems initially quite difficult — since the protocol fundamentally relies on the Verifier picking a random challenge. Fortunately there is a clever trick we can use.

This technique was developed by Fiat and Shamir in the 1980s. What they observed was that if you have a decent hash function lying around, you can convert an interactive protocol into a non-interactive one by simply using the hash function to pick the challenge.

Specifically, the revised protocol for proving knowledge of a with respect to a public key g^a looks like this:

  1. The Prover picks g^k (just as in the interactive protocol).
  2. Now, the prover computes the challenge as c = H(g^k || M) where H() is a hash function, and M is an (optional) and arbitary message string.
  3. Compute ac + k~mod~q (just as in the interactive protocol).

The upshot here is that the hash function is picking the challenge c without any interaction with the Verifier. In principle, if the hash function is “strong enough” (meaning, it’s a random oracle) then the result is a completely non-interactive proof of knowledge of the value a that the Prover can send to the Verifier. The proof of this is relatively straightforward.

The particularly neat thing about this protocol is that it isn’t just a proof of knowledge, it’s also a signature scheme. That is, if you put a message into the (optional) value M, you obtain a signature on M, which can only be produced by someone who knows the secret key a. The resulting protocol is called the Schnorr signature scheme, and it’s the basis of real-world protocols like EdDSA.

Phew.

Yes, this has been a long post and there’s probably a lot more to be said. Hopefully there will be more time for that in a third post — which should only take me another three years.
Notes:


* In this definition, it’s necessary that the statement be literally true.

The limitations of Android N Encryption

Over the past few years pixelphonewe’ve heard more about smartphone encryption than, quite frankly, most of us expected to hear in a lifetime. We learned that proper encryption can slow down even sophisticated decryption attempts if done correctly. We’ve also learned that incorrect implementations can undo most of that security.

In other words, phone encryption is an area where details matter. For the past few weeks I’ve been looking a bit at Android Nougat’s new file-based encryption to see how well they’ve addressed some of those details in their latest release. The answer, unfortunately, is that there’s still lots of work to do. In this post I’m going to talk about a bit of that.

(As an aside: the inspiration for this post comes from Grugq, who has been loudly and angrily trying to work through these kinks to develop a secure Android phone. So credit where credit is due.)

Background: file and disk encryption 

Disk encryption is much older than smartphones. Indeed, early encrypting filesystems date back at least to the early 1990s and proprietary implementations may go back before that. Even in the relatively new area of PCs operating systems, disk encryption has been a built-in feature since the early 2000s.

The typical PC disk encryption system operates as follows. At boot time you enter a password. This is fed through a key derivation function to derive a cryptographic key. If a hardware co-processor is available (e.g., a TPM), your key is further strengthened by “tangling” it with some secrets stored in the hardware. This helps to lock encryption to a particular device.

The actual encryption can be done in one of two different ways:

  1. Full Disk Encryption (FDE) systems (like TruecryptBitLocker and FileVault) encrypt disks at the level of disk sectors. This is an all-or-nothing approach, since the encryption drivers won’t necessarily have any idea what files those sectors represent. At the same time, FDE is popular — mainly because it’s extremely easy to implement.
  2. File-based Encryption (FBE) systems (like EncFS and eCryptFS) encrypt individual files. This approach requires changes to the filesystem itself, but has the benefit of allowing fine grained access controls where individual files are encrypted using different keys.

Most commercial PC disk encryption software has historically opted to use the full-disk encryption (FDE) approach. Mostly this is just a matter of expediency: FDE is just significantly easier to implement. But philosophically, it also reflects a particular view of what disk encryption was meant to accomplish.

In this view, encryption is an all-or-nothing proposition. Your machine is either on or off; accessible or inaccessible. As long as you make sure to have your laptop stolen only when it’s off, disk encryption will keep you perfectly safe.

So what does this have to do with Android?

Android’s early attempts at adding encryption to their phones followed the standard PC full-disk encryption paradigm. Beginning in Android 4.4 (Kitkat) through Android 6.0 (Marshmallow), Android systems shipped with a kernel device mapper called dm-crypt designed to encrypt disks at the sector level. This represented a quick and dirty way to bring encryption to Android phones, and it made sense — if you believe that phones are just very tiny PCs.

The problem is that smartphones are not PCs.

The major difference is that smartphone users are never encouraged to shut down their device. In practice this means that — after you enter a passcode once after boot — normal users spend their whole day walking around with all their cryptographic keys in RAM. Since phone batteries live for a day or more (a long time compared to laptops) encryption doesn’t really offer much to protect you against an attacker who gets their hands on your phone during this time.

Of course, users do lock their smartphones. In principle, a clever implementation could evict sensitive cryptographic keys from RAM when the device locks, then re-derive them the next time the user logs in. Unfortunately,  Android doesn’t do this — for the very simple reason that Android users want their phones to actually work. Without cryptographic keys in RAM, an FDE system loses access to everything on the storage drive. In practice this turns it into a brick.

For this very excellent reason, once you boot an Android FDE phone it will never evict its cryptographic keys from RAM. And this is not good.

So what’s the alternative?

Android is not the only game in town when it comes to phone encryption. Apple, for its part, also gave this problem a lot of thought and came to a subtly different solution.

Starting with iOS 4, Apple included a “data protection” feature to encrypt all data stored a device. But unlike Android, Apple doesn’t use the full-disk encryption paradigm. Instead, they employ a file-based encryption approach that individually encrypts each file on the device.

In the Apple system, the contents of each file is encrypted under a unique per-file key (metadata is encrypted separately). The file key is in turn encrypted with one of several “class keys” that are derived from the user passcode and some hardware secrets embedded in the processor.

apple
iOS data encryption. Source: iOS Security Guide.

The main advantage of the Apple approach is that instead of a single FDE key to rule them all, Apple can implement fine-grained access control for individual files. To enable this, iOS provides an API developers can use to specify which class key to use in encrypting any given file. The available “protection classes” include:

  • Complete protection. Files encrypted with this class key can only be accessed when the device is powered up and unlocked. To ensure this, the class key is evicted from RAM a few seconds after the device locks.
  • Protected Until First User Authentication. Files encrypted with this class key are protected until the user first logs in (after a reboot), and the key remains in memory.
  • No protection. These files are accessible even when the device has been rebooted, and the user has not yet logged in.

By giving developers the option to individually protect different files, Apple made it possible to build applications that can work while the device is locked, while providing strong protection for files containing sensitive data.

Apple even created a fourth option for apps that simply need to create new encrypted files when the class key has been evicted from RAM. This class uses public key encryption to write new files. This is why you can safely take pictures even when your device is locked.

Apple’s approach isn’t perfect. What it is, however, is the obvious result of a long and careful thought process. All of which raises the following question…

Why the hell didn’t Android do this as well?

The short answer is Android is trying to. Sort of. Let me explain.

As of Android 7.0 (Nougat), Google has moved away from full-disk encryption as the primary mechanism for protecting data at rest. If you set a passcode on your device, Android N systems can be configured to support a more Apple-like approach that uses file encryption. So far so good.

The new system is called Direct Boot, so named because it addresses what Google obviously saw as fatal problem with Android FDE — namely, that FDE-protected phones are useless bricks following a reboot. The main advantage of the new model is that it allows phones to access some data even before you enter the passcode. This is enabled by providing developers with two separate “encryption contexts”:

  • Credential encrypted storage. Files in this area are encrypted under the user’s passcode, and won’t be available until the user enters their passcode (once).
  • Device encrypted storage. These files are not encrypted under the user’s passcode (though they may be encrypted using hardware secrets). Thus they are available after boot, even before the user enters a passcode.

Direct Boot even provides separate encryption contexts for different users on the phone — something I’m not quite sure what to do with. But sure, why not?

If Android is making all these changes, what’s the problem?

One thing you might have noticed is that where Apple had four categories of protection, Android N only has two. And it’s the two missing categories that cause the problems. These are the “complete protection” categories that allow the user to lock their device following first user authentication — and evict the keys from memory.

Of course, you might argue that Android could provide this by forcing application developers to switch back to “device encrypted storage” following a device lock. The problem with this idea is twofold. First, Android documentation and sample code is explicit that this isn’t how things work:

credentialenc

Moreover, a quick read of the documentation shows that even if you wanted to, there is no unambiguous way for Android to tell applications when the system has been re-locked. If keys are evicted when the device is locked, applications will unexpectedly find their file accesses returning errors. Even system applications tend to do badly when this happens.

And of course, this assumes that Android N will even try to evict keys when you lock the device. Here’s how the current filesystem encryption code handles locks:

lockuserkey

While the above is bad, it’s important to stress that the real problem here is not really in the cryptography. The problem is that since Google is not giving developers proper guidance, the company may be locking Android into years of insecurity. Without (even a half-baked) solution to define a “complete” protection class, Android app developers can’t build their apps correctly to support the idea that devices can lock. Even if Android O gets around to implementing key eviction, the existing legacy app base won’t be able to handle it — since this will break a million apps that have implemented their security according to Android’s current recommendations.

In short: this is a thing you get right from the start, or you don’t do at all. It looks like — for the moment — Android isn’t getting it right.

Are keys that easy to steal?

Of course it’s reasonable to ask whether it’s having keys in RAM is that big of concern in the first place. Can these keys actually be accessed?

The answer to that question is a bit complicated. First, if you’re up against somebody with a hardware lab and forensic expertise, the answer is almost certainly “yes”. Once you’ve entered your passcode and derived the keys, they aren’t stored in some magically secure part of the phone. People with the ability to access RAM or the bus lines of the device can potentially nick them.

But that’s a lot of work. From a software perspective, it’s even worse. A software attack would require a way to get past the phone’s lockscreen in order to get running code on the device. In older (pre-N) versions of Android the attacker might need to then escalate privileges to get access to Kernel memory. Remarkably, Android N doesn’t even store its disk keys in the Kernel — instead they’re held by the “vold” daemon, which runs as user “root” in userspace. This doesn’t make exploits trivial, but it certainly isn’t the best way to handle things.

Of course, all of this is mostly irrelevant. The main point is that if the keys are loaded you don’t need to steal them. If you have a way to get past the lockscreen, you can just access files on the disk.

What about hardware?

Although a bit of a tangent, it’s worth noting that many high-end Android phones use some sort of trusted hardware to enable encryption. The most common approach is to use a trusted execution environment (TEE) running with ARM TrustZone.

This definitely solves a problem. Unfortunately it’s not quite the same problem as discussed above. ARM TrustZone — when it works correctly, which is not guaranteed — forces attackers to derive their encryption keys on the device itself, which should make offline dictionary attacks on the password much harder. In some cases, this hardware can be used to cache the keys and reveal them only when you input a biometric such as a fingerprint.

The problem here is that in Android N, this only helps you at the time the keys are being initially derived. Once that happens (i.e., following your first login), the hardware doesn’t appear to do much. The resulting derived keys seem to live forever in normal userspace RAM. While it’s possible that specific phones (e.g., Google’s Pixel, or Samsung devices) implement additional countermeasures, on stock Android N phones hardware doesn’t save you.

So what does it all mean?

How you feel about this depends on whether you’re a “glass half full” or “glass half empty” kind of person.

If you’re an optimistic type, you’ll point out that Android is clearly moving in the right direction. And while there’s a lot of work still to be done, even a half-baked implementation of file-based implementation is better than the last generation of dumb FDE Android encryption. Also: you probably also think clowns are nice.

On the other hand, you might notice that this is a pretty goddamn low standard. In other words, in 2016 Android is still struggling to deploy encryption that achieves (lock screen) security that Apple figured out six years ago. And they’re not even getting it right. That doesn’t bode well for the long term security of Android users.

And that’s a shame, because as many have pointed out, the users who rely on Android phones are disproportionately poorer and more at-risk. By treating encryption as a relatively low priority, Google is basically telling these people that they shouldn’t get the same protections as other users. This may keep the FBI off Google’s backs, but in the long term it’s bad judgement on Google’s part.

Attack of the week: 64-bit ciphers in TLS

A few months ago it was starting to seem like you couldn’t go a week without a new attack on TLS. In that context, this summer has been a blessed relief. Sadly, it looks like our vacation is over, and it’s time to go back to school.

Today brings the news that Karthikeyan Bhargavan and Gaëtan Leurent out of INRIA have a new paper that demonstrates a practical attack on legacy ciphersuites in TLS (it’s called “Sweet32”, website here). What they show is that ciphersuites that use 64-bit blocklength ciphers — notably 3DES — are vulnerable to plaintext recovery attacks that work even if the attacker cannot recover the encryption key.

While the principles behind this attack are well known, there’s always a difference between attacks in principle and attacks in practice. What this paper shows is that we really need to start paying attention to the practice.

So what’s the matter with 64-bit block ciphers?

Block ciphers are one of the most widely-used cryptographic primitives. As the nameimplies, these are schemes designed to encipher data in blocks, rather than a single bit at a time.

The two main parameters that define a block cipher are its block size (the number of bits it processes in one go), and its key size. The two parameters need not be related. So for example, DES has a 56-bit key and a 64-bit block. Whereas 3DES (which is built from DES) can use up to a 168-bit key and yet still has the same 64-bit block. More recent ciphers have opted for both larger blocks and larger keys.

When it comes to the security provided by a block cipher, the most important parameter is generally the key size. A cipher like DES, with its tiny 56-bit key, is trivially vulnerable to brute force attacks that attempt decryption with every possible key (often using specialized hardware). A cipher like AES or 3DES is generally not vulnerable to this sort of attack, since the keys are much longer.

However, as they say: key size is not everything. Sometimes the block size matters too.

You see, in practice, we often need to encrypt messages that are longer than a single block. We also tend to want our encryption to be randomized. To accomplish this, most protocols use a block cipher in a scheme called a mode of operation. The most popular mode used in TLS is CBC mode. Encryption in CBC looks like this:

Source: Wikipedia

The nice thing about CBC is that (leaving aside authentication issues) it can be proven (semantically) secure if we make various assumptions about the security of the underlying block cipher. Yet these security proofs have one important requirement. Namely, the attacker must not receive too much data encrypted with a single key.

The reason for this can be illustrated via the following simple attack.

Imagine that an honest encryptor is encrypting a bunch of messages using CBC mode. Following the diagram above, this involves selecting a random Initialization Vector (IV) of size equal to the block size of the cipher, then XORing IV with the first plaintext block (P), and enciphering the result (P \oplus IV). The IV is sent (in the clear) along with the ciphertext.

Most of the time, the resulting ciphertext block will be unique — that is, it won’t match any previous ciphertext block that an attacker may have seen. However, if the encryptor processes enough messages, sooner or later the attacker will see a collision. That is, it will see a ciphertext block that is the same as some previous ciphertext block. Since the cipher is deterministic, this means the cipher’s input (P \oplus IV) must be identical to the cipher’s previous input (P' \oplus IV') that created the previous block.

In other words, we have (P \oplus IV) = (P' \oplus IV'), which can be rearranged as (P \oplus P') = (IV \oplus IV'). Since the IVs are random and known to the attacker, the attacker has (with high probability) learned the XOR of two (unknown) plaintexts!

What can you do with the XOR of two unknown plaintexts? Well, if you happen to know one of those two plaintext blocks — as you might if you were able to choose some of the plaintexts the encryptor was processing — then you can easily recover the other plaintext. Alternatively, there are known techniques that can sometimes recover useful data even when you don’t know both blocks.

The main lesson here is that this entire mess only occurs if the attacker sees a collision. And the probability of such a collision is entirely dependent on the size of the cipher block. Worse, thanks to the (non-intuitive) nature of the birthday bound, this happens much more quickly than you might think it would. Roughly speaking, if the cipher block is b bits long, then we should expect a collision after roughly 2^{b/2} encrypted blocks.

In the case of a 64-bit blocksize cipher like 3DES, this is somewhere in the vicinity of 2^{32}, or around 4 billion enciphered blocks.

(As a note, the collision does not really need to occur in the first block. Since all blocks in CBC are calculated in the same way, it could be a collision anywhere within the messages.)

Whew. I thought this was a practical attack. 4 billion is a big number!

It’s true that 4 billion blocks seems like an awfully large number. In a practical attack, the requirements would be even larger — since the most efficient attack is for the attacker to know a lot of the plaintexts, in the hope that she will be able to recover one unknown plaintext when she learns the value (P ⊕ P’).

However, it’s worth keeping in mind that these traffic numbers aren’t absurd for TLS. In practice, 4 billion 3DES blocks works out to 32GB of raw ciphertext. A lot to be sure, but not impossible. If, as the Sweet32 authors do, we assume that half of the plaintext blocks are known to the attacker, we’d need to increase the amount of ciphertext to about 64GB. This is a lot, but not impossible.

The Sweet32 authors take this one step further. They imagine that the ciphertext consists of many HTTPS connections, consisting of 512 bytes of plaintext, in each of which is embedded the same secret 8-byte cookie — and the rest of the session plaintext is known. Calculating from these values, they obtain a requirement of approximately 256GB of ciphertext needed to recover the cookie with high probability.

That is really a lot.

But keep in mind that TLS connections are being used to encipher increasingly more data. Moreover, a single open browser frame running attacker-controlled Javascript can produce many gigabytes of ciphertext in a single hour. So these attacks are not outside of the realm of what we can run today, and presumably will be very feasible in the future.

How does the TLS attack work?

While the cryptographic community has been largely pushing TLS away from ciphersuites like CBC, in favor of modern authenticated modes of operation, these modes still exist in TLS. And they exist not only for use not only with modern ciphers like AES, but they are often available for older ciphersuites like 3DES. For example, here’s a connection I just made to Google:


Of course, just because a server supports 3DES does not mean that it’s vulnerable to this attack. In order for a particular connection to be vulnerable, both the client and server must satisfy three main requirements:

    1. The client and server must negotiate a 64-bit cipher. This is a relatively rare occurrence, but can happen in cases where one of the two sides is using an out-of-date client. For example, stock Windows XP does not support any of the AES-based ciphersuites. Similarly, SSL3 connections may negotiate 3DES ciphersuites.
    2. The server and client must support long-lived TLS sessions, i.e., encrypting a great deal of data with the same key. Unfortunately, most web browsers place no limit on the length of an HTTPS session if Keep-Alive is used, provided that the server allows the session. The Sweet32 authors scanned and discovered that many servers (including IIS) will allow sessions long enough to run their attack. Across the Internet, the percentage of vulnerable servers is small (less than 1%), but includes some important sites.
    3. The client must encipher a great deal of known data, including a secret session cookie. This is generally achieved by running adversarial Javascript code in the browser, although it could be done using standard HTML as well.

      Sites vulnerable to Sweet32. (source)

These caveats aside, the authors were able to run their attack using Firefox, sending at a rate of about 1500 connections per second. With a few optimizations, they were able to recover a 16-byte secret cookie in about 30 hours (a lucky result, given an expected 38 hour run time).The client must encipher a great deal of known data, including a secret session cookie. This is generally achieved by running adversarial Javascript code in the browser, although it could be done using standard HTML as well.

So what do we do now?

While this is not an earthshaking result, it’s roughly comparable to previous results we’ve seen with legacy ciphers like RC4.

In short, while these are not the easiest attacks to run, it’s a big problem that there even exist semi-practical attacks that undo the encryption used in standard encryption protocols. This is a problem that we should address, and these attack papers help to make those problems more clear.

Is Apple’s Cloud Key Vault a crypto backdoor?

TL;DR: No, it isn’t. If that’s all you wanted to know, you can stop reading.

Still, as you can see there’s been some talk on Twitter about the subject, and I’m afraid it could lead to a misunderstanding. That would be too bad, since Apple’s new technology is kind of a neat experiment.

So while I promise that this blog is not going to become all-Apple-all-the-time, I figured I’d take a minute to explain what I’m talking about. This post is loosely based on an explanation of Apple’s new escrow technology that Ivan Krstic gave at BlackHat. You should read the original for the fascinating details.

What is Cloud Key Vault (and what is iCloud Keychain)?

A few years ago Apple quietly introduced a new service called iCloud Keychain. This service is designed to allow you to back up your passwords and secret keys to the cloud. Now, if backing up your sensitive passwords gives you the willies, you aren’t crazy. Since these probably include things like bank and email passwords, you really want these to be kept extremely secure.

And — at least going by past experience — security is not where iCloud shines:

The problem here is that passwords need to be secured at a much higher assurance level than most types of data backup. But how can Apple ensure this? We can’t simply upload our secret passwords the way we upload photos of our kids. That would create a number of risks, including:

  1. The risk that someone will guess, reset or brute-force your iCloud password. Password resets are a particular problem. Unfortunately these seem necessary for normal iCloud usage, since people do forget their passwords. But that’s a huge risk when you’re talking about someone’s entire password collection.
  2. The risk that someone will break into Apple’s infrastructure. Even if Apple gets their front-end brute-forcing protections right (and removes password resets), the password vaults themselves are a huge target. You want to make sure that even someone who hacks Apple can’t get them out of the system.
  3. The risk that a government will compel Apple to produce data. Maybe you’re thinking of the U.S. government here. But that’s myopic: Apple stores iCloud data all over the world.

So clearly Apple needs a better way to protect these passwords. How do to it?

Why not just encrypt the passwords?

It is certainly possible for an Apple device to encrypt your password vault before sending it to iCloud. The problem here is that Apple doesn’t necessarily have a strong encryption key to do this with. Remember that the point of a backup is to survive the loss of your device, and thus we can’t assume the existence of a strong recovery key stored on your phone.

This leaves us with basically one option: a user password. This could be either the user’s iCloud password or their device passcode. Unfortunately for the typical user, these tend to be lousy. They may be strong enough to use as a login password — in a system that allows only a very limited number of login attempts. But the kinds of passwords typical users choose to enter on mobile devices are rarely strong enough to stand up to an offline dictionary attack, which is the real threat when using passwords as encryption keys.

(Even using a strong memory-hard password hash like scrypt — with crazy huge parameters — probably won’t save a user who chooses a crappy password. Blame phone manufacturers for making it painful to type in complicated passwords by forcing you to type them so often.)

So what’s Apple to do?

So Apple finds itself in a situation where they can’t trust the user to pick a strong password. They can’t trust their own infrastructure. And they can’t trust themselves. That’s a problem. Fundamentally, computer security requires some degree of trust — someone has to be reliable somewhere.

Apple’s solution is clever: they decided to make something more trustworthy than themselves. To create a new trust anchor, Apple purchased a bunch of fancy devices called Hardware Security Modules, or HSMs. These are sophisticated, tamper-resistant specialized computers that store and operate with cryptographic keys, while preventing even malicious users from extracting them. The high-end HSMs Apple uses also allow the owner to include custom programming.

Rather than trusting Apple, your phone encrypts its secrets under a hardcoded 2048-bit RSA public key that belongs to Apple’s HSM. It also encrypts a function of your device passcode, and sends the resulting encrypted blob to iCloud. Critically, only the HSM has a copy of the corresponding RSA decryption key, thus only the HSM can actually view any of this information. Apple’s network sees only an encrypted blob of data, which is essentially useless.

When a user wishes to recover their secrets, they authenticate themselves directly to the HSM. This is done using a user’s “iCloud Security Code” (iCSC), which is almost always your device passcode — something most people remember after typing it every day. This authentication is done using the Secure Remote Password protocol, ensuring that Apple (outside of the HSM) never sees any function of your password.

Now, I said that device passcodes are lousy secrets. That’s true when we’re talking about using them as encryption keys — since offline decryption attacks allow the attacker to make an unlimited number of attempts. However, with the assistance of an HSM, Apple can implement a common-sense countermeasure to such attacks: they limit you to a fixed number of login attempts. This is roughly the same protection that Apple implements on the devices themselves.

The encrypted contents of the data sent to the HSM (source).

The upshot of all these ideas is that — provided that the HSM works as designed, and that it can’t be reprogrammed — even Apple can’t access your stored data except by logging in with a correct passcode. And they only get a limited number of attempts to guess correctly, after which the account locks.

This rules out both malicious insiders and government access, with one big caveat.

What stops Apple from just reprogramming its HSM?

This is probably the biggest weakness of the system, and the part that’s driving the “backdoor’ concerns above. You see, the HSMs Apple uses are programmable. This means that — as long as Apple still has the code signing keys — the company can potentially update the custom code it includes onto the HSM to do all sort sorts of things.

These things might include: programming the HSM to output decrypted escrow keys. Or disabling the maximum login attempt counting mechanism. Or even inserting a program that runs a brute-force dictionary attack on the HSM itself. This would allow Apple to brute-force your passcode and/or recover your passwords.

Fortunately Apple has thought about this problem and taken steps to deal with it. Note that on HSMs like the one Apple is using, the code signing keys live on a special set of admin smartcards. To remove these keys as a concern, once Apple is done programming the HSM, they run these cards through a process that they call a “physical one-way hash function”.

If that sounds complicated, here’s Ivan’s slightly simpler explanation.

So, with the code signing keys destroyed, updating the HSM to allow nefarious actions should not be possible. Pretty much the only action Apple can take is to  wipe the HSM, which would destroy the HSM’s RSA secret keys and thus all of the encrypted records it’s responsible for. To make sure all admin cards are destroyed, the company has developed a complex ceremony for controlling the cards prior to their destruction. This mostly involves people making assertions that they haven’t made copies of the code signing key — which isn’t quite foolproof. But overall it’s pretty impressive.

The downside for Apple, of course, is that there had better not be a bug in any of their programming. Because right now there’s nothing they can do to fix it — except to wipe all of their HSMs and start over.

Couldn’t we use this idea to implement real crypto backdoors?

A key assertion I’ve heard is that if Apple can do this, then surely they can do something similar to escrow your keys for law enforcement. But looking at the system shows isn’t true at all.

To be sure, Apple’s reliance on a Hardware Security Module indicates a great deal of faith in a single hardware/software solution for storing many keys. Only time will tell if that faith is really justified. To be honest, I think it’s an overly-strong assumption. But iCloud Keychain is opt-in, so individuals can decide for themselves whether or not to take the risk. That wouldn’t be true of a mandatory law enforcement backdoor.

But the argument that Apple has enabled a law enforcement backdoor seems to miss what Apple has actually done. Instead of building a system that allows the company to recover your secret information, Apple has devoted enormous resources to locking themselves out. Only customers can access their own information. In other words, Apple has decided that the only way they can hold this information is if they don’t even trust themselves with it.

That’s radically different from what would be required to build a mandatory key escrow system for law enforcement. In fact, one of the big objections to such a backdoor — which my co-authors and I recently outlined in a report — is the danger that any of the numerous actors in such a system could misuse it. By eliminating themselves from the equation, Apple has effectively neutralized that concern.

If Apple can secure your passwords this way, then why don’t they do the same for your backed up photos, videos, and documents?

That’s a good question. Maybe you should ask them?

Statement on DMCA lawsuit

My name is Matthew Green. I am a professor of computer science and a researcher at Johns Hopkins University in Baltimore. I focus on computer security and applied cryptography.

Today I filed a lawsuit against the U.S. government, to strike down Section 1201 of the Digital Millennium Copyright Act. This law violates my First Amendment right to gather information and speak about an urgent matter of public concern: computer security. I am asking a federal judge to strike down key parts of this law so they cannot be enforced against me or anyone else.

A large portion of my work involves building and analyzing the digital security systems that make our modern technological world possible. These include security systems like the ones that protect your phone calls, instant messages, and financial transactions – as well as more important security mechanisms that safeguard property and even human life.

I focus a significant portion of my time on understanding the security systems that have been deployed by industry. In 2005, my team found serious flaws in the automotive anti-theft systems used in millions of Ford, Toyota and Nissan vehicles. More recently, my co-authors and I uncovered flaws in the encryption that powers nearly one third of the world’s websites, including Facebook and the National Security Agency. Along with my students, I’ve identified flaws in Apple’s iMessage text messaging system that could have allowed an eavesdropper to intercept your communications. And these are just a sampling of the public research projects I’ve been involved with.

I don’t do this work because I want to be difficult. Like most security researchers, the research I do is undertaken in good faith. When I find a flaw in a security system, my first step is to call the organization responsible. Then I help to get the flaw fixed. Such independent security research is an increasingly precious commodity. For every security researcher who investigates systems in order to fix them, there are several who do the opposite – and seek to profit from the insecurity of the computer systems our society depends on.

There’s a saying that no good deed goes unpunished. The person who said this should have been a security researcher. Instead of welcoming vulnerability reports, companiesroutinely threaten good-faith security researchers with civil action, or even criminal prosecution. Companies use the courts to silence researchers who have embarrassing things to say about their products, or who uncover too many of those products’ internal details. These attempts are all too often successful, in part because very few security researchers can afford a prolonged legal battle with well-funded corporate legal team.

This might just be a sad story about security researchers, except for the fact that these vulnerabilities affect everyone. When security researchers are intimidated, it’s the public that pays the price. This is because real criminals don’t care about lawsuits and intimidation – and they certainly won’t bother to notify the manufacturer. If good-faith researchers aren’t allowed to find and close these holes, then someone else will find them, walk through them, and abuse them.

In the United States, one of the most significant laws that blocks security researchers is  Section 1201 of the Digital Millennium Copyright Act (DMCA). This 1998 copyright law instituted a raft of restrictions aimed at preventing the “circumvention of copyright protection systems.” Section 1201 provides both criminal and civil penalties for people who bypass technological measures protecting a copyrighted work. While that description might bring to mind the copy protection systems that protect a DVD or an iTunes song, the law has also been applied to prevent users from reverse-engineering software to figure out how it works. Such reverse-engineering is a necessary party of effective security research.

Section 1201 poses a major challenge for me as a security researcher. Nearly every attempt to analyze a software-based system presents a danger of running afoul of the law. As a result, the first step in any research project that involves a commercial system is never science – it’s to call a lawyer; to ask my graduate students to sign a legal retainer; and to inform them that even with the best legal advice, they still face the possibility of being sued and losing everything they have. This fear chills critical security research.

Section 1201 also affects the way that my research is conducted. In a recent project – conducted in Fall 2015 – we were forced to avoid reverse-engineering a piece of software when it would have been the fastest and most accurate way to answer a research question. Instead, we decided to treat the system as a black box, recovering its operation only by observing inputs and outputs. This approach often leads to a less perfect understanding of the system, which can greatly diminish the quality of security research. It also substantially increases the time and effort required to finish a project, which reduces the quantity of security research.

Finally, I have been luckier than most security researchers in that I have access to legal assistance from organizations such as the Electronic Frontier Foundation. Not every security researcher can benefit from this.

The risk imposed by Section 1201 and the heavy cost of steering clear of it discourage me – and other researchers — from pursuing any project that does not appear to have an overwhelming probability of success. This means many projects that would yield important research and protect the public simply do not happen.

In 2015, I filed a request with the Library of Congress for a special exemption that would have exempted good faith security researchers from the limitations of Section 1201. Representatives of the major automobile manufacturers and the Business Software Alliance (a software industry trade group) vigorously opposed the request. This indicates to me that even reasonable good faith security testing is still a risky proposition.

This risk is particularly acute given that the exemption we eventually won was much more limited than what we asked for, and leaves out many of the technologies with the greatest impact on public health, privacy, and the security of financial transactions.

Section 1201 has prevented crucial security research for far too long. That’s why I’m seeking a court order that would strike Section 1201 from the books as a violation of the First Amendment.

What is Differential Privacy?

Yesterday at the WWDC keynote, Apple announced a series of new security and privacy features, including one feature that’s drawn a bit of attention — and confusion. Specifically, Apple announced that they will be using a technique called “Differential Privacy” (henceforth: DP) to improve the privacy of their data collection practices.

The reaction to this by most people has been a big “???”, since few people have even heard of Differential Privacy, let alone understand what it means. Unfortunately Apple isn’t known for being terribly open when it comes to sharing the secret sauce that drives their platform, so we’ll just have to hope that at some point they decide to publish more. What we know so far comes from Apple’s iOS 10 Preview guide:

Starting with iOS 10, Apple is using Differential Privacy technology to help discover the usage patterns of a large number of users without compromising individual privacy. To obscure an individual’s identity, Differential Privacy adds mathematical noise to a small sample of the individual’s usage pattern. As more people share the same pattern, general patterns begin to emerge, which can inform and enhance the user experience. In iOS 10, this technology will help improve QuickType and emoji suggestions, Spotlight deep link suggestions and Lookup Hints in Notes.

To make a long story short, it sounds like Apple is going to be collecting a lot more data from your phone. They’re mainly doing this to make their services better, not to collect individual users’ usage habits. To guarantee this, Apple intends to apply sophisticated statistical techniques to ensure that this aggregate data — the statistical functions it computes over all your information — don’t leak your individual contributions. In principle this sounds pretty good. But of course, the devil is always in the details.

While we don’t have those details, this seems like a good time to at least talk a bit about what Differential Privacy is, how it can be achieved, and what it could mean for Apple — and for your iPhone.

The motivation

In the past several years, “average people” have gotten used to the idea that they’re sending a hell of a lot of personal information to the various services they use. Surveys also tell us they’re starting to feel uncomfortable about it.

This discomfort makes sense when you think about companies using our personal data to market (to) us. But sometimes there are decent motivations for collecting usage information. For example, Microsoft recently announced a tool that can diagnose pancreatic cancer by monitoring your Bing queries. Google famously runs Google Flu Trends. And of course, we all benefit from crowdsourced data that improves the quality of the services we use — from mapping applications to restaurant reviews.

Unfortunately, even well-meaning data collection can go bad. For example, in the late 2000s, Netflix ran a competition to develop a better film recommendation algorithm. To drive the competition, they released an “anonymized” viewing dataset that had been stripped of identifying information. Unfortunately, this de-identification turned out to be insufficient. In a well-known piece of work, Narayanan and Shmatikov showed that such datasets could be used to re-identify specific users — and even predict their political affiliation! — if you simply knew a little bit of additional information about a given user.

This sort of thing should be worrying to us. Not just because companies routinely share data (though they do) but because breaches happen, and because even statistics about a dataset can sometimes leak information about the individual records used to compute it. Differential Privacy is a set of tools that was designed to address this problem.

What is Differential Privacy?

Differential Privacy is a privacy definition that was originally developed by Dwork, Nissim, McSherry and Smith, with major contributions by many others over the years. Roughly speaking, what it states can summed up intuitively as follows:

Imagine you have two otherwise identical databases, one with your information in it, and one without it. Differential Privacy ensures that the probability that a statistical query will produce a given result is (nearly) the same whether it’s conducted on the first or second database.

One way to look at this is that DP provides a way to know if your data has a significant effect on the outcome of a query. If it doesn’t, then you might as well contribute to the database — since there’s almost no harm that can come of it. Consider a silly example:

Imagine that you choose to enable a reporting feature on your iPhone that tells Apple if you like to use the 💩  emoji routinely in your iMessage conversations. This report consists of a single bit of information: 1 indicates you like 💩 , and 0 doesn’t. Apple might receive these reports and fill them into a huge database. At the end of the day, it wants to be able to derive a count of the users who like this particular emoji.

It goes without saying that the simple process of “tallying up the results” and releasing them does not satisfy the DP definition, since computing a sum on the database that contains your information will potentially produce a different result from computing the sum on a database without it. Thus, even though these sums may not seem to leak much information, they reveal at least a little bit about you. A key observation of the Differential Privacy research is that in many cases, DP can be achieved if the tallying party is willing to add random noise to the result. For example, rather than simply reporting the sum, the tallying party can inject noise from a Laplace or gaussian distribution, producing a result that’s not quite exact — but that masks the contents of any given row. (For other interesting functions, there are many other techniques as well.)

Even more usefully, the calculation of “how much” noise to inject can be made without knowing the contents of the database itself (or even its size). That is, the noise calculation can be performed based only on knowledge of the function to be computed, and the acceptable amount of data leakage.

A tradeoff between privacy and accuracy

Now obviously calculating the total number of 💩 -loving users on a system is a pretty silly example. The neat thing about DP is that the same overall approach can be applied to much more interesting functions, including complex statistical calculations like the ones used by Machine Learning algorithms. It can even be applied when many different functions are all computed over the same database.

But there’s a big caveat here. Namely, while the amount of “information leakage” from a single query can be bounded by a small value, this value is not zero. Each time you query the database on some function, the total “leakage” increases — and can never go down. Over time, as you make more queries, this leakage can start to add up.

This is one of the more challenging aspects of DP. It manifests in two basic ways:

  1. The more information you intend to “ask” of your database, the more noise has to be injected in order to minimize the privacy leakage. This means that in DP there is generally a fundamental tradeoff between accuracy and privacy, which can be a big problem when training complex ML models.
  2. Once data has been leaked, it’s gone. Once you’ve leaked as much data as your calculations tell you is safe, you can’t keep going — at least not without risking your users’ privacy. At this point, the best solution may be to just to destroy the database and start over. If such a thing is possible.

The total allowed leakage is often referred to as a “privacy budget”, and it determines how many queries will be allowed (and how accurate the results will be). The basic lesson of DP is that the devil is in the budget. Set it too high, and you leak your sensitive data. Set it too low, and the answers you get might not be particularly useful.

Now in some applications, like many of the ones on our iPhones, the lack of accuracy isn’t a big deal. We’re used to our phones making mistakes. But sometimes when DP is applied in complex applications, such as training Machine Learning models, this really does matter.

Mortality vs. info disclosure, from Frederikson et al.
The red line is partient mortality.

To give an absolutely crazy example of how big the tradeoffs can be, consider this paper by Frederikson et al. from 2014. The authors began with a public database linking Warfarin dosage outcomes to specific genetic markers. They then used ML techniques to develop a dosing model based on their database — but applied DP at various privacy budgets while training the model. Then they evaluated both the information leakage and the model’s success at treating simulated “patients”.

The results showed that the model’s accuracy depends a lot on the privacy budget on which it was trained. If the budget is set too high, the database leaks a great deal of sensitive patient information — but the resulting model makes dosing decisions that are about as safe as standard clinical practice. On the other hand, when the budget was reduced to a level that achieved meaningful privacy, the “noise-ridden” model had a tendency to kill its “patients”.

Now before you freak out, let me be clear: your iPhone is not going to kill you. Nobody is saying that this example even vaguely resembles what Apple is going to do on the phone. The lesson of this research is simply that there are interesting tradeoffs between effectiveness and the privacy protection given by any DP-based system — these tradeoffs depend to a great degree on specific decisions made by the system designers, the parameters chosen by the deploying parties, and so on. Hopefully Apple will soon tell us what those choices are.

How do you collect the data, anyway?

You’ll notice that in each of the examples above, I’ve assumed that queries are executed by a trusted database operator who has access to all of the “raw” underlying data. I chose this model because it’s the traditional model used in most of the literature, not because it’s a particularly great idea.

In fact, it would be worrisome if Apple was actually implementing their system this way. That would require Apple to collect all of your raw usage information into a massive centralized database, and then (“trust us!”) calculate privacy-preserving statistics on it. At a minimum this would make your data vulnerable to subpoenas, Russian hackers, nosy Apple executives and so on.

Fortunately this is not the only way to implement a Differentially Private system. On the theoretical side, statistics can be computed using fancy cryptographic techniques (such as secure multi-party computation or fully-homomorphic encryption.) Unfortunately these techniques are probably too inefficient to operate at the kind of scale Apple needs.

A much more promising approach is not to collect the raw data at all. This approach was recently pioneered by Google to collect usage statistics in their Chrome browser. The system, called RAPPOR, is based on an implementation of the 50-year old randomized response technique. Randomized response works as follows:

  1. When a user wants to report a piece of potentially embarrassing information (made up example: “Do you use Bing?”), they first flip a coin, and if the coin comes up “heads”, they return a random answer — calculated by flipping a second coin. Otherwise they answer honestly.
  2. The server then collects answers from the entire population, and (knowing the probability that the coins will come up “heads”), adjusts for the included “noise” to compute an approximate answer for the true response rate.

Intuitively, randomized response protects the privacy of individual user responses, because a “yes” result could mean that you use Bing, or it could just be the effect of the first mechanism (the random coin flip). More formally, randomized response has been shown to achieve Differential Privacy, with specific guarantees that can adjusted by fiddling with the coin bias.

 I’ve met Craig Federighi. He actually
looks like this in person.
RAPPOR takes this relatively old technique and turns it into something much more powerful. Instead of simply responding to a single question, it can report on complex vectors of questions, and may even return complicated answers, such as strings — e.g., which default homepage you use. The latter is accomplished by first encoding the string into a Bloom filter — a bitstring constructed using hash functions in a very specific way. The resulting bits are then injected with noise, and summed, and the answers recovered using a (fairly complex) decoding process.

While there’s no hard evidence that Apple is using a system like RAPPOR, there are some small hints. For example, Apple’s Craig Federighi describes Differential Privacy as using hashing, subsampling and noise injection to enable…crowdsourced learning while keeping the data of individual users completely private.” That’s pretty weak evidence for anything, admittedly, but presence of the “hashing” in that quote at least hints towards the use of RAPPOR-like filters.

The main challenge with randomized response systems is that they can leak data if a user answers the same question multiple times. RAPPOR tries to deal with this in a variety of ways, one of which is to identify static information and thus calculate “permanent answers” rather than re-randomizing each time. But it’s possible to imagine situations where such protections could go wrong. Once again, the devil is very much in the details — we’ll just have to see. I’m sure many fun papers will be written either way.

So is Apple’s use of DP a good thing or a bad thing?

As an academic researcher and a security professional, I have mixed feelings about Apple’s announcement. On the one hand, as a researcher I understand how exciting it is to see research technology actually deployed in the field. And Apple has a very big field.

On the flipside, as security professionals it’s our job to be skeptical — to at a minimum demand people release their security-critical code (as Google did with RAPPOR), or at least to be straightforward about what it is they’re deploying. If Apple is going to collect significant amounts of new data from the devices that we depend on so much, we should really make sure they’re doing it right — rather than cheering them for Using Such Cool Ideas. (I made this mistake already once, and I still feel dumb about it.)

But maybe this is all too “inside baseball”. At the end of the day, it sure looks like Apple is honestly trying to do something to improve user privacy, and given the alternatives, maybe that’s more important than anything else.

Attack of the Week: Apple iMessage

Today’s Washington Post has a story entitled “Johns Hopkins researchers poke a hole in Apple’s encryption“, which describes the results of some research my students and I have been working on over the past few months.

As you might have guessed from the headline, the work concerns Apple, and specifically Apple’s iMessage text messaging protocol. Over the past months my students Christina Garman, Ian Miers, Gabe Kaptchuk and Mike Rushanan and I have been looking closely at the encryption used by iMessage, in order to determine how the system fares against sophisticated attackers. The results of this analysis include some very neat new attacks that allow us to — under very specific circumstances — decrypt the contents of iMessage attachments, such as photos and videos.

The research team. From left: Gabe Kaptchuk, Mike Rushanan, Ian Miers, Christina Garman

Now before I go further, it’s worth noting that the security of a text messaging protocol may not seem like the most important problem in computer security. And under normal circumstances I might agree with you. But today the circumstances are anything but normal: encryption systems like iMessage are at the center of a critical national debate over the role of technology companies in assisting law enforcement.

A particularly unfortunate aspect of this controversy has been the repeated call for U.S. technology companies to add “backdoors” to end-to-end encryption systems such as iMessage. I’ve always felt that one of the most compelling arguments against this approach — an argument I’ve made along with other colleagues — is that we just don’t know how to construct such backdoors securely. But lately I’ve come to believe that this position doesn’t go far enough — in the sense that it is woefully optimistic. The fact of the matter is that forget backdoors: we barely know how to make encryption work at all. If anything, this work makes me much gloomier about the subject.

But enough with the generalities. The TL;DR of our work is this:

Apple iMessage, as implemented in versions of iOS prior to 9.3 and Mac OS X prior to 10.11.4, contains serious flaws in the encryption mechanism that could allow an attacker — who obtains iMessage ciphertexts — to decrypt the payload of certain attachment messages via a slow but remote and silent attack, provided that one sender or recipient device is online. While capturing encrypted messages is difficult in practice on recent iOS devices, thanks to certificate pinning, it could still be conducted by a nation state attacker or a hacker with access to Apple’s servers. You should probably patch now.

For those who want the gory details, I’ll proceed with the rest of this post using the “fun” question and answer format I save for this sort of post.

What is Apple iMessage and why should I care?

Those of you who read this blog will know that I have a particular obsession with Apple iMessage. This isn’t because I’m weirdly obsessed with Apple — although it is a little bit because of that. Mostly it’s because I think iMessage is an important protocol. The text messaging service, which was introduced in 2011, has the distinction of being the first widely-used end-to-end encrypted text messaging system in the world.

To understand the significance of this, it’s worth giving some background. Before iMessage, the vast majority of text messages were sent via SMS or MMS, meaning that they were handled by your cellular provider. Although these messages are technically encrypted, this encryption exists only on the link between your phone and the nearest cellular tower. Once an SMS reaches the tower, it’s decrypted, then stored and delivered without further protection. This means that your most personal messages are vulnerable to theft by telecom employees or sophisticated hackers. Worse, many U.S. carriers still use laughably weak encryption and protocols that are vulnerable to active interception.

So from a security point of view, iMessage was a pretty big deal. In a single stroke, Apple deployed encrypted messaging to millions of users, ensuring (in principle) that even Apple itself couldn’t decrypt their communications. The even greater accomplishment was that most people didn’t even notice this happened — the encryption was handled so transparently that few users are aware of it. And Apple did this at very large scale: today, iMessage handles peak throughput of more than 200,000 encrypted messages per second, with a supported base of nearly one billion devices.

So iMessage is important. But is it any good?

Answering this question has been kind of a hobby of mine for the past couple of years. In the past I’ve written about Apple’s failure to publish the iMessage protocol, and on iMessage’s dependence on a vulnerable centralized key server. Indeed, the use of a centralized key server is still one of iMessage’s biggest weaknesses, since an attacker who controls the keyserver can use it to inject keys and conduct man in the middle attacks on iMessage users.

But while key servers are a risk, attacks on a key server seem fundamentally challenging to implement — since they require the ability to actively manipulate Apple infrastructure without getting caught. Moreover, such attacks are only useful for prospective surveillance. If you fail to substitute a user’s key before they have an interesting conversation, you can’t recover their communications after the fact.A more interesting question is whether iMessage’s encryption is secure enough to stand up against retrospective decryption attacks — that is, attempts to decrypt messages after they have been sent. Conducting such attacks is much more interesting than the naive attacks on iMessage’s key server, since any such attack would require the existence of a fundamental vulnerability in iMessage’s encryption itself. And in 2016 encryption seems like one of those things that we’ve basically figured out how to get right.

Which means, of course, that we probably haven’t.

How does iMessage encryption work?

What we know about the iMessage encryption protocol comes from a previous reverse-engineering effort by a group from Quarkslab, as well as from Apple’s iOS Security Guide. Based on these sources, we arrive at the following (simplified) picture of the basic iMessage encryption scheme:

To encrypt an iMessage, your phone first obtains the RSA public key of the person you’re sending to. It then generates a random AES key k and encrypts the message with that key using CTR mode. Then it encrypts k using the recipient’s RSA key. Finally, it signs the whole mess using the sender’s ECDSA signing key. This prevents tampering along the way.

So what’s missing here?

Well, the most obviously missing element is that iMessage does not use a Message Authentication Code (MAC) or authenticated encryption scheme to prevent tampering with the message. To simulate this functionality, iMessage simply uses an ECDSA signature formulated by the sender. Naively, this would appear to be good enough. Critically, it’s not.

The attack works as follows. Imagine that a clever attacker intercepts the message above and is able to register her own iMessage account. First, the attacker strips off the original ECDSA signature made by the legitimate sender, and replaces it with a signature of her own. Next, she sends the newly signed message to the original recipient using her own account:

The outcome is that the user receives and decrypts a copy of the message, which has now apparently originated from the attacker rather than from the original sender. Ordinarily this would be a pretty mild attack — but there’s a useful wrinkle. In replacing the sender’s signature with one of her own, the attacker has gained a powerful capability. Now she can tamper with the AES ciphertext (red) at will.

Specifically, since in iMessage the AES ciphertext is not protected by a MAC, it is therefore malleable. As long as the attacker signs the resulting message with her key, she can flip any bits in the AES ciphertext she wants — and this will produce a corresponding set of changes when the recipient ultimately decrypts the message. This means that, for example, if the attacker guesses that the message contains the word “cat” at some position, she can flip bits in the ciphertext to change that part of the message to read “dog” — and she can make this change even though she can’t actually read the encrypted message.

Only one more big step to go.

Now further imagine that the recipient’s phone will decrypt the message correctly provided that the underlying plaintext that appears following decryption is correctly formatted. If the plaintext is improperly formatted — for a silly example, our tampering made it say “*7!” instead of “pig” — then on receiving the message, the recipient’s phone might return an error that the attacker can see.

It’s well known that such a configuration capability allows our attacker the ability to learn information about the original message, provided that she can send many “mauled” variants to be decrypted. By mauling the underlying message in specific ways — e.g., attempting to turn “dog” into “pig” and observing whether decryption succeeds — the attacker can gradually learn the contents of the original message. The technique is known as a format oracle, and it’s similar to the padding oracle attack discovered by Vaudenay.

So how exactly does this format oracle work?

The format oracle in iMessage is not a padding oracle. Instead it has to do with the compression that iMessage uses on every message it sends.

You see, prior to encrypting each message payload, iMessage applies a complex formatting that happens to conclude with gzip compression. Gzip is a modestly complex compression scheme that internally identifies repeated strings, applies Huffman coding, then tacks a CRC checksum computed over the original data at the end of the compressed message. It’s this gzip-compressed payload that’s encrypted within the AES portion of an iMessage ciphertext.

It turns out that given the ability to maul a gzip-compressed, encrypted ciphertext, there exists a fairly complicated attack that allows us to gradually recover the contents of the message by mauling the original message thousands of times and sending the modified versions to be decrypted by the target device. The attack turns on our ability to maul the compressed data by flipping bits, then “fix up” the CRC checksum correspondingly so that it reflects the change we hope to see in the uncompressed data. Depending on whether that test succeeds, we can gradually recover the contents of a message — one byte at a time.

While I’m making this sound sort of simple, the truth is it’s not. The message is encoded using Huffman coding, with a dynamic Huffman table we can’t see — since it’s encrypted. This means we need to make laser-specific changes to the ciphertext such that we can predict the effect of those changes on the decrypted message, and we need to do this blind. Worse, iMessage has various countermeasures that make the attack more complex.The complete details of the attack appear in the paper, and they’re pretty eye-glazing, so I won’t repeat them here. In a nutshell, we are able to decrypt a message under the following conditions:

  1. We can obtain a copy of the encrypted message
  2. We can send approximately 2^18 (invisible) encrypted messages to the target device
  3. We can determine whether or not those messages decrypted successfully or not
The first condition can be satisfied by obtaining ciphertexts from a compromise of Apple’s Push Notification Service servers (which are responsible for routing encrypted iMessages) or by intercepting TLS connections using a stolen certificate — something made more difficult due to the addition of certificate pinning in iOS 9. The third element is the one that initially seems the most challenging. After all, when I send an iMessage to your device, there’s no particular reason that your device should send me any sort of response when the message decrypts. And yet this information is fundamental to conducting the attack!

It turns out that there’s a big exception to this rule: attachment messages.

How do attachment messages differ from normal iMessages?

When I include a photo in an iMessage, I don’t actually send you the photograph through the normal iMessage channel. Instead, I first encrypt that photo using a random 256-bit AES key, then I compute a SHA1 hash and upload the encrypted photo to iCloud. What I send you via iMessage is actually just an iCloud.com URL to the encrypted photo, the SHA1 hash, and the decryption key.

Contents of an “attachment” message.

 

When you successfully receive and decrypt an iMessage from some recipient, your Messages client will automatically reach out and attempt to download that photo. It’s this download attempt, which happens only when the phone successfully decrypts an attachment message, that makes it possible for an attacker to know whether or not the decryption has succeeded.

One approach for the attacker to detect this download attempt is to gain access to and control your local network connections. But this seems impractical. A more sophisticated approach is to actually maul the URL within the ciphertext so that rather than pointing to iCloud.com, it points to a related URL such as i8loud.com. Then the attacker can simply register that domain, place a server there and allow the client to reach out to it. This requires no access to the victim’s local network.

By capturing an attachment message, repeatedly mauling it, and monitoring the download attempts made by the victim device, we can gradually recover all of the digits of the encryption key stored within the attachment. Then we simply reach out to iCloud and download the attachment ourselves. And that’s game over. The attack is currently quite slow — it takes more than 70 hours to run — but mostly because our code is slow and not optimized. We believe with more engineering it could be made to run in a fraction of a day.

Result of decrypting the AES key for an attachment. Note that the ? symbol represents a digit we could not recover for various reasons, typically due to string repetitions. We can brute-force the remaining digits.

The need for an online response is why our attack currently works against attachment messages only: those are simply the messages that make the phone do visible things. However, this does not mean the flaw in iMessage encryption is somehow limited to attachments — it could very likely be used against other iMessages, given an appropriate side-channel.

How is Apple fixing this?

Apple’s fixes are twofold. First, starting in iOS 9.0 (and before our work), Apple began deploying aggressive certificate pinning across iOS applications. This doesn’t fix the attack on iMessage crypto, but it does make it much harder for attackers to recover iMessage ciphertexts to decrypt in the first place.

Unfortunately even if this works perfectly, Apple still has access to iMessage ciphertexts. Worse, Apple’s servers will retain these messages for up to 30 days if they are not delivered to one of your devices. A vulnerability in Apple Push Network authentication, or a compromise of these servers could read them all out. This means that pinning is only a mitigation, not a true fix.

As of iOS 9.3, Apple has implemented a short-term mitigation that my student Ian Miers proposed. This relies on the fact that while the AES ciphertext is malleable, the RSA-OAEP portion of the ciphertext is not. The fix maintains a “cache” of recently received RSA ciphertexts and rejects any repeated ciphertexts. In practice, this shuts down our attack — provided the cache is large enough. We believe it probably is.

In the long term, Apple should drop iMessage like a hot rock and move to Signal/Axolotl.

So what does it all mean?

As much as I wish I had more to say, fundamentally, security is just plain hard. Over time we get better at this, but for the foreseeable future we’ll never be ahead. The only outcome I can hope for is that people realize how hard this process is — and stop asking technologists to add unacceptable complexity to systems that already have too much of it.