On Monday a team of researchers from Münster, RUB and NXP disclosed serious cryptographic vulnerabilities in a number of encrypted email clients. The flaws, which go by the cute vulnerability name of “Efail”, potentially allow an attacker to decrypt S/MIME or PGP-encrypted email with only minimal user interaction.
By the standards of cryptographic vulnerabilities, this is about as bad as things get. In short: if an attacker can intercept and alter an encrypted email — say, by sending you a new (altered) copy, or modifying a copy stored on your mail server — they can cause many GUI-based email clients to send the full plaintext of the email to an attacker controlled-server. Even worse, most of the basic problems that cause this flaw have been known for years, and yet remain in clients.
The big (and largely under-reported) story of EFail is the way it affects S/MIME. That “corporate” email protocol is simultaneously (1) hated by the general crypto community because it’s awful and has a slash in its name, and yet (2) is probably the most widely-used email encryption protocol in the corporate world. The table at the right — excerpted from the paper — gives you a flavor of how Efail affects S/MIME clients. TL;DR it affects them very badly.
Efail also happens to affect a smaller, but non-trivial number of OpenPGP-compatible clients. As one might expect (if one has spent time around PGP-loving folks) the disclosure of these vulnerabilities has created something of a backlash on HN, and among people who make and love OpenPGP clients. Mostly for reasons that aren’t very defensible.
So rather than write about fun things — like the creation of CFB and CBC gadgets — today, I’m going to write about something much less exciting: the problem of vulnerabilitydisclosure in ecosystems like PGP. And how bad reactions to disclosure can hurt us all.
How Efail was disclosed to the PGP community
Putting together a comprehensive timeline of the Efail disclosure process would probably be a boring, time-intensive project. Fortunately Thomas Ptacek loves boring and time-intensive projects, and has already done this for us.
Briefly, the first Efail disclosures to vendors began last October, more than 200 days prior to the agreed publication date. The authors notified a large number of vulnerable PGP GUI clients, and also notified the GnuPG project (on which many of these projects depend) by February at the latest. From what I can tell every major vendor agreed to make some kind of patch. GnuPG decided that it wasn’t their fault, and basically stopped corresponding.
All parties agreed not to publicly discuss the vulnerability until an agreed date in April, which was later pushed back to May 15. The researchers also notified the EFF and some journalists under embargo, but none of them leaked anything. On May 14 someone dumped the bug onto a mailing list. So the EFF posted a notice about the vulnerability (which we’ll discuss a bit more below), and the researchers put up a website. That’s pretty much the whole story.
There are three basic accusations going around about the Efail disclosure. They can be summarized as (1) maintaining embargoes in coordinated disclosures is really hard, (2) the EFF disclosure “unfairly” made this sound like a serious vulnerability “when it isn’t”, and (3) everything was already patched anyway so what’s the big deal.
Disclosures are hard; particularly coordinated ones
I’ve been involved in two disclosures of flaws in open encryption protocols. (Both were TLS issues.) Each one poses an impossible dilemma. You need to simultaneously (a) make sure every vendor has as much advance notice as possible, so they can patch their software. But at the same time (b)you need to avoid tellingliterallyanyone, because nothing on the Internet stays secret. At some point you’ll notify some FOSS project that uses an open development mailing list or ticket server, and the whole problem will leak out into the open.
Disclosing bugs that affect PGP is particularly fraught. That’s because there’s no such thing as “PGP”. What we have instead is a large and distributed community that revolves around the OpenPGP protocol. The pillar of this community is the GnuPG project, which maintains the core GnuPG tool and libraries that many clients rely on. Then there are a variety of niche GUI-based clients and email plugin projects. Finally, there are commercial vendors like Apple and Microsoft. (Who are mostly involved in the S/MIME side of things, and may reluctantly allow PGP plugins.)
Then, of course there are thousands of end-users, who will generally fail to update their software unless something really bad and newsworthy happens.
The obvious solution to the disclosure problem to use a staged disclosure. You notify the big commercial vendors first, since that’s where most of the affected users are. Then you work your way down the “long tail” of open source projects, knowing that inevitably the embargo could break and everyone will have to patch in a hurry. And you keep in mind that no matter what happens, everyone will blame you for screwing up the disclosure.
For the PGP issues in Efail, the big client vendors are Mozilla (Thunderbird), Microsoft (Outlook) and maybe Apple (Mail). The very next obvious choice would be to patch the GnuPG tool so that it no longer spits out unauthenticated plaintext, which is the root of many of the problems in Efail.
The Efail team appears to have pursued exactly this approach for the client-side vulnerabilities. Sadly, the GnuPG team made the decision that it’s not their job to pre-emptively address problems that they view as ‘clients misusing the GnuPG API’ (my paraphrase), even when that misuse appears to be rampant across many of the clients that use their tool. And so the most obvious fix for one part of the problem was not available.
This is probably the most unfortunate part of the Efail story, because in this case GnuPG is very much at fault. Their API does something that directly violates cryptographic best practices— namely, releasing unauthenticated plaintext prior to producing an error message. And while this could be understood as a reasonable API design at design time, continuing to support this API even as clients routinely misuse it has now led to flaws across the ecosystem. The refusal of GnuPG to take a leadership role in preemptively safeguarding these vulnerabilities both increases the difficulty of disclosing these flaws, and increases the probability of future issues.
So what went wrong with the Efail disclosure?
Despite what you may have heard, given the complexity of this disclosure, very little went wrong. The main issues people have raised seem to have to do with the contents of an EFF post. And with some really bad communications from Robert J. Hansen at the Enigmail (and GnuPG) project.
The EFF post. The Efail researchers chose to use the Electronic Frontier Foundation as their main source for announcing the existence of the vulnerability to the privacy community. This hardly seems unreasonable, because the EFF is generally considered a trusted broker, and speaks to the right community (at least here in the US).
The EFF post doesn’t give many details, nor does it give a list of affected (or patched) clients. It does give two pretty mild recommendations:
Temporarily disable or uninstall your existing clients until you’ve checked that they’re patched.
Maybe consider using a more modern cryptosystem like Signal, at least until you know that your PGP client is safe again.
This naturally led to a huge freakout by many in the PGP community. Some folks, including vendors, have misrepresented the EFF post as essentially pushing people to “permanently” uninstall PGP, which will “put lives at risk” because presumably these users (whose lives are at risk, remember) will immediately fall back to sending incriminating information via plaintext emails — rather than temporarily switching their communications to one of several modern, well-studied secure messengers, or just not emailing for a few hours.
The most reasonable criticism I’ve heard of the EFF post is that it doesn’t give many details about which clients are patched, and which are vulnerable. This could presumably give someone the impression that this vulnerability is still present in their email client, and thus would cause them to feel less than secure in using it.
I have to be honest that to me that sounds like a really good outcome. The problem with Efail is that it doesn’t matter if your client is secure. The Efail vulnerability could affect you if even a single one of your communication partners is using an insecure client.
So needless to say I’m not very sympathetic to the reaction around the EFF post. If you can’t be sure whether your client is secure, you probably should feel insecure.
Bad communications from GnuPG and Enigmail. On the date of the disclosure, anyone looking for accurate information about security from two major projects — GnuPG and Enigmail — would not have been able to find it.
They wouldn’t have found it because developers from both Enigmail and GnuPGwere on mailing lists and Twitter claiming that they had never heard of Efail, and hadn’t been notified by the researchers. Needless to say, these allegations took off around the Internet, sometimes in place of real information that could have helped users (like, whether either project had patched.)
It goes without saying that neither allegation was actually true. In fact, both project members soon checked with their fellow developers (and their memories) and found out that they’d both been given months of notice by the researchers, and that Enigmail had even developed a patch. (However, it turned out that even this patch may not perfectly address the issue, and the community is still working to figure out exactly what still needs to be done.)
This is an understandable mistake, perhaps. But it sure is a bad one.
PGP is bad technology and it’s making a bad community
Now that I’ve made it clear that neither the researchers nor the EFF is out to get the PGP community, let me put on my mask and horns and tell you why someone should be.
I’ve written extensively about PGP on this blog, but in the past I’ve written mostly from a technical point of view about the problems with PGP. But what’s really problematic about PGP is not just the cryptography; it’s the story it tells about path dependence and how software communities work.
The fact of the matter is that OpenPGP is not really a cryptography project. That is, it’s not held together by cryptography. It’s held together by backwards-compatibility and (increasingly) a kind of an obsession with the idea of PGP as an end in and of itself, rather than as a means to actually make end-users more secure.
Let’s face it, as a protocol, PGP/OpenPGP is just not what we’d develop if we started over today. It was formed over the years out of mostly experimental parts, which were in turn replaced, bandaged and repaired — and then worked into numerous implementations, which all had to be insanely flexible and yet compatible with one another. The result is bad, and most of the software implementing it is worse. It’s the equivalent of a beloved antique sports car, where the electrical system is totally shot, but it still drives. You know, the kind of car where the owner has to install a hand-switch so he can turn the reverse lights on manually whenever he wants to pull out of a parking space.
Unfortunately for us all, (Open)PGP does exist. And that means that even fancy greenfield email projects feel like they need to support OpenPGP, or at least some subset of it. This in turn perpetuates the PGP myth, and causes other clients to use it. And as a direct result, even if some clients re-implement OpenPGP from scratch, other clients will end up using tools like GnuPG which will support unauthenticated encryption with bad APIs. And the cycle will go round and around, like a spaceship stuck near the event horizon of a black hole.
And as the standard perpetuates itself, largely for the sake of being a standard, it will fail to attract new security people. It will turn away exactly the type of people who should be working on these tools. Those people will go off and build encryption systems in a totally different area, or they’ll get into cryptocurrency. And — with some exceptions — the people who work in the community will increasingly work in that community because they’re supporting PGP, and not because they’re trying to seek out the best security technologies for their users. And the serious (email) users of PGP will be using it because they like the idea of using PGP better than they like using an actual, secure email standard.
And as things get worse, and fail to develop, people who work on it will become more dogmatic about its importance, because it’s something threatened and not a real security protocol that anyone’s using. To me that’s where PGP is going today, and that is why the community has such a hard time motivating itself to take these vulnerabilities seriously, and instead reacts defensively.
Maybe that’s a random, depressing way to end a post. But that’s the story I see in OpenPGP. And it makes me really sad.
In general I try to limit this blog to posts that focus on generally-applicable techniques in cryptography. That is, I don’t focus on the deeply wonky. But this post is going to be an exception. Specifically, I’m going to talk about a topic that most “typical” implementers don’t — and shouldn’t — think about.
Specifically: I’m going to talk about various techniques for making public key encryption schemes chosen ciphertext secure. I see this as the kind of post that would have saved me ages of reading when I was a grad student, so I figured it wouldn’t hurt to write it all down.
Background: CCA(1/2) security
Early (classical) ciphers used a relatively weak model of security, if they used one at all. That is, the typical security model for an encryption scheme was something like the following:
I generate an encryption key (or keypair for public-key encryption)
I give you the encryption of some message of my choice
You “win” if you can decrypt it
This is obviously not a great model in the real world, for several reasons. First off, in some cases the attacker knows a lot about the message to be decrypted. For example: it may come from a small space (like a set of playing cards). For this reason we require a stronger definition like “semantic security” that assumes the attacker can choose the plaintext distribution, and can also obtain the encryption of messages of his/her own choice. I’ve written more about this here.
More relevant to this post, another limitation of the above game is that — in some real-world examples — the attacker has even more power. That is: in addition to obtaining the encryption of chosen plaintexts, they may be able to convince the secret keyholder to decrypt chosen ciphertexts of their choice.
At first blush this seems like a really stupid model. If you can ask the keyholder to decrypt chosen ciphertexts, then isn’t the scheme just obviously broken? Can’t you just decrypt anything you want?
The answer, it turns out, is that there are many real-life examples where the attacker has decryption capability, but the scheme isn’t obviously broken. For example:
Sometimes an attacker can decrypt a limited set of ciphertexts (for example, because someone leaves the decryption machine unattended at lunchtime.) The question then is whether they can learn enough from this access to decrypt other ciphertexts that are generated after she loses access to the decryption machine — for example, messages that are encrypted after the operator comes back from lunch.
Sometimes an attacker can submit any ciphertext she wants — but will only obtain a partial decryption of the ciphertext. For example, she might learn only a single bit of information such as “did this ciphertext decrypt correctly”. The question, then, is whether she can leverage this tiny amount of data to fully decrypt some ciphertext of her choosing.
The first example is generally called a “non-adaptive” chosen ciphertext attack, or a CCA1 attack (and sometimes, historically, a “lunchtime” attack). There are a few encryption schemes that totally fall apart under this attack — the most famous textbook example is Rabin’s public key encryption scheme, which allows you to recover the full secret key from just a single chosen-ciphertext decryption.
The more powerful second example is generally referred to as an “adaptive” chosen ciphertext attack, or a CCA2 attack. The term refers to the idea that the attacker can select the ciphertexts they try to decrypt based on seeing a specific ciphertext that they want to attack, and by seeing the answers to specific decryption queries.
In this article we’re going to use the more powerful “adaptive” (CCA2) definition, because that subsumes the CCA1 definition. We’re also going to focus primarily on public-key encryption.
With this in mind, here is the intuitive definition of the experiment we want a CCA2 public-key encryption scheme to be able to survive:
I generate an encryption keypair for a public-key scheme and give you the public key.
You can send me (sequentially and adaptively) many ciphertexts, which I will decrypt with my secret key. I’ll give you the result of each decryption.
Eventually you’ll send me a pair of messages (of equal length) and I’ll pick a bit at random, and return to you the encryption of , which I will denote as .
You’ll repeat step (2), sending me ciphertexts to decrypt. If you send me I’ll reject your attempt. But I’ll decrypt any other ciphertext you send me, even if it’s only slightly different from .
The attacker outputs their guess . They “win” the game if .
We say that our scheme is secure if the attacker wins only with a significantly greater probability than they would win with if they simply guessed at random. Since they can win this game with probability 1/2 just by guessing randomly, that means we want (Probability attacker wins the game) – 1/2 to be “very small” (typically a negligible function of the security parameter).
You should notice two things about this definition. First, it gives the attacker the full decryption of any ciphertext they send me. This is obviously much more powerful than just giving the attacker a single bit of information, as we mentioned in the example further above. But note that powerful is good. If our scheme can remain secure in this powerful experiment, then clearly it will be secure in a setting where the attacker gets strictly less information from each decryption query.
The second thing you should notice is that we impose a single extra condition in step (4), namely that the attacker cannot ask us to decrypt . We do this only to prevent the game from being “trivial” — if we did not impose this requirement, the attacker could always just hand us back to decrypt, and they would always learn the value of .
(Notice as well that we do not give the attacker the ability to request encryptions of chosen plaintexts. We don’t need to do that in the public key encryption version of this game, because we’re focusing exclusively on public-key encryption here — since the attacker has the public key, she can encrypt anything she wants without my help.)
With definitions out of the way, let’s talk a bit about how we achieve CCA2 security in real schemes.
A quick detour: symmetric encryption
This post is mainly going to focus on public-key encryption, because that’s actually the problem that’s challenging and interesting to solve. It turns out that achieving CCA2 for symmetric-key encryption is really easy. Let me briefly explain why this is, and why the same ideas don’t work for public-key encryption.
(To explain this, we’ll need to slightly tweak the CCA2 definition above to make it work in the symmetric setting. The changes here are small: we won’t give the attacker a public key in step (1), and at steps (2) and (4) we will allow the attacker to request the encryption of chosen plaintexts as well as the decryption.)
The first observation is that many common encryption schemes — particularly, the widely-used cipher modes of operation like CBC and CTR — are semantically secure in a model where the attacker does not have the ability to decrypt chosen ciphertexts. However, these same schemes break completely in the CCA2 model.
The simple reason for this is ciphertext malleability. Take CTR mode, which is particularly easy to mess with. Let’s say we’ve obtained a ciphertext at step (4) (recall that is the encryption of ), it’s trivially easy to “maul” the ciphertext — simply by flipping, say, a bit of the message (i.e., XORing it with “1”). This gives us a new ciphertext that we are now allowed to submit for decryption. We are now allowed (by the rules of the game) to submit this ciphertext, and obtain , which we can use to figure out .
So how do we fix this problem? The straightforward observation is that we need to prevent the attacker from mauling the ciphertext . The generic approach to doing this is to modify the encryption scheme so that it includes a Message Authentication Code (MAC) tag computed over every CTR-mode ciphertext. The key for this MAC scheme is generated by the encrypting party (me) and kept with the encryption key. When asked to decrypt a ciphertext, the decryptor first checks whether the MAC is valid. If it’s not, the decryption routine will output “ERROR”. Assuming an appropriate MAC scheme, the attacker can’t modify the ciphertext (including the MAC) without causing the decryption to fail and produce a useless result.
So in short: in the symmetric encryption setting, the answer to CCA2 security is simply for the encrypting parties to authenticate each ciphertext using a secret authentication (MAC) key they generate. Since we’re talking about symmetric encryption, that extra (secret) authentication key can be generated and stored with the decryption key. (Some more efficient schemes make this all work with a single key, but that’s just an engineering convenience.) Everything works out fine.
So now we get to the big question.
CCA security is easy in symmetric encryption. Why can’t we just do the same thing for public-key encryption?
As we saw above, it turns out that strong authenticated encryption is sufficient to get CCA(2) security in the world of symmetric encryption. Sadly, when you try this same idea generically in public key encryption, it doesn’t always work. There’s a short reason for this, and a long one. The short version is: it matters whois doing the encryption.
Let’s focus on the critical difference. In the symmetric CCA2 game above, there is exactly one person who is able to (legitimately) encrypt ciphertexts. That person is me. To put it more clearly: the person who performs the legitimate encryption operations (and has the secret key) is also the same person who is performing decryption.
Even if the encryptor and decryptor aren’t literally the same person, the encryptor still has to be honest. (To see why this has to be the case, remember that the encryptor has shared secret key! If that party was a bad guy, then the whole scheme would be broken, since they could just output the secret key to the bad guys.) And once you’ve made the stipulation that the encryptor is honest, then you’re almost all the way there. It suffices simply to add some kind of authentication (a MAC or a signature) to any ciphertext she encrypts. At that point the decryptor only needs to determine whether any given ciphertexts actually came from the (honest) encryptor, and avoid decrypting the bad ones. You’re done.
Public key encryption (PKE) fundamentally breaks all these assumptions.
In a public-key encryption scheme, the main idea is that anyone can encrypt a message to you, once they get a copy of your public key. The encryption algorithm may sometimes be run by good, honest people. But it can also be run by malicious people. It can be run by parties who are adversarial. The decryptor has to be able to deal with all of those cases. One can’t simply assume that the “real” encryptor is honest.
Let me give a concrete example of how this can hurt you. A couple of years ago I wrote a post about flaws in Apple iMessage, which (at the time) used simple authenticated (public key) encryption scheme. The basic iMessage encryption algorithm used public key encryption (actually a combination of RSA with some AES thrown in for efficiency) so that anyone could encrypt a message to my key. For authenticity, it required that every message be signed with an ECDSA signature by the sender.
When I received a message, I would look up the sender’s public key and first make sure the signature was valid. This would prevent bad guys from tampering with the message in flight — e.g., executing nasty stuff like adaptive chosen ciphertext attacks. If you squint a little, this is almost exactly a direct translation of the symmetric crypto approach we discussed above. We’re simply swapping the MAC for a digital signature.
The problems with this scheme start to become apparent when we consider that there might be multiple people sending me ciphertexts. Let’s say the adversary is on the communication path and intercepts a signed message from you to me. They want to change (i.e., maul) the message so that they can execute some kind of clever attack. Well, it turns out this is simple. They simply rip off the honest signature and replace it one they make themselves:
The new message is identical, but now appears to come from a different person (the attacker). Since the attacker has their own signing key, they can maul the encrypted message as much as they want, and sign new versions of that message. If you plug this attack into (a version) of the public-key CCA2 game up top, you see they’ll win quite easily. All they have to do is modify the challenge ciphertext at step (4) to be signed with their own signing key, then they can change it by munging with the CTR mode encryption, and request the decryption of that ciphertext.
Of course if I only accept messages from signed by some original (guaranteed-to-be-honest) sender, this scheme might work out fine. But that’s not the point of public key encryption. In a real public-key scheme — like the one Apple iMessage was trying to build — I should be able to (safely) decrypt messages from anyone, and in that setting this naive scheme breaks down pretty badly.
Ok, this post has gotten a bit long, and so far I haven’t actually gotten to the various “tricks” for adding chosen ciphertext security to real public key encryption schemes. That will have to wait until the next post, to come shortly.
Over the past several years I’ve been privileged to observe two contradictory and fascinating trends. The first is that we’re finally starting to use the cryptography that researchers have spent the past forty years designing. We see this every day in examples ranging from encrypted messaging to phone security to cryptocurrencies.
But before I get to all of that — much further below — let me stress that this is not a post about the quantum computing apocalypse, nor is it about the success of cryptography in the 21st century. Instead I’m going to talk about something much more wonky. This post will be about one of the simplest (and coolest!) cryptographic technologies ever developed: hash-based signatures.
Hash-based signature schemes were first invented in the late 1970s by Leslie Lamport, and significantly improved by Ralph Merkle and others. For many years they were largely viewed as an interesting cryptographic backwater, mostly because they produce relatively large signatures (among other complications). However in recent years these constructions have enjoyed something of a renaissance, largely because — unlike signatures based on RSA or the discrete logarithm assumption — they’re largely viewed as resistant to serious quantum attacks like Shor’s algorithm.
First some background.
Background: Hash functions and signature schemes
In order to understand hash-based signatures, it’s important that you have some familiarity with cryptographic hash functions. These functions take some input string (typically or an arbitrary length) and produce a fixed-size “digest” as output. Common cryptographic hash functions like SHA2, SHA3 or Blake2 produce digests ranging from 256 bits to 512 bits.
In order for a function to be considered a ‘cryptographic’ hash, it must achieve some specific security requirements. There are a number of these, but here we’ll just focus on three common ones:
1. Pre-image resistance (sometimes known as “one-wayness”): given some output , it should be time-consuming to find an input such that . (There are many caveats to this, of course, but ideally the best such attack should require a time comparable to a brute-force search of whatever distribution is drawn from.)
2. Second-preimage resistance: This is subtly different than pre-image resistance. Given some input , it should be hard for an attacker to find a different input such that .
3. Collision resistance: It should be hard to find any two values such that . Note that this is a much stronger assumption than second-preimage resistance, since the attacker has complete freedom to find any two messages of its choice.
The example hash functions I mentioned above are believed to provide all of these properties. That is, nobody has articulated a meaningful (or even conceptual) attack that breaks any of them. That could always change, of course, in which case we’d almost certainly stop using them. (We’ll discuss the special case of quantum attacks a bit further below.)
Since our goal is to use hash functions to construct signature schemes, it’s also helpful to briefly review that primitive.
A digital signature scheme is a public key primitive in which a user (or “signer”) generates a pair of keys, called the public key and private key. The user retains the private key, and can use this to “sign” arbitrary messages — producing a resulting digital signature. Anyone who has possession of the public key can verify the correctness of a message and its associated signature.
From a security perspective, the main property we want from a signature scheme is unforgeability, or “existential unforgeability“. This requirement means that an attacker (someone who does not possess the private key) should not be able to forge a valid signature on a message that you did not sign. For more on the formal definitions of signature security, see this page.
The Lamport One-Time Signature
The first hash-based signature schemes was invented in 1979 by a mathematician named Leslie Lamport. Lamport observed that given only simple hash function — or really, a one-way function — it was possible to build an extremely powerful signature scheme.
Powerful that is, provided that you only need to sign one message! More on this below.
For the purposes of this discussion, let’s suppose we have the following ingredient: a hash function that takes in, say, 256-bit inputs and produces 256-bit outputs. SHA256 would be a perfect example of such a function. We’ll also need some way to generate random bits.
Let’s imagine that our goal is to sign 256 bit messages. To generate our secret key, the first thing we need to do is generate a series of 512 separate random bitstrings, each of 256 bits in length. For convenience, we’ll arrange those strings into two separate lists and refer to each one by an index as follows:
The lists represent the secret key that we’ll use for signing. To generate the public key, we now simply hash every one of those random strings using our function . This produces a second pair of lists:
We can now hand out our public key to the entire world. For example, we can send it to our friends, embed it into a certificate, or post it on Keybase.
Now let’s say we want to sign a 256-bit message using our secret key. The very first thing we do is break up and represent as a sequence of 256 individual bits:
The rest of the signing algorithm is blindingly simple. We simply work through the message from the first bit to the last bit, and select a string from one of the two secret key list. The list we choose from depends on value of the message bit we’re trying to sign.
Concretely, for i=1 to 256: if the message bit , we grab the secret key string from the list, and output that string as part of our signature. If the message bit we copy the appropriate string from the list. Having done this for each of the message bits, we concatenate all of the strings we selected. This forms our signature.
Here’s a toy illustration of the process, where (for simplicity) the secret key and message are only eight bits long. Notice that each colored box below represents a different 256-bit random string:
When a user — who already has the public key — receives a message and a signature, she can verify the signature easily. Let represent the component of the signature: for each such string. She simply checks the corresponding message bit and computes hash . If the result should match the corresponding element from . If the result should match the element in .
The signature is valid if every single element of the signature, when hashed, matches the correct portion of the public key. Here’s an (admittedly) sketchy illustration of the verification process, for at least one signature component:
If your initial impression of Lamport’s scheme is that it’s kind of insane, you’re both a bit right and a bit wrong.
Let’s start with the negative. First, it’s easy to see that Lamport signatures and keys are be quite large: on the order of thousands of bits. Moreover — and much more critically — there is a serious security limitation on this scheme: each key can only be used to sign one message. This makes Lamport’s scheme an example of what’s called a “one time signature”.
To understand why this restriction exists, recall that every Lamport signature reveals exactly one of the two possible secret key values at each position. If I only sign one message, the signature scheme works well. However, if I ever sign two messages that differ at any bit position , then I’m going to end up handing out both secret key values for that position. This can be a problem.
Imagine that an attacker sees two valid signatures on different messages. She may be able to perform a simple “mix and match” forgery attack that allows her to sign a third message that I never actually signed. Here’s how that might look in our toy example:
The degree to which this hurts you really depends on how different the messages are and how many of them you’ve given the attacker to play with. But it’s rarely good news.
So to sum up our observations about the Lamport signature scheme. It’s simple. It’s fast. And yet for various practical reasons it kind of sucks. Maybe we can do a little better.
From one-time to many-time signatures: Merkle’s tree-based signature
While the Lamport scheme is a good start, our inability to sign many messages with a single key is a huge drawback. Nobody was more inspired by this than Martin Hellman’s student Ralph Merkle. He quickly came up with a clever way to address this problem.
While we can’t exactly retrace Merkle’s steps, let’s see if we can recover some of the obvious ideas.
Let’s say our goal is to use Lamport’s signature to sign many messages — say of them. The most obvious approach is to simply generate different keypairs for the original Lamport scheme, then concatenate all the public keys together into one mega-key.
(Mega-key is a technical term I just invented).
If the signer holds on to all secret key components, she can now sign different messages by using exactly one secret Lamport key per message. This seems to solve the problem without ever requiring her to re-use a secret key. The verifier has all the public keys, and can verify all the received messages. No Lamport keys are ever used to sign twice.
Obviously this approach sucks big time.
Specifically, in this naive approach, signing times requires the signer to distribute a public key that is times as large as a normal Lamport public key. (She’ll also need to hang on to a similar pile of secret keys.) At some point people will get fed up with this, and probably won’t every get to be very large. Enter Merkle.
What Merkle proposed was a way to retain the ability to sign different messages, but without the linear-cost blowup of public keys. Merkle’s idea worked like this:
First, generate separate Lamport keypairs. We can call those .
Next, place each public key at one leaf of a Merkle hash tree (see below), and compute the root of the tree. This root will become the “master” public key of the new Merkle signature scheme.
The signer retains all of the Lamport public and secret keys for use in signing.
Merkle trees are described here. Roughly speaking, what they provide is a way to collect many different values such that they can be represented by a single “root” hash (of length 256 bits, using the hash function in our example). Given this hash, it’s possible to produce a simple “proof” that an element is in a given hash tree. Moreover, this proof has size that is logarithmic in the number of leaves in the tree.
To sign the message, the signer simply selects the public key from the tree, and signs the message using the corresponding Lamport secret key. Next, she concatenates the resulting signature to the Lamport public key and tacks on a “Merkle proof” that shows that this specific Lamport public key is contained within the tree identified by the root (i.e., the public key of the entire scheme). She then transmits this whole collection as the signature of the message.
(To verify a signature of this form, the verifier simply unpacks this “signature” as a Lamport signature, Lamport public key, and Merkle Proof. She verifies the Lamport signature against the given Lamport public key, and uses the Merkle Proof to verify that the Lamport public key is really in the tree. With these three objectives achieved, she can trust the signature as valid.)
This approach has the disadvantage of increasing the “signature” size by more than a factor of two. However, the master public key for the scheme is now just a single hash value, which makes this approach scale much more cleanly than the naive solution above.
As a final optimization, the secret key data can itself be “compressed” by generating all of the various secret keys using the output of a cryptographic pseudorandom number generator, which allows for the generation of a huge number of (apparently random) bits from a single short ‘seed’.
Making signatures and keys (a little bit) more efficient
Merkle’s approach allows any one-time signature to be converted into an -time signature. However, his construction still requires us to use some underlying one-time signature like Lamport’s scheme. Unfortunately the (bandwidth) costs of Lamport’s scheme are still relatively high.
There are two major optimizations that can help to bring down these costs. The first was also proposed by Merkle. We’ll cover this simple technique first, mainly because it helps to explain the more powerful approach.
If you recall Lamport’s scheme, in order sign a 256-bit message we required a vector consisting of 512 separate secret key (and public key) bitstrings. The signature itself was a collection of 256 of the secret bitstrings. (These numbers were motivated by the fact that each bit of the message to be signed could be either a “0” or a “1”, and thus the appropriate secret key element would need to be drawn from one of two different secret key lists.)
But here’s a thought: what if we don’t sign all of the message bits?
Let’s be a bit more clear. In Lamport’s scheme we sign every bit of the message — regardless of its value — by outputting one secret string. What if, instead of signing both zero values and one values in the message, we signed only the message bits were equal to one? This would cut the public and secret key sizes in half, since we could get rid of the list entirely.
We would now have only a single list of bitstrings in our secret key. For each bit position of the message where we would output a string . For every position where we would output… zilch. (This would also tend to reduce the size of signatures, since many messages contain a bunch of zero bits, and those would now ‘cost’ us nothing!)
An obvious problem with this approach is that it’s horrendously insecure. Please do not implement this scheme!
As an example, let’s say an attacker observes a (signed) message that begins with “1111…”, and she want to edit the message so it reads “0000…” — without breaking the signature. All she has to do to accomplish this is to delete several components of the signature! In short, while it’s very difficult to “flip” a zero bit into a one bit, it’s catastrophically easy to do the reverse.
But it turns out there’s a fix, and it’s quite elegant.
You see, while we can’t prevent an attacker from editing our message by turning one bits into zero bits, we can catch them. To do this, we tack on a simple “checksum” to the message,then sign the combination of the original message and the checksum. The signature verifier must verify the entire signature over both values, and also ensure that the received checksum is correct.
The checksum we use is trivial: it consists of a simple binary integer that represents the total number of zero bits in the original message.
If the attacker tries to modify the content of the message (excluding the checksum) in order to turn some one bit into a zero bit, the signature scheme won’t stop her. But this attack have the effect of increasing the number of zero bits in the message. This will immediately make the checksum invalid, and the verifier will reject the signature.
Of course, a clever attacker might also try to mess with the checksum (which is also signed along with the message) in order to “fix it up” by increasing the integer value of the checksum. However — and this is critical — since the checksum is a binary integer, in order to increase the value of the checksum, she would always need to turn some zero bit of the checksum into a one bit. But since the checksum is also signed, and the signature scheme prevents this kind of change, the attacker has nowhere to go.
(If you’re keeping track at home, this does somewhat increase the size of the ‘message’ to be signed. In our 256-bit message example, the checksum will require an additional eight bits and corresponding signature cost. However, if the message has many zero bits, the reduced signature size will typically still be a win.)
Winternitz: Trading space for time
The trick above reduces the public key size by half, and reduces the size of (some) signatures by a similar amount. That’s nice, but not really revolutionary. It still gives keys and signatures that are thousands of bits long.
It would be nice if we could make a bigger dent in those numbers.
The final optimization we’ll talk about was proposed by Robert Winternitz as a further optimization of Merkle’s technique above. In practical use it gives a 4-8x reduction in the size of signatures and public keys — at a cost of increasing both signing and verification time.
Winternitz’s idea is an example of a technique called a “time-space tradeoff“. This term refers to a class of solutions in which space is reduced at the cost of adding more computation time (or vice versa). To explain Winternitz’s approach, it helps to ask the following question:
What if, instead of signing messages composed of bits (0 or 1), we treated our messages as though they were encoded using larger symbol alphabets? For example, what if we signed four-bit ‘nibbles’? Or eight-bit bytes?
In Lamport’s original scheme, we had two lists of bitstrings as part of the signing (and public) key. One was for signing zero message bits, and the other was for one bits.
Now let’s say we want to sign bytes rather than bits. An obvious idea would be to increase the number of secret key lists (and public key) from two such list to 256 such lists — one list for each possible value of a message byte. The signer could work through the message one byte at a time, and pick from the much larger menu of key values.
Unfortunately, this solution really stinks. It reduces the size of the signature by a factor of eight, at a cost of increasing the public and secret key size by a factor of 256. Even this might be fine if the public keys could be used for many signatures, but they can’t — when it comes to key re-use, this “byte signing version of Lamport” suffers from the same limitations as the original Lamport signature.
All of which brings us to Winternitz’s idea.
Since it’s too expensive to store and distribute 256 truly random lists, what if we generated those lists programatically only when we needed them?
Winternitz’s idea was to generate single list of random seeds for our initial secret key. Rather than generating additional lists randomly, he proposed to use the hash function on each element of that initial secret key, in order to derive the next such list for the secret key: . And similarly, one can use the hash function again on that list to get the next list . And so on for many possible lists.
This is helpful in that we now only need to store a single list of secret key values , and we can derive all the others lists on-demand just by applying the hash function.
But what about the public key? This is where Winternitz gets clever.
Specifically, Winternitz proposed that the public key could be derived by applying the hash function one more time to the final secret key list. This would produce a single public key list . (In practice we only need 255 secret key lists, since we can treat the final secret key list as the public key.) The elegance of this approach is that given any one of the possible secret key values it’s always possible to check it against the public key, simply by hashing forward multiple times and seeing if we reach a public key element.
The whole process of key generation is illustrated below:
To sign the first byte of a message, we would pick a value from the appropriate list. For example, if the message byte was “0”, we would output a value from in our signature. If the message byte was “20”, we would output a value from . For bytes with the maximal value “255” we don’t have a secret key list. That’s ok: in this case we can output an empty string, or we can output the appropriate element of .
Note as well that in practice we don’t really need to store each of these secret key lists. We can derive any secret key value on demand given only the original list . The verifier only holds the public key vector and (as mentioned above) simply hashes forward an appropriate number of times — depending on the message byte — to see whether the result is equal to the appropriate component of the public key.
Like the Merkle optimization discussion in the previous section, the scheme as presented so far has a glaring vulnerability. Since the secret keys are related (i.e., ), anyone who sees a message that signs the message “0” can easily change the corresponding byte of the message to a “1”, and update the signature to match. In fact, an attacker can increment the value of any byte(s) in the message. Without some check on this capability, this would allow very powerful forgery attacks.
The solution to this problem is similar to the we discussed just above. To prevent an attacker from modifying the signature, the signer calculates and also signs a checksum of the original message bytes. The structure of this checksum is designed to prevent the attacker from incrementing any of the bytes, without invalidating the checksum. I won’t go into the gory details right now, but you can find them here.
It goes without saying that getting this checksum right is critical. Screw it up, even a little bit, and some very bad things can happen to you. This would be particularly unpleasant if you deployed these signatures in a production system.
Illustrated in one terrible picture, a 4-byte toy example of the Winternitz scheme looks like this:
What are hash-based signatures good for?
Throughout this entire discussion, we’ve mainly been talking about the how of hash-based signatures rather than the why of them. It’s time we addressed this. What’s the point of these strange constructions?
One early argument in favor of hash-based signatures is that they’re remarkably fast and simple. Since they only require only the evaluation of a hash function and some data copying, from a purely computational cost perspective they’re highly competitive with schemes like ECDSA and RSA. This could hypothetically be important for lightweight devices. Of course, this efficiency comes at a huge tradeoff in bandwidth efficiency.
More concretely: the imminent arrival of quantum computers is going to have a huge impact on the security of nearly all of our practical signature schemes, ranging from RSA to ECDSA and so on. This is due to the fact that Shor’s algorithm (and its many variants) provides us with a polynomial-time algorithm for solving the discrete logarithm and factoring problems, which is likely to render most of these schemes insecure.
Most implementations of hash-based signatures are not vulnerable to Shor’s algorithm. That doesn’t mean they’re completely immune to quantum computers, of course. The best general quantum attacks on hash functions are based on a search technique called Grover’s algorithm, which reduces the effective security of a hash function. However, the reduction in effective security is nowhere near as severe as Shor’s algorithm (it ranges between the square root and cube root), and so security can be retained by simply increasing the internal capacity and output size of the hash function. Hash functions like SHA3 were explicitly developed with large digest sizes to provide resilience against such attacks.
So at least in theory, hash-based signatures are interesting because they provide us with a line of defense against future quantum computers — for the moment, anyway.
What about the future?
Note that so far I’ve only discussed some of the “classical” hash-based signature schemes. All of the schemes I described above were developed in the 1970s or early 1980s. This hardly brings us up to present day.
After I wrote the initial draft of this article, a few people asked for pointers on more recent developments in the field. I can’t possibly give an exhaustive list here, but let me describe just a couple of the more recent ideas that others brought up (thanks to Zooko and Claudio Orlandi):
Signatures without state. A limitation of all the signature schemes above is that they require the signer to keep state between signatures. In the case of one-time signatures the reasoning is obvious: you have to avoid using any key more than once. But even in the multi-time Merkle signature, you have to remember which leaf public key you’re using, so you can avoid using any leaf twice. Even worse, the Merkle scheme requires the signer to construct all the keypairs up front, so the total number of signatures is bounded.
In the 1980s, Oded Goldreich pointed out that one can build signatures without these limitations. The idea is as follows: rather than generate all signatures up front, one can generate a short “certification tree” of one-time public keys. Each of these keys can be used to sign additional one-time public keys at a lower layer of the tree, and so on and so forth. Provided all of the private keys are generated deterministically using a single seed, this means that the full tree need not exist in full at key generation time, but can be built on-demand whenever a new key is generated. Each signature contains a “certificate chain” of signatures and public keys starting from the root and going down to a real signing keypair at the bottom of the tree.
This technique allows for the construction of extremely “deep” trees with a vast (exponential) number of possible signing keys. This allows us to construct so many one-time public keys that if we pick a signing key randomly (or pseudorandomly), then with high probability the same signing will never be used twice. This is intuition, of course. For a highly optimized and specific instantiation of this idea, see the SPHINCS proposal by Bernstein et al. The concrete SPHINCS-256 instantiation gives signatures that are approximately 41KB in size.
Picnic: post-quantum zero-knowledge based signatures. In a completely different direction lies Picnic. Picnic is based on a new non-interactive zero-knowledge proof system called ZKBoo. ZKBoo is a new ZK proof system that works on the basis of a technique called “MPC in the head”, where the prover uses multi-party computation to calculate a function with the prover himself. This is too complicated to explain in a lot of detail, but the end result is that one can then prove complicated statements using only hash functions.
The long and short of it is that Picnic and similar ZK proof systems provide a second direction for building signatures out of hash functions. The cost of these signatures is still quite large — hundreds of kilobytes. But future improvements in the technique could substantially reduce this size.
Epilogue: the boring security details
If you recall a bit earlier in this article, I spent some time describing the security properties of hash functions. This wasn’t just for show. You see, the security of a hash-based signature depends strongly on which properties a hash function is able to provide.
(And by implication, the insecurity of a hash-based signature depends on which properties of a hash function an attacker has managed to defeat.)
Most original papers discussing hash-based signatures generally hang their security arguments on the preimage-resistance of the hash function. Intuitively, this seems pretty straightforward. Let’s take the Lamport signature as an example. Given a public key element , an attacker who is able to compute hash preimages can easily recover a valid secret key for that component of the signature. This attack obviously renders the scheme insecure.
However, this argument considers only the case where an attacker has the public key but has not yet seen a valid signature. In this case the attacker has a bit more information. They now have (for example) both the public key and a portion of the secret key: and . If such an attacker can find a second pre-image for the public key she can’t sign a different message. But she has produced a new signature. In the strong definition of signature security (SUF-CMA) this is actually considered a valid attack. So SUF-CMA requires the slightly stronger property of second-preimage resistance.
Of course, there’s a final issue that crops up in most practical uses of hash-based signature schemes. You’ll notice that the description above assumes that we’re signing 256-bit messages. The problem with this is that in real applications, many messages are longer than 256 bits. As a consequence, most people use the hash function to first hash the message as and then the sign the resulting value instead of the message.
This leads to a final attack on the resulting signature scheme, since the existential unforgeability of the scheme now depends on the collision-resistance of the hash function. An attacker who can find two different messages such that has now found a valid signature on two different messages. This leads to a trivial break of EUF-CMA security.
Yesterday, David Benjamin posted a pretty esoteric note on the IETF’s TLS mailing list. At a superficial level, the post describes some seizure-inducingly boring flaws in older Canon printers. To most people that was a complete snooze. To me and some of my colleagues, however, it was like that scene in X-Files where Mulder and Scully finally learn that aliens are real.
Those fossilized printers confirmed a theory we’d developed in 2014, but had been unable to prove: namely, the existence of a specific feature in RSA’s BSAFE TLS library called “Extended Random” — one that we believe to be evidence of a concerted effort by the NSA to backdoor U.S. cryptographic technology.
Before I get to the details, I want to caveat this post in two different ways. First, I’ve written about the topic of cryptographic backdoors way too much. In 2013, the Snowden revelations revealed the existence of a campaign to sabotage U.S. encryption systems. Since that time, cryptographers have spent thousands of hours identifying, documenting, and trying to convince people to care about these backdoors. We’re tired and we want to do more useful things.
The second caveat covers a problem with any discussion of cryptographic backdoors. Specifically, you never really get absolute proof. There’s always some innocent or coincidental explanation that could sort of fit the evidence — maybe it was all a stupid mistake. So you look for patterns of unlikely coincidences, and use Occam’s razor a lot. You don’t get a Snowden every day.
With all that said, let’s talk about Extended Random, and what this tells us about the NSA. First some background.
Dual_EC_DRBG and RSA BSAFE
To understand the context of this discovery, you need to know about a standard called Dual EC DRBG. This was a proposed random number generator that the NSA developed in the early 2000s. It was standardized by NIST in 2007, and later deployed in some important cryptographic products — though we didn’t know it at the time.
Dual EC has a major problem, which is that it likely contains a backdoor. This was pointed out in 2007 by Shumow and Ferguson, and effectively confirmed by the Snowden leaks in 2013. Drama ensued. NIST responded by pulling the standard. (For an explainer on the Dual EC backdoor, see here.)
Somewhere around this time the world learned that RSA Security had made Dual EC the default random number generator in their popular cryptographic library, which was called BSAFE. RSA hadn’t exactly kept this a secret, but it was such a bonkers thing to do that nobody (in the cryptographic community) had known. So for years RSA shipped their library with this crazy algorithm, which made its way into all sorts of commercial devices.
The RSA drama didn’t quite end there, however. In late 2013, Reuters reported that RSA had taken $10 million to backdoor their software. RSA sort of denies this. Or something. It’s not really clear.
Regardless of the intention, it’s known that RSA BSAFE did incorporate Dual EC. This could have been an innocent decision, of course, since Dual EC was a NIST standard. To shed some light on that question, in 2014 my colleagues and I decided to reverse-engineer the BSAFE library to see if it the alleged backdoor in Dual EC was actually exploitable by an attacker like the NSA. We figured that specific engineering decisions made by the library designers could be informative in tipping the scales one way or the other.
It turns out they were.
In the course of reverse engineering the Java version of BSAFE, we discovered a funny inclusion. Specifically, we found that BSAFE supports a non-standard extension to the TLS protocol called “Extended Random”.
The Extended Random extension is an IETF Draft proposed by an NSA employee named Margaret Salter (at some point the head of NSA’s Information Assurance Directorate, which worked on “defensive” crypto for DoD) along with Eric Rescorla as a contractor. (Eric was very clearly hired to develop a decent proposal that wouldn’t hurt TLS, and would primarily be used on government machines. The NSA did not share their motivations with him.)
It’s important to note that Extended Random by itself does not introduce any cryptographic vulnerabilities. All it does is increase the amount of random data (“nonces”) used in a TLS protocol connection. This shouldn’t hurt TLS at all, and besides it was largely intended for U.S. government machines.
The only thing that’s interesting about Extended Random is what happens when that random data is generated using the Dual EC algorithm. Specifically, this extra data acts as “rocket fuel”, significantly increasing the efficiency of exploiting the Dual EC backdoor to decrypt TLS connections.
In short, if you’re an agency like the NSA that’s trying to use Dual EC as a backdoor to intercept communications, you’re muchbetter off with a system that uses both Dual EC DRBG and Extended Random. Since Extended Random was never standardized by the IETF, it shouldn’t be in any systems. In fact, to the best of our knowledge, BSAFE is the only system in the world that implements it.
In addition to Extended Random, we discovered a variety of features that, combined with the Dual EC backdoor, could make RSA BSAFE fairly easy to exploit. But Extended Random is by far the strangest and hardest to justify.
So where did this standard come from? For those who like technical mysteries, it turns out that Extended Random isn’t the only funny-smelling proposal the NSA made. It’s actually one of four failed IETF proposals made by NSA employees, or contractors who work closely with the NSA, all of which try to boost the amount of randomness in TLS. Thomas Ptacek has a mind-numbingly detailed discussion of these proposals and his view of their motivation in this post.
Oh my god I never thought spies could be so boring. What’s the new development?
Despite the fact that we found Extended Random in RSA BSAFE (a free version we downloaded from the Internet), a fly in the ointment was that it didn’t actually seem to be enabled. That is: the code was there but the switches to enable it were hard-coded to “off”.
This kind of put a wrench in our theory that RSA might have included Extended Random to make BSAFE connections more exploitable by the NSA. There might be some commercial version of BSAFE out there with this code active, but we were never able to find it or prove it existed. And even worse, it might appear only in some special “U.S. government only” version of BSAFE, which would tend to undermine the theory that there was something intentional about including this code — after all, why would the government spy on itself?
Which finally brings us to the news that appeared on the TLS mailing list the other day. It turns out that certain Canon printers are failing to respond properly to connections made using the new version of TLS (which is called 1.3), because they seem to have implemented an unauthorized TLS extension using the same number as an extension that TLS 1.3 needs in order to operate correctly. Here’s the relevant section of David’s post:
The web interface on some Canon printers breaks with 1.3-capable ClientHello messages. We have purchased one and confirmed this with a PIXMA MX492. User reports suggest that it also affects PIXMA MG3650 and MX495 models. It potentially affects a wide range of Canon printers.
These printers use the RSA BSAFE library to implement TLS and this library implements the extended_random extension and assigns it number 40. This collides with the key_share extension and causes 1.3-capable handshakes to fail.
So in short, this news appears to demonstrate that commercial (non-free) versions of RSA BSAFE did deploy the Extended Random extension, and made it active within third-party commercial products. Moreover, they deployed it specifically to machines — specifically off-the-shelf commercial printers — that don’t seem to be reserved for any kind of special government use.
(If these turn out to be special Department of Defense printers, I will eat my words.)
Ironically, the printers are now the only thing that still exhibits the features of this (now deprecated) version of BSAFE. This is not because the NSA was targeting printers. Whatever devices they were targeting are probably gone by now. It’s because printer firmware tends to be obsolete and yet highly persistent. It’s like a remote pool buried beneath the arctic circle that preserves software species that would otherwise vanish from the Internet.
Which brings us to the moral of the story: not only are cryptographic backdoors a terrible idea, but they totally screw up the assigned numbering system for future versions of your protocol.
Actually no, that’s a pretty useless moral. Instead, let’s just say that you can deploy a cryptographic backdoor, but it’s awfully hard to control where it will end up.
The big news in crypto today is the KRACK attack on WPA2 protected WiFi networks. Discovered by Mathy Vanhoef and Frank Piessens at KU Leuven, KRACK (Key Reinstallation Attack)leverages a vulnerability in the 802.11i four-way handshake in order to facilitate decryption and forgery attacks on encrypted WiFi traffic.
I don’t want to spend much time talking about KRACK itself, because the vulnerability is pretty straightforward. Instead, I want to talk about why this vulnerability continues to exist so many years after WPA was standardized. And separately, to answer a question: how did this attack slip through, despite the fact that the 802.11i handshake was formally proven secure?
A quick TL;DR on KRACK
For a detailed description of the attack, see the KRACK website or the paper itself. Here I’ll just give a brief, high level description.
The 802.11i protocol (also known as WPA2) includes two separate mechanisms to ensure the confidentiality and integrity of your data. The first is a record layer that encrypts WiFi frames, to ensure that they can’t be read or tampered with. This encryption is (generally) implemented using AES in CCM mode, although there are newer implementations that use GCM mode, and older ones that use RC4-TKIP (we’ll skip these for the moment.)
The key thing to know is that AES-CCM (and GCM, and TKIP) is a stream cipher, which means it’s vulnerable to attacks that re-use the same key and “nonce”, also known as an initialization vector. 802.11i deals with this by constructing the initialization vector using a “packet number” counter, which initializes to zero after you start a session, and always increments (up to 2^48, at which point rekeying must occur). This should prevent any nonce re-use, provided that the packet number counter can never be reset.
The second mechanism you should know about is the “four way handshake” between the AP and a client (supplicant) that’s responsible for deriving the key to be used for encryption. The particular message KRACK cares about is message #3, which causes the new key to be “installed” (and used) by the client.
The key vulnerability in KRACK (no pun intended) is that the acknowledgement to message #3 can be blocked by adversarial nasty people.* When this happens, the AP re-transmits this message, which causes (the same) key to be reinstalled into the client (note: see update below*). This doesn’t seem so bad. But as a side effect of installing the key, the packet number counters all get reset to zero. (And on some implementations like Android 6, the key gets set to zero — but that’s another discussion.)
The implication is that by forcing the AP to replay this message, an adversary can cause a connection to reset nonces and thus cause keystream re-use in the stream cipher. With a little cleverness, this can lead to full decryption of traffic streams. And that can lead to TCP hijacking attacks. (There are also direct traffic forgery attacks on GCM and TKIP, but this as far as we go for now.)
How did this get missed for so long?
If you’re looking for someone to blame, a good place to start is the IEEE. To be clear, I’m not referring to the (talented) engineers who designed 802.11i — they did a pretty good job under the circumstances. Instead, blame IEEE as an institution.
One of the problems with IEEE is that the standards are highly complex and get made via a closed-door process of private meetings. More importantly, even after the fact, they’re hard for ordinary security researchers to access. Go ahead and google for the IETF TLS or IPSec specifications — you’ll find detailed protocol documentation at the top of your Google results. Now go try to Google for the 802.11i standards. I wish you luck.
The IEEE has been making a few small steps to ease this problem, but they’re hyper-timid incrementalist bullshit. There’s an IEEE program called GET that allows researchers to access certain standards (including 802.11) for free, but only after they’ve been public for six months — coincidentally, about the same time it takes for vendors to bake them irrevocably into their hardware and software.
This whole process is dumb and — in this specific case — probably just cost industry tens of millions of dollars. It should stop.
The second problem is that the IEEE standards are poorly specified. As the KRACK paper points out, there is no formal description of the 802.11i handshake state machine. This means that implementers have to implement their code using scraps of pseudocode scattered around the standards document. It happens that this pseudocode leads to the broken implementation that enables KRACK. So that’s bad too.
And of course, the final problem is implementers. One of the truly terrible things about KRACK is that implementers of the WPA supplicant (particularly on Linux) managed to somehow make Lemon Pledge out of lemons. On Android 6 in particular, replaying message #3 actually sets an all-zero key. There’s an internal logic behind why this happens, but Oy Vey. Someone actually needs to look at this stuff.
What about the security proof?
The fascinating thing about the 802.11i handshake is that despite all of the roadblocks IEEE has thrown in people’s way, it (the handshake, at least) has been formally analyzed. At least, for some definition of the term.
(This isn’t me throwing shade — it’s a factual statement. In formal analysis, definitions really, really matter!)
A paper by He, Sundararajan, Datta, Derek and Mitchell (from 2005!) looked at the 802.11i handshake and tried to determine its security properties. What they determined is that yes, indeed, it did produce a secret and strong key, even when an attacker could tamper with and replay messages (under various assumptions). This is good, important work. The proof is hard to understand, but this is par for the course. It seems to be correct.
Even better, there are other security proofs showing that — provided the nonces are never repeated — encryption modes like CCM and GCM are highly secure. This means that given a secure key, it should be possible to encrypt safely.
So what went wrong?
The critical problem is that while people looked closely at the two components — handshake and encryption protocol — in isolation, apparently nobody looked closely at the two components as they were connected together. I’m pretty sure there’s an entire geek meme about this.
Of course, the reason nobody looked closely at this stuff is that doing so is just plainhard. Protocols have an exponential number of possible cases to analyze, and we’re just about at the limit of the complexity of protocols that human beings can truly reason about, or that peer-reviewers can verify. The more pieces you add to the mix, the worse this problem gets.
In the end we all know that the answer is for humans to stop doing this work. We need machine-assisted verification of protocols, preferably tied to the actual source code that implements them. This would ensure that the protocol actually does what it says, and that implementers don’t further screw it up, thus invalidating the security proof.
This needs to be done urgently, but we’re so early in the process of figuring out how to do it that it’s not clear what it will take to make this stuff go live. All in all, this is an area that could use a lot more work. I hope I live to see it.
* Update: An early version of this post suggested that the attacker would replay the third message. This can indeed happen, and it does happen in some of the more sophisticated attacks. But primarily, the paper describes forcing the AP to resend it by blocking the acknowledgement from being received at the AP. Thanks to Nikita Borisov and Kyle Birkeland for the fix!
One of the saddest and most fascinating things about applied cryptography is how little cryptography we actually use. This is not to say that cryptography isn’t widely used in industry — it is. Rather, what I mean is that cryptographic researchers have developed so many useful technologies, and yet industry on a day to day basis barely uses any of them. In fact, with a few minor exceptions, the vast majority of the cryptography we use was settled by the early-2000s.*
Most people don’t sweat this, but as a cryptographer who works on the boundary of research and deployed cryptography it makes me unhappy. So while I can’t solve the problem entirely, what I can do is talk about some of these newer technologies. And over the course of this summer that’s what I intend to do: talk. Specifically, in the next few weeks I’m going to write a series of posts that describe some of the advanced cryptography that we don’t generally see used.
Today I’m going to start with a very simple question: what lies beyond public key cryptography? Specifically, I’m going to talk about a handful of technologies that were developed in the past 20 years, each of which allows us to go beyond the traditional notion of public keys.
This is a wonky post, but it won’t be mathematically-intense. For actual definitions of the schemes, I’ll provide links to the original papers, and references to cover some of the background. The point here is to explain what these new schemes do — and how they can be useful in practice.
Identity Based Cryptography
In the mid-1980s, a cryptographer named Adi Shamir proposed a radical new idea. The idea, put simply, was to get rid of public keys.
To understand where Shamir was coming from, it helps to understand a bit about public key encryption. You see, prior to the invention of public key crypto, all cryptography involved secret keys. Dealing with such keys was a huge drag. Before you could communicate securely, you needed to exchange a secret with your partner. This process was fraught with difficulty and didn’t scale well.
Public key encryption (beginning with Diffie-Hellman and Shamir’s RSA cryptosystem) hugely revolutionized cryptography by dramatically simplifying this key distribution process. Rather than sharing secret keys, users could now transmit their public key to other parties. This public key allowed the recipient to encrypt to you (or verify your signature) but it could not be used to perform the corresponding decryption (or signature generation) operations. That part would be done with a secret key you kept to yourself.
While the use of public keys improved many aspects of using cryptography, it also gave rise to a set of new challenges. In practice, it turns out that having public keys is only half the battle — people still need to use distribute them securely.
For example, imagine that I want to send you a PGP-encrypted email. Before I can do this, I need to obtain a copy of your public key. How do I get this? Obviously we could meet in person and exchange that key on physical media — but nobody wants to do this. It would much more desirable to obtain your public key electronically. In practice this means either (1) we have to exchange public keys by email, or (2) I have to obtain your key from a third piece of infrastructure, such as a website or key server. And now we come to the problem: if that email or key server is untrustworthy (or simply allows anyone to upload a key in your name),I might end up downloading a malicious party’s key by accident. When I send a message to “you”, I’d actually be encrypting it to Mallory.
Solving this problem — of exchanging public keys and verifying their provenance — has motivated a huge amount of practical cryptographic engineering, including the entire web PKI. In most cases, these systems work well. But Shamir wasn’t satisfied. What if, he asked, we could do it better? More specifically, he asked: could we replace those pesky public keys with something better?
Shamir’s idea was exciting. What he proposed was a new form of public key cryptography in which the user’s “public key” could simply be their identity. This identity could be a name (e.g., “Matt Green”) or something more precise like an email address. Actually, it didn’t realy matter. What did matter was that the public key would be some arbitrary string — and not a big meaningless jumble of characters like “7cN5K4pspQy3ExZV43F6pQ6nEKiQVg6sBkYPg1FG56Not”.
Of course, using an arbitrary string as a public key raises a big problem. Meaningful identities sound great — but I don’t own them. If my public key is “Matt Green”, how do I get the corresponding private key? And ifI can get out that private key, what stops some other Matt Green from doing the same, and thus reading my messages? And ok, now that I think about this, what stops some random person who isn’t named Matt Green from obtaining it? Yikes. We’re headed straight into Zooko’s triangle.
Shamir’s idea thus requires a bit more finesse. Rather than expecting identities to be global, he proposed a special server called a “key generation authority” that would be responsible for generating the private keys. At setup time, this authority would generate a single master public key (MPK), which it would publish to the world. If you wanted to encrypt a message to “Matt Green” (or verify my signature), then you could do so using my identity and the single MPK of an authority we’d both agree to use. To decrypt that message (or sign one), I would have to visit the same key authority and ask for a copy of my secret key. The key authority would compute my key based on a master secret key (MSK), which it would keep very secret.
With all algorithms and players specified, whole system looks like this:
This design has some important advantages — and more than a few obvious drawbacks. On the plus side, it removes the need for any key exchange at all with the person you’re sending the message to. Once you’ve chosen a master key authority (and downloaded its MPK), you can encrypt to anyone in the entire world. Even cooler: at the time you encrypt, your recipient doesn’t even need to have contacted the key authority yet. She can obtain her secret key after I send her a message.
Of course, this “feature” is also a bug. Because the key authority generates all the secret keys, it has an awful lot of power. A dishonest authority could easily generate your secret key and decrypt your messages. The polite way to say this is that standard IBE systems effectively “bake in” key escrow.**
Putting the “E” in IBE
All of these ideas and more were laid out by Shamir in his 1984 paper. There was just one small problem: Shamir could only figure out half the problem.
Specifically, Shamir’s proposed a scheme for identity-based signature (IBS) — a signature scheme where the public verification key is an identity, but the signing key is generated by the key authority. Try as he might, he could not find a solution to the problem of building identity-based encryption (IBE). This was left as an open problem.***
It would take more than 16 years before someone answered Shamir’s challenge. Surprisingly, when the answer came it came not once but three times.
The first, and probably most famous realization of IBE was developed by Dan Boneh and Matthew Franklin much later — in 2001. The timing of Boneh and Franklin’s discovery makes a great deal of sense. The Boneh-Franklin scheme relies fundamentally on elliptic curves that support an efficient “bilinear map” (or “pairing”).**** The algorithms needed to compute such pairings were not known when Shamir wrote his paper, and weren’t employed constructively — that is, as a useful thing rather than an attack — until about 2000. The same can be said about a second scheme called Sakai-Kasahara, which would be independently discovered around the same time.
(For a brief tutorial on the Boneh-Franklin IBE scheme, see this page.)
The third realization of IBE was not as efficient as the others, but was much more surprising. This scheme was developed by Clifford Cocks, a senior cryptologist at Britain’s GCHQ. It’s noteworthy for two reasons. First, Cocks’ IBE scheme does not require bilinear pairings at all — it is based in the much older RSA setting, which means in principle it spent all those years just waiting to be found. Second, Cocks himself had recently become known for something even more surprising: discovering the RSA cryptosystem, nearly five years before RSA did. To bookend that accomplishment with a second major advance in public key cryptography was a pretty impressive accomplishment.
In the years since 2001, a number of additional IBE constructions have been developed, using all sorts of cryptographic settings. Nonetheless, Boneh and Franklin’s early construction remains among the simplest and most efficient.
Even if you’re not interested in IBE for its own sake, it turns out that this primitive is really useful to cryptographers for many things beyond simple encryption. In fact, it’s more helpful to think of IBE as a way of “pluralizing” a single public/secret master keypair into billions of related keypairs. This makes it useful for applications as diverse as blocking chosen ciphertext attacks,forward-secure public key encryption, and short signature schemes.
Attribute Based Encryption
Of course, if you leave cryptographers alone with a tool like IBE, the first thing they’re going to do is find a way to make things more complicated improve on it.
One of the biggest such improvements is due to Sahai and Waters. It’s called Attribute-Based Encryption, or ABE.
The origin of this idea was not actually to encrypt with attributes. Instead Sahai and Waters were attempting to develop an Identity-Based encryption scheme that could encrypt using biometrics. To understand the problem, imagine I decide to use a biometric like your iris scan as the “identity” to encrypt you a ciphertext. Later on you’ll ask the authority for a decryption key that corresponds to your own iris scan — and if everything matches up and you’ll be able to decrypt.
The problem is that this will almost never work.
The issue here is that biometric readings (like iris scans or fingerprint templates) are inherently error-prone. This means every scan will typically be very close, but often there will be a few bits that disagree. With standard IBE
this is fatal: if the encryption identity differs from your key identity by even a single bit, decryption will not work. You’re out of luck.
Sahai and Waters decided that the solution to this problem was to develop a form of IBE with a “threshold gate”. In this setting, each bit of the identity is represented as a different “attribute”. Think of each of these as components you’d encrypt under — something like “bit 5 of your iris scan is a 1” and “bit 23 of your iris scan is a 0”. The encrypting party lists all of these bits and encrypts under each one. The decryption key generated by the authority embeds a similar list of bit values. The scheme is defined so that decryption will work if and only if the number of matching attributes (between your key and the ciphertext) exceeds some pre-defined threshold: e.g., any 2024 out of 2048 bits must be identical in order to decrypt.
The beautiful thing about this idea is not fuzzy IBE. It’s that once you have a threshold gate and a concept of “attributes”, you can more interesting things. The main observation is that a threshold gate can be used to implement the boolean AND and OR gates, like so:
Even better, you can stack these gates on top of one another to assign a fairly complex boolean formula — which will itself determine what conditions your ciphertext can be decrypted under. For example, switching to a more realistic set of attributes, you could encrypt a medical record so that either a pediatrician in a hospital could read it, or an insurance adjuster could. All you’d need is to make sure people received keys that correctly described their attributes (which are just arbitrary strings, like identities).
The other direction can be implemented as well. It’s possible to encrypt a ciphertext under a long list of attributes, such as creation time, file name, and even GPS coordinates indicating where the file was created. You can then have the authority hand out keys that correspond to a very precise slice of your dataset — for example, “this key decrypts any radiology file encrypted in Chicago between November 3rd and December 12th that is tagged with ‘pediatrics’ or ‘oncology'”.
Once you have a related of primitives like IBE and ABE, the researchers’ instinct is to both extend and generalize. Why stop at simple boolean formulae? Can we make keys (or ciphertexts) that embed arbitrary computer programs? The answer, it turns out, is yes — though not terribly efficiently. A set of recentworks show that it is possible to build ABE that works over arbitrary polynomial-size circuits, using various lattice-based assumptions. So there is certainly a lot of potential here.
This potential has inspired researchers to generalize all of the above ideas into a single class of encryption called “functional encryption“. Functional encryption is more conceptual than concrete — it’s just a way to look at all of these systems as instances of a specific class. The basic idea is to represent the decryption procedure as an algorithm that computes an arbitary function F over (1) the plaintext inside of a ciphertext, and (2) the data embedded in the key. This function has the following profile:
output = F(key data, plaintext data)
In this model IBE can be expressed by having the encryption algorithm encrypt (identity, plaintext) and defining the function F such that, if “key input == identity”, it outputs the plaintext, and outputs an empty string otherwise. Similarly, ABE can be expressed by a slightly more complex function. Following this paradigm, once can envision all sorts of interesting functions that might be computed by different functions and realized by future schemes.
But those will have to wait for another time. We’ve gone far enough for today.
So what’s the point of all this?
For me, the point is just to show that cryptography can do some pretty amazing things. We rarely see this on a day-to-day basis when it comes to industry and “applied” cryptography, but it’s all there waiting to be used.
Perhaps the perfect application is out there. Maybe you’ll find it.
* An earlier version of this post said “mid-1990s”. In comments below, Tom Ristenpart takes issue with that and makes the excellent point that many important developments post-date that. So I’ve moved the date forward about five years, and I’m thinking about how to say this in a nicer way.
** There is also an intermediate form of encryption known as “certificateless encryption“. Proposed by Al-Riyami and Paterson, this idea uses a combination of standard public key encryption and IBE. The basic idea is to encrypt each message using both a traditional public key (generated by the recipient) and an IBE identity. The recipient must then obtain a copy of the secret key from the IBE authority to decrypt. The advantages here are twofold: (1) the IBE key authority can’t decrypt the message by itself, since it does not have the corresponding secret key, which solves the “escrow” problem. And (2) the sender does not need to verify that the public key really belongs to the sender (e.g., by checking a certificate), since the IBE portion prevents imposters from decrypting the resulting message. Unfortunately this system is more like traditional public key cryptography than IBE, and does not have the neat usability features of IBE.
*** A part of the challenge of developing IBE is the need to make a system that is secure against “collusion” between different key holders. For example, imagine a very simple system that has only 2-bit identities. This gives four possible identities: “00”, “01”, “10”, “11”. If I give you a key for the identity “01” and I give Bob a key for “10”, we need to ensure that you two cannot collude to produce a key for identities “00” and “11”. Many earlier proposed solutions have tried to solve this problem by gluing together standard public encryption keys in various ways (e.g., by having a separate public key for each bit of the identity and giving out the secret keys as a single “key”). However these systems tend to fail catastrophically when just a few users collude (or their keys are stolen). Solving the collusion problem is fundamentally what separates real IBE from its faux cousins.
**** A full description of Boneh and Franklin’s scheme can be found here, or in the original paper. Some code is here and here and here. I won’t spend more time on it, except to note that the scheme is very efficient. It was patented and implemented by Voltage Security, now part of HPE.
This post is the second in a two-part series on zero-knowledge proofs. Click here to 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:
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:
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:
(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:
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 for some public key — but we don’t actually know the value . Our Simulator assumes that the Verifier will choose some value as its challenge, and moreover, it knows that the honest Verifier will choose the value only based on its random number generator — and not based on any inputs the Prover has provided.
First, output some initial as the Prover’s first message, and find out what challenge the Verifier chooses.
Rewind the Verifier, and pick a random integer in the range .
Compute and output as the Prover’s new initial message.
When the Verifier challenges on again, output .
Notice that the transcript will verify correctly as a perfectly valid, well-distributed proof of knowledge of the value . The Verifier will accept this output as a valid proof of knowledge of , even though the Simulator does not know 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 that corresponds to a public key . 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 with respect to a public key looks like this:
The Prover picks (just as in the interactive protocol).
Now, the prover computes the challenge as where is a hash function, and M is an (optional) and arbitary message string.
Compute (just as in the interactive protocol).
The upshot here is that the hash function is picking the challenge 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 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 , you obtain a signature on , which can only be produced by someone who knows the secret key . The resulting protocol is called the Schnorr signature scheme, and it’s the basis of real-world protocols like EdDSA.
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.
* In this definition, it’s necessary that the statement be literally true.