Psychic Signatures in Java

This is probably the cryptography bug of the year. It's easy to exploit and bypasses signature verification on anything using ECDSA in Java, including SAML and JWT (if you're using ECDSA in either).

The bug is simple: like a lot of number-theoretic asymmetric cryptography, the core of ECDSA is algebra on large numbers modulo some prime. Algebra in this setting works for the most part like the algebra you learned in 9th grade; in particular, zero times any algebraic expression is zero. An ECDSA signature is a pair of large numbers (r, s) (r is the x-coordinate of a randomly selected curve point based on the infamous ECDSA nonce; s is the signature proof that combines x, the hash of the message, and the secret key). The bug is that Java 15+ ECDSA accepts (0, 0).

For the same bug in a simpler setting, just consider finite field Diffie Hellman, where we agree on a generator G and a prime P, Alice's secret key is `a mod P` and her public key is `G^a mod P`; I do the same with B. Our shared secret is `A^b mod P` or `B^a mod P`. If Alice (or a MITM) sends 0 (or 0 mod P) in place of A, then they know what the result is regardless of anything else: it's zero. The same bug recurs in SRP (which is sort of a flavor of DH) and protocols like it (but much worse, because Alice is proving that she knows a key and has an incentive to send zero).

The math in ECDSA is more convoluted but not much more; the kernel of ECDSA signature verification is extracting the `r` embedded into `s` and comparing it to the presented `r`; if `r` and `s` are both zero, that comparison will always pass.

It is much easier to mess up asymmetric cryptography than it is to mess up most conventional symmetric cryptography, which is a reason to avoid asymmetric cryptography when you don't absolutely need it. This is a devastating bug that probably affects a lot of different stuff. Thoughts and prayers to the Java ecosystem!

Interestingly, EdDSA (generally known as Ed25519) does not need as many checks as ECDSA, and assuming the public key is valid, an all-zero signature will be rejected with the main checks. All you need to do is verify the following equation:

R = SB - Hash(R || A || M) A

Where R and S are the two halves of the signature, A is the public key, and M is the message (and B is the curve's base point). If the signature is zero, the equation reduces to Hash(R || A || M)A = 0, which is always false with a legitimate public key.

And indeed, TweetNaCl does not explicitly check that the signature is not zero. It doesn't need to.

However.

There are still ways to be clever and shoot ourselves in the foot. In particular, there's the temptation to convert the Edwards point to Montgomery, perform the scalar multiplication there, then convert back (doubles the code's speed compared to a naive ladder). Unfortunately, doing that introduces edge cases that weren't there before, that cause the point we get back to be invalid. So invalid in fact that adding it to another point gives us zero half the time or so, causing the verification to succeed even though it should have failed!

(Pro tip: don't bother with that conversion, variable time double scalarmult https://loup-vaillant.fr/tutorials/fast-scalarmult is even faster.)

A pretty subtle error, though with eerily similar consequences. It looked like a beginner-nuclear-boyscout error, but my only negligence there was messing with maths I only partially understood. (A pretty big no-no, but I have learned my lesson since.)

Now if someone could contact the Whycheproof team and get them to fix their front page so people know they have EdDSA test vectors, that would be great. https://github.com/google/wycheproof/pull/79 If I had known about those, the whole debacle could have been avoided. Heck, I bet my hat their ECDSA test vectors could have avoided the present Java vulnerability. They need to be advertised better.

This is the sort of dumb mistake that ought to get caught by unit testing. A junior, assigned the task of testing this feature, ought to see that in the cryptographic signature design these values are checked as not zero, try setting them to zero, and... watch it burn to the ground.

Except that, of course, people don't actually do unit testing, they're too busy.

Somebody is probably going to mention fuzz testing. But, if you're "too busy" to even write the unit tests for the software you're about to replace, you aren't going to fuzz test it are you?

Thank you for that, that was a great explanation.