jnz@riseup.net transcribed 2.0K bytes:

Hey Isis,

I have a few crypto questions for you. The bonus question is purely out of interest (I searched for the answer, but didn't get a good/understandable answer, if you are busy don't worry about it).

Hey Jake!

I'm CCing the conversation to tor-dev, since these are all really good questions. (Also you found another problem in our specs!)

Q1. What does the bit argument in the the ntor-onion-key-crosscert field do?

The following is an example ntor-onion-key-crosscert field. The bit argument is the "1" following ntor-onion-key-crosscert keyword.

...

ntor-onion-key-crosscert 1 -----BEGIN ED25519 CERT----- AQoABl2EAdjttTuvK9jQzdyi+Mh46ky1Jk9+Hw+FJ6eos5zUxcBDAJF+TGU6oZij DOToeg+/lIaEkv0RwIrl5aJta/jf2EdTgHRfCsMOLEWMfqjOA6pmkwc3jnWvWNms 7NVBzUcMWgA= -----END ED25519 CERT-----

Q2. The following sentence is the bit argument's documentation from the dir-spec. Could you possibly explain to me what this means? What is a public key's "sign"?

...

Bit must be "0" or "1". It indicates the sign of the ed25519 public key corresponding to the ntor onion key.

The following paragraph is also in the dir-spec, I know that it is relevant, but I don't understand what it means.

...

C. Converting a curve25519 public key to an ed25519 public key

Given a curve25519 x-coordinate (u), we can get the y coordinate of the ed25519 key using

y = (u-1)/(u+1)

and then we can apply the usual ed25519 point decompression algorithm to find the x coordinate of the ed25519 point to check signatures with.

Note that we need the sign of the X coordinate to do this operation; otherwise, we'll have two possible X coordinates that might have correspond to the key. Therefore, we need the 'sign' of the X coordinate, as used by the ed25519 key expansion algorithm.

To get the sign, the easiest way is to take the same private key, feed it to the ed25519 public key generation algorithm, and see what the sign is.

BONUS. What is the difference between a curve25519 public key and an ed25519 public key? Why would you want to convert back and forth between the two?

Okay, this is going to be a long answer. Let me know if parts don't make sense.

So, as you noted, dir-spec.txt says [0]:

Bit must be "0" or "1".  It indicates the sign of the ed25519
public key corresponding to the ntor onion key.

To compute the ed25519 public key corresponding to a
curve25519 key, see appendix C.

There are a few different representations of the curve over the Galois field GF(2²⁵⁵ - 19) called curve25519. The (arguably) most useful is the twisted Edwards form of the curve (see our Rust documentation [1] for further details on point forms and more literature) defined for affine points P = (x,y) by the equation

-x²+y² = 1 + dx²y² (1)

where

d = - 121665/121666

This is due to the fact that efficient, complete formulae (meaning that the same formulae, e.g. for point addition, can be used for all points, so you don't have messy if/then statements to deal with exceptional points) exist for Edwards curves. (Recently, complete formulae for Weierstrass curves and other prime-order curves have also been described. [2] However, they are still slightly less-efficient for most tasks.)

There's a useful technique called a Montgomery ladder, which is an approach to constant-time (if implemented in a sane manner) point multiplication (i.e. computing nP by taking the point P and adding it to itself n times). This is particularly useful for elliptic curve Diffie-Hellman (ECDH) implementations, but it requires first converting a point from a certain representation (a.k.a. within a specific spatial coordinate system), usually "extended homogeneous twisted Edwards coordinates" (see Hişil's thesis [3], which is the explanation of all the different forms and group laws), into the affine point Q = (u,v) as it would lie on the Montgomery form of the curve:

bv² = u³ + au² + u (2)

where

a = 486662 b = 1

The conversion is rather straighforward, as you can see from mine and Henry de Valence's code [4], but there's a problem in that the compressed form (i.e. "give me the serialised version that is just an array of 32 bytes") of points on the Montgomery curve are just the u-coordinate. You can use this u-coordinate to recover the v-coordinate, [5] and then use both u and v to recover the original Edwards x-coordinate, [6] but there's two possible x-coordinates, one positive and one negative.

So, if you want to be able to use a key for both purposes, then, when using Montgomery form (e.g. an X25519 key a.k.a. a "curve25519 key") then you have to also pass around the sign of x. It's kind of icky.

In this case, the "ntor-onion-key-crosscert" is an Ed25519 signature (so the public key is a point in twisted Edwards form) of the relay's public master identity key, computed using the "ntor-onion-key" (which is an X25519 key, so a point in Montgomery form). The verifier knows the "ntor-onion-key", but needs to know the sign of x in order to be able to convert it into the Edwards-form public key to check this signature.

Following through the code tor is using, it appears that if "Bit" is "0", then x will be positive. If "Bit" is "1" then x will be negative. Tor's behaviour w.r.t. the sign bit (using either curve25519-donna [7] or ref10 [8]) appears to be identical to other implementations, namely Open Whispersystems' code [9] and curve25519-dalek. [10]

It appears it isn't specified anywhere what the "Bit" is supposed to mean in terms of parsing the ntor-onion-cert into an Ed25519 key though. :( I made an attempt at specifying it better. [11]

As for the question of why one would want to convert back and forth between the two… normally, you wouldn't want to do this. There's nothing concretely horribly wrong or insecure about it, but there are plenty of more theoretical concerns regarding any type of key reuse for different protocols, primarily because it removes any sense of domain separation, such that an attack on one protocol might be assisted by or contribute to attacking the other. With separate keys for separate purposes, you can breathe a lot easier knowing that if, e.g. your ECDH keys were compromised, that your signing key (being a totally different key) is (probably) unaffected. The only time where is does make sense to share a key between two protocols is what Nick's design for the cross-certifications, i.e.:

a) Prove that you control the private part of an encryption key by converting it to a signing key, then signing the public master identity key, and b) Authenticate the signature by using the master identity key to sign the entire document (including the other signature).

Thanks for all the help, I hope that you are having a nice day and... Happy Hacking.

• Jake

No worries! Thanks for the excellent questions and catching more bugs in our specs!

[0]: https://gitweb.torproject.org/torspec.git/tree/dir-spec.txt?id=ce09d266#n536 [1]: https://docs.rs/curve25519-dalek/0.10.0/curve25519_dalek/curve/index.html [2]: https://ellipticnews.wordpress.com/2016/05/30/complete-group-laws-for-prime-... [3]: https://github.com/isislovecruft/library--/blob/master/cryptography%20%26%20... [4]: https://github.com/isislovecruft/curve25519-dalek/blob/f2883028d/src/curve.r... [5]: https://github.com/isislovecruft/curve25519-dalek/blob/f2883028d/src/curve.r... [6]: https://github.com/isislovecruft/curve25519-dalek/blob/f2883028d/src/curve.r... [7]: https://gitweb.torproject.org/tor.git/tree/src/ext/ed25519/donna/ed25519_tor... [8]: https://gitweb.torproject.org/tor.git/tree/src/ext/ed25519/ref10/keyconv.c?i... [9]: https://github.com/WhisperSystems/curve25519-java/blob/master/android/jni/ed... [10]: https://github.com/isislovecruft/curve25519-dalek/blob/f2883028d/src/curve.r... [11]: https://gitweb.torproject.org/torspec.git/commit/?id=2395f34affbe97c19d7bb9e...

Best,