Ian Goldberg iang@cs.uwaterloo.ca writes:
On Mon, Apr 24, 2017 at 04:47:44PM +0300, George Kadianakis wrote:
Ian Goldberg tor@cypherpunks.ca writes:
On Thu, Apr 20, 2017 at 03:40:58PM +0300, George Kadianakis wrote:
Hey Ian,
so the current problem with ed25519-donna is that when I'm doing the above, I'm getting the following 32-byte key after ge25519_pack(). Here it is hexed up:
0x0100000000000000000000000000000000000000000000000000000000000000I don't see any methods in ed25519-donna for checking a point or a packed point to see if it's the point at infinity (identity element).
There actually is such a function, but it's buried in a weird place. ge25519_is_neutral_vartime in ed25519-donna-batchverify.h does what you want, but it's static, so you would need to copy it (and remove the line involving batch_point_buffer). Probably not worthwhile.
I can easily believe the representation of the identity (neutral) element in ed25519 is the hex value above. (And it is; see below.)
I've been assuming that the point at infinity would be an all-zeroes 32-byte array, because that's what we are doing in curve25519, but I actually have no idea how the array representation of ge25519_pack() works. Here is our process for curve25519: https://gitweb.torproject.org/tor.git/tree/src/common/crypto_curve25519.c#n1...
Curve25519 only outputs the x coordinate, which is indeed 0, so you get the all-zero value. ed25519 outputs the y coordinate (which is the 255-bit value 0x000...0001 in little-endian format) and the single-bit parity of the x coordinate (which is 0), so you do get the hex you give above. (The identity element is the point (0,1) in Edwards curves, not actually the point at infinity.)
The above packed point is an all zeroes 32-byte string, apart from the very first bit which is set. Could it be that the first bit is the compressed sign of the 'x' coordinate or something, and that's actually the point at infinity?
But then what's the right way to check for the point at infinity? Check that everything is 0 apart from the first sign bit?
Yes, pack the point, and compare it to the above 32-byte value.
And after I figure this out, I need to do the same for the reference implementation of ed25519, because as it seems Tor uses two simultaneous implementations of ed25519: https://gitweb.torproject.org/tor.git/tree/src/ext/ed25519
Yes, hopefully that implementation packs points in the same way!
Thanks for the help Ian :) Very much appreciated!
No worries.
[CCing tor-dev since people might find it interesting]
Thanks for the advice Ian :)
With your help, I have now implemented the validation check. Check out the ticket #22006 and the branch `bug22006` here: https://gitweb.torproject.org/user/asn/tor.git/commit/?h=bug22006&id=628...
You should also check that the point it not *itself* the identity element.
One last thing I'd like to do here is test our validation function against a pubkey with torsion component to make sure that it gets rejected. How would you go about generating such a pubkey?
I was thinking of using the CRT to find an integer a \in Z, s.t a == 0 (mod l) && a == 1 (mod 8) but then I didn't know how to go from that scalar with a torsion component to an ed25519 point...
It turns out the point whose packed representation is 32 bytes of 0x00 is a torsion point; it is the point (-1,0).
Indeed, these are the 7 pure torsion points in ed25519:
26e8958fc2b227b045c3f489f2ef98f0d5dfac05d3c63339b13802886d53fc05 0000000000000000000000000000000000000000000000000000000000000000 =(-1,0) c7176a703d4dd84fba3c0b760d10670f2a2053fa2c39ccc64ec7fd7792ac037a ecffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f =(0,-1) c7176a703d4dd84fba3c0b760d10670f2a2053fa2c39ccc64ec7fd7792ac03fa 0000000000000000000000000000000000000000000000000000000000000080 =(1,0) 26e8958fc2b227b045c3f489f2ef98f0d5dfac05d3c63339b13802886d53fc85
So just take any of the above points, and add it to a valid pubkey to get an invalid pubkey.
You should probably also check points not on the curve at all, such as:
e19c65de75c68cf3b7643ea732ba9eb1a3d20d6d57ba223c2ece1df66feb5af0
If you generate a 32-byte string at random, about 1/2 the time it won't be on the curve at all (that is, if P is the unpack of those 32 bytes, 8*l*P is *not* the identity), about 7/16 of the time it is on the curve, but has a torsion component (8*l*P is the identity, but l*P is not), and 1/16 of the time it's a valid pubkey (l*P is the identity, but P is not).
Good stuff Ian.
I pushed a new branch `bug22006` that: - Checks that the pubkey is not the identity element itself. - Adds tests based on the points you listed above.
Check it out here: https://gitweb.torproject.org/user/asn/tor.git/log/?h=bug22006_v2
Another thing:
My understanding is that this is a special-case validation and it's expensive, so I opted to not perform it everytime we generate a new ed25519 keypair or when we receive one from the internet. So for example, I'm not doing it when we extract the ed25519 signing pubkey that signs the HS descriptor, since we don't care if there are equivalent forms of that key.
So this validation function is currently unused, and I plan to only use it on the prop224 client-side when a client handles received onion addresses.
Finally, how did you derive the list of points above? o.o
Cheers!