Re: [Cfrg] Mishandling twist attacks

Samuel Neves <sneves@dei.uc.pt> Sat, 29 November 2014 01:20 UTC

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Date: Sat, 29 Nov 2014 01:19:45 +0000
From: Samuel Neves <sneves@dei.uc.pt>
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Subject: Re: [Cfrg] Mishandling twist attacks
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On 28-11-2014 17:22, Watson Ladd wrote:
> What exactly is wrong with telling everyone to multiply by 8, not 4,
> even if the cofactor is 4?

If your protocol is tightly coupled with an elliptic curve, nothing wrong with that, I suppose. But schemes and
protocols are often specified in terms of generic groups, where order and cofactor always exist, but the notion of twist
security may not.

> So if we add this requirement to have the curve have larger cofactor
> then the twist, then we still get E-521, and we will get Curve25519 at
> the low end. It seems to me like we should make this change to the
> generation method, and run it on 2^389-21 to get the intermediate size
> curve.

All this bickering further convinces me that complete Edwards curves over 3 (mod 4) primes are the way to go:

 - Square root computations are the simplest. 1 (mod 4) is too lenient, by the way: I don't think anybody is interested
in computing square roots over 1 (mod 8) primes.

 - Edwards curves over 3 (mod 4) primes can have both order and twist with cofactor 4.

 - For users obsessed with speed, Mike Hamburg has described how to use an isogeny to get twisted Edwards-speed out of
these curves [1].

We already have an excellent candidate in this space, namely Curve1174 over 2^251-9. E-521 is also such a curve. Since
2^389-21---which appears to be one of the nicest primes in that range (the other one being 2^379-19)---is also congruent
to 3 (mod 4), it seems logical to keep things consistent and choose 3 (mod 4) primes for every work factor.

[1] https://eprint.iacr.org/2014/027