[curves] How to find another generator on decaf_448？
Mike Hamburg
mike at shiftleft.org
Fri Jan 20 14:42:55 PST 2017
Hi Fan,
Decaf’s cofactor is 1, so all non-identity points are generators.
For Cramer-Shoup you will need a random point, such that it’s hard to figure out its discrete log (base g). You will need to be able to argue that the point was really generated in a way that would make it hard to embed a back door. A straightforward way to get this property is by hashing a random seed, and then applying Elligator. Since Cramer-Shoup is specified as using a *uniformly* random point (even though it’s probably secure with something slightly less than uniform), you should use point_from_hash_uniform. Since Cramer-Shoup is designed to be secure in the standard model, you should include a uniformly random seed, perhaps 512 bits long. To prevent a theoretical backdoor mentioned by Stanislav Smyshlaev, you should hash the base point as well.
Overall, the computation would then be elligator(hash(base_point, seed)). In C++, that’s something like:
std::string seed = [a fixed 512-bit constant which you chose at random];
Point::from_hash(SHAKE<256>::Hash(std::string(Point::base()) + seed, Point::HASH_BYTES*2))
If you’re using two random generators instead of random + base point, then hashing in Point::base() above isn’t necessary, but the hash itself is still required.
You might as well check that the resulting point isn’t the identity. You can check that orderQ * P == identity if you like, but that should be true for any output of Point::from_hash.
Cheers,
— Mike
> On Jan 20, 2017, at 1:01 PM, Fan Jiang <fan.torchz at gmail.com> wrote:
>
> Hi,
> I'm currently working on a CramerShoup implementation using decaf_448,
> Whereas decaf is to eliminate the cofactor by compression,
> Should I still use the equation "orderQ*cofactor*P == identity" to check the candidate generator P?
> Or, What should be a "valid" generator mean in this use case?
>
> Thanks,
> Fan
>
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