[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.

— 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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