I believe those gadgets are specifically for curves where the scalar
field is the base field of the curve you're working with, so they
probably wouldn't be that useful for arbitrary fields. Most of the
complexity here is the bignum arithmetic inside the circuit, though.

> Is there any more clever way to do this than just providing splitting into 
> bits to implement modular arithmetic in a different field?

Not that I know of. I explored the feasibility of this kind of stuff
in the past and concluded each point addition would be around the cost
of a SHA256 invocation. You can minimize the number of additions using
window tables. The best approach seemed to be giant window tables
queried with merkle tree lookups using something like MiMC. The
additions are most efficient when working with affine formulas
(inversions can be witnessed as efficiently as multiplications). You
may be able to get this down to 2^20 constraints for ~256-bit scalars,
which might be around 10-20 second proving time.

Sean

On Wed, Jan 3, 2018 at 1:36 PM, Andrew Miller <soc1...@illinois.edu> wrote:
> Suppose one did want to build a secp256k1 gadget. I notice that libsnark
> already provides a general gadget for  weierstrass form elliptic curves,
> parameterized by a field. So all we'd have to do is define the secp256k1
> operations in the alt_bn128 or in bls12 fields. Is there any more clever way
> to do this than just providing splitting into bits to implement modular
> arithmetic in a different field?
>
> On Jan 3, 2018 2:11 PM, "Sean Bowe" <s...@z.cash> wrote:
>>
>> If any curve is acceptable, I would encourage Jubjub, which we'll be
>> using for the next version of Zcash. In which case you will be able to
>> leverage our Sapling crypto code once it is more mature over the next
>> month or so. https://github.com/zcash-hackworks/sapling-crypto
>>
>> Sean
>>
>> On Wed, Jan 3, 2018 at 1:02 PM, James Prestwich via zapps-wg
>> <zapps...@lists.z.cash.foundation> wrote:
>> > I'd prefer sha256 or bitcoin-style hash160. I'm interested in a few
>> > different curves, including secp256k1. Eventually for EdDSA keys as
>> > well. Is
>> > there a list of supported curve operations?
>> >
>> > On Wed, Jan 3, 2018 at 12:57 PM Andrew Miller <soc1...@illinois.edu>
>> > wrote:
>> >>
>> >> Thank you so much for expressing your question in Camenisch-Stadler
>> >> notation! That makes it very clear what you're going for.
>> >>
>> >> What hash function H do you have in mind, would SHA2 work? Also what
>> >> group
>> >> G do you have in mind, secp256k1?
>> >>
>> >> If so, I do not know of any existing implementation of secp256k1
>> >> operations specifically in libsnark, so that would presumably be the
>> >> biggest
>> >> challenge.
>> >>
>> >>
>> >> On Jan 3, 2018 1:47 PM, "James Prestwich via zapps-wg"
>> >> <zapps...@lists.z.cash.foundation> wrote:
>> >>
>> >> I'd like to participate in the setup ceremony.
>> >>
>> >> I also have an app I'd like to build using a zk-proof of knowledge of
>> >> an
>> >> ECC private key. {(a) : A = a * G, B = H(a)}. Can anyone point me to
>> >> good
>> >> resources on getting started?
>> >>
>> >>
>> >

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