Secp256r1

[SherLzp]

Hi sir @gbotrel @yelhousni , recently I want to add support for secp256r1 curve for gnark-crypto.

I'm fully understanding how the following value comes, could you please explain it, appreciate it:

1. What's the q and r here?
   https://github.com/ConsenSys/gnark-crypto/blob/develop/ecc/secp256k1/fp/element.go#L82

2. How about the values here:
   https://github.com/ConsenSys/gnark-crypto/blob/develop/ecc/secp256k1/fp/element.go#L298-L301

And is there any other parts should I take care?

[yelhousni]

Hi @SherLzp !

1. q is the modulus 115792089237316195423570985008687907853269984665640564039457584007908834671663and r is 2^64. So -1/q % 2^64 is 15580212934572586289.
2. These are the 64-bit limbs of (q-1)/2 in decimal. e.g. write (q-1)/2 in binary and take the first 64 bits. This is 1111111111111111111111111111111111111111111111111111111111111111 in binary and 18446744073709551615 in decimal.

You shouldn't care about these as they are auto-generated. You just need to put the curve moduli in the config file (e.g. for secp256k1) and the fp/ and fr/ packages are auto-generated.

[SherLzp]

thanks!