Reduce code duplication in field implementations

[str4d]

Currently this library has two field implementations, the Pallas field and the Vesta field. They only differ in their prime modulus, and are otherwise almost identically implemented. This makes the library harder to maintain, as every change needs to be duplicated.

There are several possible approaches we could take to improve the situation:

• Use a declarative macro.
  • We already use a declarative macro for implementing the curves, so it would be in keeping with the rest of the library.
  • This has been proposed and implemented here: https://github.com/zcash/pasta_curves/pull/44/commits/e150cc5670ba3559dd9a89609ffb41495ee560e3
• Use a procedural macro, i.e. ff_derive
  • This is how we originally implemented BLS12-381 in the pairing crate, but we replaced that with a direct implementation of the field in the bls12_381 crate, and ff_derive hasn't been as effectively maintained since. If we switched to it, the majority of our field maintenance effort could go into ff_derive instead of here.
• Use crypto-bigint to implement the internal field details.
  • This would reduce the amount of code we need to maintain, which would just be the mapping from a big integer element to a prime field element.
  • This would be quite a divergence from previous approaches. I also don't know how feature-complete crypto-bigint is, though progress is being made.
  • If this is a practical and performant approach, we could also combine it with either the declarative macro or ff_derive approach to make it easier to maintain as well.

[str4d]

Something I want to consider for this is how easy each of the approaches makes implementing "backends": specialisations for different targets for improved performance (e.g. AVX2).

[ashWhiteHat]

And I also have idea about code duplication.

We can separate interface and arithmetic(algorithm). The arithmetic is often the same for field and curve. We can implement it separately like following. https://github.com/zero-network/zero/tree/master/primitive/crypto/src/arithmetic And interface is just passing its arguments to arithmetic function. https://github.com/zero-network/zero/blob/master/primitive/crypto/src/dress/curve/edwards/group.rs#L76

If we apply the optimization to arithmetic, it affects for every component. How about creating zkcrypto/crypto-arithmetic and import arithmetic from components? It will be a big refactoring but we can reduce code duplication and synchronizing workload.

[ashWhiteHat]

I think macro is not good way to reduce code because it makes visibility bad. but we can use macro for the parts where visibility is not necessary as in RefOps. We can also implement it as module with the same way with above arithmetic.