Ark-ff Small Field Support

[benbencik]

Description

The primary motivation of this PR is to create a generalized path toward vectorized/ SIMD instruction optimizations for finite fields in Arkworks.

In the process, we pick up a non-trivial performance boost in serial.

• We introduce SmallFp, a procedural macro for instantiating prime fields with modulus < 2^127
• In the place of BigInt, this macro uses native integer types (u8, u16, u32, u64, u128)
• For a summary of results and benchmarking please see section "results" in slides
• There a no breaking changes in this effort

SLIDES: https://andrewzitek.xyz/images/small_fp_slides.pdf

Closes: #1038

This work was done in collaboration with @z-tech

[z-tech]

@Pratyush @weikengchen @mmaker I've updated the PR and Issue with more information and the presentation slides: https://andrewzitek.xyz/images/small_fp_slides.pdf

We present this to Ale at 17h GMT+1 today. You are very welcome to attend and we are happy to schedule this 15 min presentation with you at another time.

Thanks.

[weikengchen]

I feel that the current shape is okay-enough for a review... let me take a look

[weikengchen]

I am just curious why not just always do

SmallFp::new(other % P::MODULUS)

or the like (due to types), since I feel it would be faster than the try_from and many other things that involve function calls.

[z-tech]

We know that other is u16 but the argument to SmallFp::new() is type T in {u8, u16, u32, u64, u128} so the cleanest I can figure out how to make it would be something like:

fn from(other: u16) -> Self {
        let modulus_as_u128: u128 = P::MODULUS.into();
        let other_as_u128: u128 = other.into();
        let reduced = other_as_u128 % modulus_as_u128;
        // let reduced_as_t = P::T::try_from(reduced).unwrap(); <-- this fails bc the error doesn't implement debug
        let reduced_as_t = P::T::try_from(reduced).unwrap_or_else(|_| panic!("Reduced value should fit in T"));
        SmallFp::new(reduced_as_t)
}

But by including the match, being that other is u16 we'd probably fall into the Ok() branch a huge amount of the time (but not all the time) and skip the u128 reduction.

[weikengchen]

I think I am in favor of the new one because it offers another advantage --- it seems to be mostly constant-time, while the previous one doesn't.

[z-tech]

Hey, I did another pass I think it's this that's needed to match the behavior of the BigInt impl: benbencik#10

[weikengchen]

(I think I have 6 files left but would love to hear from you on the ones above first)

[weikengchen]

We probably will find a way to bypass the nightly compilation failures in another PR... so ignore that for now
One way is to add unused_unsafe for now.

[weikengchen]

oh there is already a PR in the work for that: #1059 (comment)