feat(field): add mersenne31 prime field

[PatStiles]

Adds Mersenne31 Prime Field

Description

Description of the pull request changes and motivation.

Starts #540

Adds an implementation of the Mersenne31 prime field following the implementation in Plonky 3.

Please delete options that are not relevant.

• New feature

Checklist

• Linked to Github Issue
• Unit tests added

[codecov-commenter]

Codecov Report

Merging #565 (f9e614a) into main (2c2e0f7) will decrease coverage by 0.04%.
The diff coverage is 93.33%.

@@            Coverage Diff             @@
##             main     #565      +/-   ##
==========================================
- Coverage   95.61%   95.58%   -0.04%     
==========================================
  Files         111      113       +2     
  Lines       19310    19595     +285     
==========================================
+ Hits        18464    18730     +266     
- Misses        846      865      +19     

Files	Coverage Δ
math/src/field/fields/u32_mersenne_prime_field.rs	98.51% <98.51%> (ø)
...elds/fft_friendly/u64_mersenne_montgomery_field.rs	0.00% <0.00%> (ø)

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[PatStiles]

@MauroToscano @schouhy Bench results for the Mersenne31 backend in comparison to a U64 montgomery backend

[MauroToscano]

@MauroToscano @schouhy Bench results for the Mersenne31 backend in comparison to a U64 montgomery backend

The benchmark seems wrong, it's based in the old benchmarks of the Math library which are also not good.

When using Criterion, the compiler may optimize when constants that are always the same. That's why you see one Pow taking 300 ps, which is highly suspicious. If you blackbox the paramater with "bench.iter(|| x.pow(black_box(*y)));", now the mersenne's backend is faster. Which is nice!

If you want to copy the ones of Math, you will have to fix this kind of issues. If you see some difference that big it may be the compiler optimizing something that it shouldn't be optimizing.

The ones used here are more solid: https://github.com/lambdaclass/lambdaworks/tree/main/benches. We can improve the ones in math and fix this kind of issues.

[PatStiles]

@MauroToscano Appreciate the feedback will get on it!

[MauroToscano]

Looks good now

[PatStiles]

@MauroToscano @schouhy I finished adding the field extensions and hooking into libfuzzer. One note is the lambdaworks quadratic extension differed slightly from the implementation of the inv() operation for the Mersenne31 fields complex extension in plonky3