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Fuentes et al. algorithm for computing hard part of final exponentiation #40

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Fuentes et al. algorithm for computing hard part of final exponentiation #40

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@Raneet10 Raneet10 commented Sep 16, 2023

This PR leverages the Fuentes et al. algorithm for computing the hard part of the final exponentiation based on the fact that f ^ d' is also a pairing and can be computed at least as efficiently as f ^ d where d' is a multiple of d and d = Φk(p)/r = (p⁴ − p² + 1)/r

goos: darwin
goarch: arm64
pkg: github.com/cloudflare/bn256
           │   old.txt   │              new.txt               │
           │   sec/op    │   sec/op     vs base               │
Pairing-10   695.0µ ± 0%   680.5µ ± 0%  -2.09% (p=0.000 n=50)

           │   old.txt    │            new.txt             │
           │     B/op     │     B/op      vs base          │
Pairing-10   38.25Ki ± 0%   38.25Ki ± 0%  ~ (p=1.000 n=50)

           │  old.txt   │            new.txt             │
           │ allocs/op  │ allocs/op   vs base            │
Pairing-10   352.0 ± 0%   352.0 ± 0%  ~ (p=1.000 n=50) ¹
¹ all samples are equal

The benchmark indicates slight improvement but I'd really love to get reviews to see whether further optimizations can be made.
Thank you!

@armfazh armfazh self-requested a review September 19, 2023 19:39
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