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New implementation of sqrt_ratio using the modified version of the Tonelli--Shanks algorithm from https://eprint.iacr.org/2020/1497.pdf the gain is made mostly when (p - 1) is 2-adically large (3 mod 4, 5 mod 8, etc). In these cases the 1 inversion, 2 sqrts (= 3 exponentiations) of the previous implementation is replaced with 1 merged sqrt-ratio (= 1 exponentiation). Performance should become similar between the old and new implementations when p - 1 = 2^s * t with t small. Main changes: - Refactor `sqrt_impl` to separate out the Tonelli--Shanks loop logic - Write new `sqrt_ratio_impl` to generate an implementation for each prime. - Write new test to check `sqrt_ratio` behaviour. Note: Old function `sqrt_ratio_generic` no longer plays a role if derive is used.
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Implement in ff_derive the "merged" version of
sqrt_ratiousing the Tonelli--Shanks algorithm as described in Scott's Tricks of the trade article.In broad terms if$\mathbb{F}_p$ , and the same is true for inversions in $\mathbb{F}_p$ . The current implementation $2^{255} - 19$ ) when
p - 1 = 2^s * twithtlarge-ish the bulk of the Tonelli--Shanks algorithm is bundled in a single exponentiation insqrt_ratio_genericrequires 1 inversion and 2 square-roots (hence 3 exponentiations). Writingx = num^3 * divone can bundle all of these into a single "projenitor" calculation. This leads to a speed-up (e.g., ~2.5x forp-1is "not-too-2-adically-small" (with respect top). Whentis small (e.g., the fields used by JubJub, Bls381, Pallas, Vesta) performance is more-or-less comparable (or maybe very slightly worse) since most of the time is soaked up in the "loop" part of Tonelli--Shanks (which, in both cases, is called twice).Some (naive) comparisons can be found at this link.