I wrote an article/blog about how to do fast modular multiplication:
https://simonlindholm.github.io/files/bincoef.pdf
tl;dr:
- avoid latency-bound loops
- dynamic modulo is slow, constant modulo is fast
- if you perform many multiplications with the same dynamic modulo, you can do what the compiler does and use Barrett reduction (involves some precomputation)
- it is actually possible to beat the compiler if you accept a result in [0, 2*MOD):
uint64_t reduce(uint64_t a) {
return a - (uint64_t)((__uint128_t(-1ULL / MOD) * a) >> 64) * MOD;
}
Same goes for addition and subtraction: if you can live with a result in [0, 2*MOD), just do a + b
or a - b + MOD
and skip the range correction that brings the result into [0, MOD). Delay modular reductions far as possible, ideally combining them with multiplications. While being mindful of overflows, of course.
- on 32-bit, use Montgomery multiplication instead, to avoid __uint128_t
- if really desperate, combine Montgomery multiplication with SIMD; this runs 3x faster than Barrett reduction when AVX2 is available
- for larger numbers (e.g. multiplication of 64-bit numbers), use floating-point based methods