num-modular

A generic implementation of integer division and modular arithmetics in Rust. It provide basic operators and an type to represent integers in a modulo ring. Specifically the following features are supported:

• Common modular arithmetics: add, sub, mul, div, neg, double, square, inv, pow
• Optimized modular arithmetics in Montgomery form
• Optimized modular arithmetics with pseudo Mersenne primes as moduli
• Fast integer divisibility check
• Legendre, Jacobi and Kronecker symbols

It also support various integer type backends, including primitive integers and num-bigint. Note that this crate also supports [no_std]. To enable std related functionalities, enable the std feature of the crate.

lib.rs:

This crate provides efficient Modular arithmetic operations for various integer types, including primitive integers and num-bigint. The latter option is enabled optionally.

To achieve fast modular arithmetics, convert integers to any [ModularInteger] implementation use static new() or associated [ModularInteger::convert()] functions. Some builtin implementations of [ModularInteger] includes [MontgomeryInt] and [FixedMersenneInt].

Example code:

use num_modular::{ModularCoreOps, ModularInteger, MontgomeryInt};

// directly using methods in ModularCoreOps
let (x, y, m) = (12u8, 13u8, 5u8);
assert_eq!(x.mulm(y, &m), x * y % m);

// convert integers into ModularInteger
let mx = MontgomeryInt::new(x, &m);
let my = mx.convert(y); // faster than static MontgomeryInt::new(y, m)
assert_eq!((mx * my).residue(), x * y % m);

Comparison of fast division / modular arithmetics

Several fast division / modulo tricks are provided in these crate, the difference of them are listed below:

• [PreModInv]: pre-compute modular inverse of the divisor, only applicable to exact division
• Barret (to be implemented): pre-compute (rational approximation of) the reciprocal of the divisor, applicable to fast division and modulo
• [Montgomery]: Convert the dividend into a special form by shifting and pre-compute a modular inverse, only applicable to fast modulo, but faster than Barret reduction
• [FixedMersenne]: Specialization of modulo in form 2^P-K under 2^127.