finitely -- Optimized Finite Field Arithmetic

This crate implements arithmetic on rings of the form (Z/nZ)[x]/(p(x)) with arbitrary n and p. It aims to be performant and feature-rich.

Example usage:

use finitely::make_ring;

make_ring! {
  pub Field25 = { Z % 5, x^2 = [2] };
}

let x = Field25::from_coeffs(&[1, 0]);

assert_eq!(x * x, 2);

// Notice that 3 * 2 = 6, which is 1 modulo 5.
// Therefore x * (3 * x) = x * x * 3 = 2 * 3 = 1.
// Therefore the inverse of x is 3x.
assert_eq!(x.invert(), Some(x * 3));

let x_plus_one = x + 1;
assert_eq!(x_plus_one.invert(), Some(x + 4));
assert_eq!(x_plus_one / x, x * 3 + 1);

What does this implement (but for non-mathematicians)?

This crate lets you represent polynomials that look like a[0] + a[1]x + a[2]x^2 + ... + a[k]x^k, with equivalences enforced to make it constant-size. Namely, each one of the coefficients a[i] is taken modulo n (the equivalence here is to say that n is equivalent to zero), and that the polynomial p is equivalent to zero.

What does the second equivalence mean? Consider a polynomial that is of this shape: p(x) = x^m + b[m-1]x^(m-1) + ... + b[2]x^2 + b[1]x + b[0] (it is important that the coefficient of x^m be one). If we declare that p(x) is equivalent to zero, then we have essentially said that:

x^m = -(b[m-1]x^(m-1) + ... + b[2]x^2 + b[1]x + b[0])

So every polynomial with degree (highest power of x) which is greater than or equal to m can be rewritten as a polynomial of smaller degree.

Why do mathematicians care?

If we pick n carefully (prime), and p carefully (irreducible), then the mathematical structure we get out is what is known as a field. A field is a mathematical structure where you have the four typical operations you are used to from f32: +, -, *, and /. If n and p are not chosen carefully, then / does not exist (but the other three still do).

How is this useful to a non-mathematician?

This crate can be used to represent constant-length vectors of integers modulo n. If you do not use * or / (by another polynomial, and instead just regular integers), then you have an array of length m (where m is the degree of p) of integers modulo n.