Constant galois_2p8::field::PRIMITIVES

pub const PRIMITIVES: [IrreducablePolynomial; 16];

Contains the primitive polynomials of GF(2^8).

In GF(2^x) wherein arithmetic operations are performed modulo a polynomial, the polynomial is said to be primitive if for some alpha, each nonzero member value of the field can be uniquely represented by alpha ^ p for p < 2^x, where alpha is a root of the polynomial. That is, for some root of the polynomial, every member of the field can be represented as the exponentiation of said root.

In GF(2^x), the only nontrivial prime root of any given IrreducablePolynomial is two. We say “is primitive” as a shorthand for meaning that the IrreducablePolynomial is primitive assuming a root of two.

The use of primitive polynomials confers an immediate performance benefit for single values: we can represent multiplication and division as addition and subtraction within logarithms and exponents. Additionally, some usages of Galois fields, e.g. Reed-Solomon syndrome coding, require primitive polynomials to function properly.

This table is used by the is_primitive method. Specifically,

PRIMITIVES.binary_search(&poly).is_ok() == poly.is_primitive()