Galois Fields

This section contains classes and functions for creating Galois field arrays.

Galois field classes

Class factory functions

GF(order[, irreducible_poly, ...])	Creates a FieldArray subclass for \(\mathrm{GF}(p^m)\).
Field(order[, irreducible_poly, ...])	Alias of GF().

Abstract base classes

FieldArray(x[, dtype, copy, order, ndmin])

A ndarray subclass over \(\mathrm{GF}(p^m)\).

Pre-made FieldArray subclasses

GF2(x[, dtype, copy, order, ndmin])

A ndarray subclass over \(\mathrm{GF}(2)\).

Prime field functions

Primitive roots

primitive_root(n[, start, stop, method])	Finds a primitive root modulo \(n\) in the range [start, stop).
primitive_roots(n[, start, stop, reverse])	Iterates through all primitive roots modulo \(n\) in the range [start, stop).
is_primitive_root(g, n)	Determines if \(g\) is a primitive root modulo \(n\).

Extension field functions

Irreducible polynomials

irreducible_poly(order, degree[, method])	Returns a monic irreducible polynomial \(f(x)\) over \(\mathrm{GF}(q)\) with degree \(m\).
irreducible_polys(order, degree[, reverse])	Iterates through all monic irreducible polynomials \(f(x)\) over \(\mathrm{GF}(q)\) with degree \(m\).

Primitive polynomials

primitive_poly(order, degree[, method])	Returns a monic primitive polynomial \(f(x)\) over \(\mathrm{GF}(q)\) with degree \(m\).
primitive_polys(order, degree[, reverse])	Iterates through all monic primitive polynomials \(f(x)\) over \(\mathrm{GF}(q)\) with degree \(m\).
conway_poly(characteristic, degree)	Returns the Conway polynomial \(C_{p,m}(x)\) over \(\mathrm{GF}(p)\) with degree \(m\).
matlab_primitive_poly(characteristic, degree)	Returns Matlab's default primitive polynomial \(f(x)\) over \(\mathrm{GF}(p)\) with degree \(m\).

Primitive elements

primitive_element(irreducible_poly[, method])	Finds a primitive element \(g\) of the Galois field \(\mathrm{GF}(q^m)\) with degree-\(m\) irreducible polynomial \(f(x)\) over \(\mathrm{GF}(q)\).
primitive_elements(irreducible_poly)	Finds all primitive elements \(g\) of the Galois field \(\mathrm{GF}(q^m)\) with degree-\(m\) irreducible polynomial \(f(x)\) over \(\mathrm{GF}(q)\).
is_primitive_element(element, irreducible_poly)	Determines if \(g\) is a primitive element of the Galois field \(\mathrm{GF}(q^m)\) with degree-\(m\) irreducible polynomial \(f(x)\) over \(\mathrm{GF}(q)\).

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Last update: May 18, 2022