Berlekamp Q-matrix

Related Concepts

Finite Field; Irreducible Polynomial

Definition

The Q-matrix is the key component in Berlekamp’s algorithm for factoring a polynomial over finite field.

Theory

Let ${ \bf \text {F}}_{q}$ be a finite field and let f(x) be a monic polynomial of degree d over ${ \bf \text {F}}_{q}$:

$$ f(x) = {x}^{d} + {f}_{ d-1}{x}^{d-1} + \cdots + {f}_{ 1}x + {f}_{0}, $$

where the coefficients ${f}_{0},\,\ldots,\,{f}_{d-1}$ are elements of ${ \bf \text {F}}_{q}$. The factorization of f(x) has the form

$$ f(x) ={ \prod \limits_{i}}{h}_{i}{(x)}^{{e}_{i} }, $$

where each factor ${h}_{i}(x)$ is an irreducible polynomial and ${e}_{i} \geq 1$ is the multiplicity of the factor ${h}_{i}(x)$.

Berlekamp’s algorithm exploits the fact that for any polynomial g(x) over ${ \bf \text {F}}_{q}$,

$$g{(x)}^{q} - g(x) ={ \prod \limits_{c\in {{\bf F}}_{ q}}}(g(x) - c).$$

Accordingly, given a polynomial g(x) such that

$$g{(x)}^{q} - g(x) \equiv 0{\rm mod}\,\,f(x),$$

one can find factors of f(...