Factorization of polynomials over finite fields: Difference between revisions

Revision as of 08:50, 24 July 2024

Revision as of 12:52, 7 May 2025

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The correctness of the algorithm is based on the following:

<blockquote>{{Anchor|lemma}}'''Lemma.''' For ''i'' ≥ 1 the polynomial

: <math>x^{q^i}-x \in \mathbf{F}_q[x]</math>

is the product of all monic irreducible polynomials in '''F'''_''q''[''x''] whose degree divides ''i''.</blockquote>

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== Rabin's test of irreducibility ==

Like distinct-degree factorization algorithm, Rabin's algorithm<ref>{{cite journal |last1=Rabin |first1=Michael |year=1980 |title=Probabilistic algorithms in finite fields |journal=SIAM Journal on Computing |volume=9 |issue=2 |pages=273–280 |doi=10.1137/0209024 |citeseerx=10.1.1.17.5653 }}</ref> is based on the Lemma[[#lemma|lemma]] stated above. Distinct-degree factorization algorithm tests every ''d'' not greater than half the degree of the input polynomial. Rabin's algorithm takes advantage that the factors are not needed for considering fewer ''d''. Otherwise, it is similar to distinct-degree factorization algorithm. It is based on the following fact.

Let ''p''₁, ..., ''p_k'', be all the prime divisors of ''n'', and denote <math>n/p_i=n_i</math>, for 1 ≤ ''i'' ≤ ''k'' polynomial ''f'' in '''F'''_''q''[''x''] of degree ''n'' is irreducible in '''F'''_''q''[''x''] if and only if <math> \gcd \left (f,x^{q^{n_i}}-x \right )=1</math>, for 1&nbsp;≤&nbsp;''i''&nbsp;≤&nbsp;''k'', and ''f'' divides <math>x^{q^n}-x</math>. In fact, if ''f'' has a factor of degree not dividing ''n'', then ''f'' does not divide <math>x^{q^n}-x</math>; if ''f'' has a factor of degree dividing ''n'', then this factor divides at least one of the <math>x^{q^{n_i}}-x.</math>