Some algorithms for skew polynomials over finite fields

Xavier Caruso (1) , Jérémy Le Borgne (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

In this paper, we study the arithmetics of skew polynomial rings over finite fields, mostly from an algorithmic point of view. We give various algorithms for fast multiplication, division and extended Euclidean division. We give a precise description of quotients of skew polynomial rings by a left principal ideal, using results relating skew polynomial rings to Azumaya algebras. We use this description to give a new factorization algorithm for skew polynomials, and to give other algorithms related to factorizations of skew polynomials, like counting the number of factorizations as a product of irreducibles.

Mots clés: skew polynomials; algorithmics

Domaines: Théorie des nombres [math.NT]

Xavier Caruso, Jérémy Le Borgne. Some algorithms for skew polynomials over finite fields. 2012. ⟨hal-00765577⟩