The Complete Analysis of a Polynomial Factorization Algorithm over Nite Elds the Complete Analysis of a Polynomial Factorization Algorithm over Nite Elds the Complete Analysis of a Polynomial Factorization Algorithm over Finite Fields

A uniied treatment of parameters relevant to factoring polynomials over nite elds is given. The framework is based on generating functions for describing parameters of interest and on singularity analysis for extracting asymptotic values. An outcome is a complete analysis of the standard polynomial factorization chain that is based on elimination of repeated factors, distinct degree factorization, and equal degree separation. Several basic statistics on polynomials over nite elds are obtained in the course of the analysis. L'analyse compl ete d'un algorithme de factorisation de polyn^ omes sur les corps nis R esum e : Cet article propose un cadre unii e pour l'analyse des param etres qui intervien-nent dans la factorisation de polyn^ omes sur un corps ni. L' etude repose d'une part sur les s eries g en eratrices utilis ees pour d ecrire les principaux param etres, d'autre part sur l'analyse de singularit e qui fournit les informations asymptotiques correspondantes. Est ainsi obtenue une analyse compl ete d'une cha^ ne classique de factorisation fond ee sur l' elimination des fac-teurs r ep et es, la factorisation en degr e disticts, et la s eparation des degr es egaux. Plusieurs statistiques de base relatives aux polyn^ omes sur les corps nis r esultent de cette analyse. Abstract. A uniied treatment of parameters relevant to factoring polynomials over nite elds is given. The framework is based on generating functions for describing parameters of interest and on singularity analysis for extracting asymptotic values. An outcome is a complete analysis of the standard polynomial factorization chain that is based on elimination of repeated factors, distinct degree factorization, and equal degree separation. Several basic statistics on polynomials over nite elds are obtained in the course of the analysis. 1. Introduction Factoring polynomials over nite elds intervenes in many areas like polynomial factorization over the integers 8, 31], cryptography 7, 33, 36], number theory 3], or coding theory 2]. The implications include nding complete partial fraction decompositions (a problem itself useful for symbolic integration), designing cyclic redundancy codes, computing the number of points on elliptic curves, and building arithmetic public key cryptosystems. In particular, the factorization of random polynomials over nite elds is needed in the index calculus method for computing discrete logarithms over nite elds 36]. This paper derives basic probabilistic properties of random polynomials over nite elds that are of interest in the study of polynomial factorization algorithms. We show that …