Factoring Polynomials over Special Finite Fields

Authors:
Eric Bach;Joachim Von Zur Gathen;Hendrik W. Lenstra, Jr.
Affiliations:
Computer Sciences Department, University of Wisconsin, Madison, Wisconsin, 53706, f1bach@cs.wisc.eduf1;Fachbereich Mathematik-Informatik, Universität Paderborn, 33095, Paderborn, Germanyf2gathen@uni-paderborn.def2;Department of Mathematics # 3840, University of California, Berkeley, California, 94720-3840, Mathematisch Instituut, Universiteit Leiden, Postbus 9512, 2300 RA, Leiden, The Netherlandsf3hwl@math. ...
Venue:
Finite Fields and Their Applications
Year:
2001

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Abstract

We exhibit a deterministic algorithm for factoring polynomials in one variable over finite fields. It is efficient only if a positive integer k is known for which @F"k(p) is built up from small prime factors; here @F"k denotes the kth cyclotomic polynomial, and p is the characteristic of the field. In the case k=1, when @F"k(p)=p-1, such an algorithm was known, and its analysis required the generalized Riemann hypothesis. Our algorithm depends on a similar, but weaker, assumption; specifically, the algorithm requires the availability of an irreducible polynomial of degree r over Z/pZ for each prime number r for which @F"k(p) has a prime factor l with l=1 mod r. An auxiliary procedure is devoted to the construction of roots of unity by means of Gauss sums. We do not claim that our algorithm has any practical value.