Factorization of polynomials over finite fields and factorization of primes in algebraic number fields

Abstract

Based on Kummer Theorem, we study the deterministic complexity of two factorization problems: polynomial factorization over finite fields and prime factorization in algebraic number fields. We show that factoring polynomials of degree n in Fp[x], with p prime, is polynimially equivalent to factoring p in algebraic number field of extension degree n over Q, where p is “regular” with respect to the generating polynomials of the number fields. Part of the proof also yields an efficient polynomial time algorithm for computing the factorization pattern. Number theoretical methods are then developed to solve two important kinds of polynomials:φn(x) mod p where φn is the n-th cyclotomic polynomial, and xn. - α mod p where α e N. We show that when Extended Riemann Hypothesis is assumed, all the roots of both kinds of polynomials in Fp can be found efficiently in time polynomial in n and logp. As α consequence, when p χ 1(n), factorization of p in the n-th cyclotomic field can be computed in polynomial time. The result on finding all roots of xn χ α(p) extends α result of Adleman, Menders, and Miller, which states that the least root of xn χ α(p) can be found in polynomial time, when Extended Riemann Hypothesis is assumed.