Factorization of solvable polynomials over finite fields and the generalized Riemann hypothesis

Abstract

This article presents an algorithm that, assuming the generalized Riemann hypothesis, factors a polynomial f mod p, where f ∃Z[X] is solvable overQ, into irreducible (over the fieldFpm) factors in time polynomial in m, log p, and the length of notation of f. The following problems are also solved in time polynomial in m, n, and log p: 1) construction of the fieldFpm, 2) construction of all isomorphisms between two realizations ofFpm, and 3) computation of the roots of degree n inFpm.