Galois groups and factoring polynomials over finite fields

Abstract

Let p be a prime and F be a polynomial with integer coefficients. Suppose that the discriminant of F is not divisible by p, and denote by m the degree of the splitting field of F over Q and by L the maximal size of the coefficients of F. Then, assuming the generalized Riemann hypothesis (GRH), it is shown that the irreducible factors of F modulo p can be found in deterministic time polynomial in deg F, m, log p, and L. As an application, it is shown that it is possible under GRH to solve certain equations of the form nP=R, where R is a given and P is an unknown point of an elliptic curve defined over GF(p) in polynomial time (n is counted in unary). An elliptic analog of results obtained recently about factoring polynomials with the help of smooth multiplicative subgroups of finite field is proved. >