Factoring univariate integral polynomials in polynomial average time

Abstract

Let A be a primitive squarefree univariate integral polynomial of degree n. An irreducible factor of A can be found by forming products of lifted modulo p factors of A for a suitable small prime p. One can either form first the products consisting of the smallest numbers of lifted factors (cardinality procedure) or form first the products with smallest degrees (degree procedure). Let ∏ be the partition of n consisting of the degrees of the irreducible factors of A. The average number of products formed before finding an irreducible factor of A is a function of ∏, C(∏) or D(∏) respectively. Let C*(n) (D*(n)) be the maximum of C(∏) (D(∏)) for all partitions, ∏, of n. Subject to the validity of two conjectures, for which considerable evidence is presented, it is proved that C*(n) is dominated by n2 whereas D*(n) is exponential. If the conjectures are true then the cardinality procedure results in a complete factorization algorithm for primitive univariate integral polynomials whose average computing time, in a very strong sense, is dominated by a polynomial function of its degree n.