On the generation of multivariate polynomials which are hard to factor

In this paper we consider the difficulty of factoring multivariate polynomials F(z, y, z,...) modulo n. We consider in particular the case in which F is a product of two randomly chosen polynomials P and Q with algebraically specified coefficients, and n is the product of two randomly chosen primes p and q. The general problem of factoring 1’ is known to be at least as hard as the factorization of n, but in many restricted cases (when P or Q are known to have a particular form) the problem can be much easier. The main result of this paper is that (with one trivial exception), the problem of factoring F is at least as hard as the factorization of n whenever P and Q are randomly chosen from the same sample space, regardless of what may be known