Recovering zeros of polynomials modulo a prime

Abstract

Let $ p$ be a prime and $ \mathbb{F}_p$ the finite field with $ p$ elements. We show how, when given an irreducible bivariate polynomial $ F \in \mathbb{F}_p[X,Y]$ and an approximation to a zero, one can recover the root efficiently, if the approximation is good enough. The strategy can be generalized to polynomials in the variables $ X_1,\ldots ,X_m$ over the field $ \mathbb{F}_p$. These results have been motivated by the predictability problem for nonlinear pseudorandom number generators and other potential applications to cryptography.