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