A strategy for recovering roots of bivariate polynomials modulo a prime

Abstract

Let $p$ be a prime and \F_p the finite field with $p$ elements. We show how, when given an irreducible bivariate polynomial f \in \F_p[X,Y] and approximations to (v_0,v_1) \in \F_p^2 such that $f(v_0,v_1)=0$, one can recover $(v_0,v_1)$ efficiently, if the approximations are good enough. This result has been motivated by the predictability problem for non-linear pseudorandom number generators and, other potential applications to cryptography