Chapter

Advances in Cryptology – EUROCRYPT 2005

Volume 3494 of the series Lecture Notes in Computer Science pp 251-267

A Tool Kit for Finding Small Roots of Bivariate Polynomials over the Integers

  • Johannes BlömerAffiliated withFaculty of Computer Science, Electrical Engineering and Mathematics, University of Paderborn
  • , Alexander MayAffiliated withFaculty of Computer Science, Electrical Engineering and Mathematics, University of Paderborn

Abstract

We present a new and flexible formulation of Coppersmith’s method for finding small solutions of bivariate polynomials p(x,y) over the integers. Our approach allows to maximize the bound on the solutions of p(x,y) in a purely combinatorial way. We give various construction rules for different shapes of p(x,y)’s Newton polygon. Our method has several applications. Most interestingly, we reduce the case of solving univariate polynomials f(x) modulo some composite number N of unknown factorization to the case of solving bivariate polynomials over the integers. Hence, our approach unifies both methods given by Coppersmith at Eurocrypt 1996.

Keywords

Coppersmith’s method univariate vs. bivariate RSA