Solving modular cubic equations with Coppersmith's method

Abstract

Several cryptosystems based on Elliptic Curve Cryptography such as KMOV and Demytko process the message as a point M=(x0,y0) of an elliptic curve with an equation of the form y2≡x3+ax+b(modn) over a finite field when n is a prime number, or over a finite ring when n=pq is an RSA modulus. Other systems use singular cubic curves such as y2≡x3+ax2(modn) and y2+axy≡x3(modn). In this paper, we present a method to find the small solutions of the former modular cubic equations. Our method is based on Coppersmith's technique and enables one to find the solutions (x0,y0) when |x0|3|y0|2 is smaller than the modulus.