Solving modular cubic equations with Coppersmith's method

Several cryptosystems based on Elliptic Curve Cryptography such as KMOV and Demytko process the message as a point M = ( x 0, y 0 ) of an elliptic curve with an equation of the form y 2 ≡ x 3 + a x + b ( mod n ) over a finite field when n is a prime number, or over a finite ring when n = p q is an RSA modulus. Other systems use singular cubic curves such as y 2 ≡ x 3 + a x 2 ( mod n ) and y 2 + a x y ≡ x 3 ( mod n ). 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 ( x 0, y 0 ) when | x 0 | 3 | y 0 | 2 is smaller than the modulus.

Authors

• Virgile Dossou-Yovo
• Abderrahmane Nitaj
• Alain Togbé

Keywords

• Elliptic curve cryptography
• Cubic curves
• Coppersmith's method
• Lattice basis reduction
• Cryptanalysis