Cryptanalysis of the RSA Schemes with Short Secret Exponent from Asiacrypt ’99

At Asiacrypt’ 99, Sun, Yang and Laih proposed three RSA variants with short secret exponent that resisted all known attacks, including the recent Boneh-Durfee attack from Eurocrypt ’99 that improved Wiener’s attack on RSA with short secret exponent. The resistance comes from the use of unbalanced primes p and q. In this paper, we extend the Boneh-Durfee attack to break two out of the three proposed variants. While the Boneh-Durfee attack was based on Coppersmith’s lattice-based technique for finding small roots to bivariate modular polynomial equations, our attack is based on its generalization to trivariate modular polynomial equations. The attack is heuristic but works well in practice, as the Boneh-Durfee attack. In particular, we were able to break in a few minutes the numerical examples proposed by Sun, Yang and Laih. The results illustrate once again the fact that one should be very cautious when using short secret exponent with RSA.