New Attacks on RSA with Small Secret CRT-Exponents

Abstract

It is well-known that there is an efficient method for decrypting/signing with RSA when the secret exponent d is small modulo p–1 and q–1. We call such an exponent d a small CRT-exponent. It is one of the major open problems in attacking RSA whether there exists a polynomial time attack for small CRT-exponents, i.e. a result that can be considered as an equivalent to the Wiener and Boneh-Durfee bound for small d. At Crypto 2002, May presented a partial solution in the case of an RSA modulus N=pq with unbalanced prime factors p and q. Based on Coppersmith’s method, he showed that there is a polynomial time attack provided that q < N ^0.382. We will improve this bound to q < N ^0.468. Thus, our result comes close to the desired normal RSA case with balanced prime factors. We also present a second result for balanced RSA primes in the case that the public exponent e is significantly smaller than N. More precisely, we show that there is a polynomial time attack if $d_{p}, d_{q} \leq min\{(N/e)^{\frac{2}{5}},N^{\frac{1}{4}}\}$. The method can be used to attack two fast RSA variants recently proposed by Galbraith, Heneghan, McKee, and by Sun, Wu.