Cryptanlysis of Short Secret Exponent and Large Public Exponent of RSA

Abstract

Wiener proved that in RSA, the secret exponent d can be discovered if d＜ N1/4 and N3/4 ＜ e ＜N, where e is the public exponent and N is the modulus. However, he also presented an open problem that whether there exists an attack on RSA when d is short and e ＞ N. In this paper, we improve Wiener’s method to solve the case that d ＜ N1/4 and N 7/4 ＜ e ＜ N2. Furthermore, we show that the secret exponent d can be easily recovered when d ＜ N 1/4 / (3t)1/2 and Nt-1/4 ＜ e ＜ Nt, where it is a small integer.