Generalization of Some Attacks on RSA with Small Prime Combination and Small Private Exponent

In 2002, de Weger proposed an attack on small RSA secret decryption exponent with small prime difference |p-q|, which improved the insecure private key bound of Wiener and Boneh-Durfee attacks. Recently, Maitra and Sarkar found new weak keys in another case when 2q-p is small. In this paper, several results on attacking RSA with small prime differences q-p, 2p-q and small private exponent are generalized. Our observation is that when |ip-jq| is small for some integers i,j, we can explore new insecure keys in RSA. One of our results also makes some statement in literature more accurate.