Common modulus attacks on small private exponent RSA and some fast variants (in practice)

In this work we re-examine two common modulus attacks on RSA. First, we show that Guo's continued fraction attack works much better in practice than previously expected. Given three instances of RSA with a common modulus N and private exponents each smaller than N 33, the attack can factor the modulus about 93% of the time in practice. The success rate of the attack can be increased up to almost 100% by including a relatively small exhaustive search. Next, we consider F-Iowgrave-Graham and Seifert's lattice -based attack and show that a second necessary condition for the attack exists that limits the hounds (beyond the original bounds) once n >= 7 instances of RSA are used. In particular, by construction, the attack is limited to private exponents at most N0.5-is an element of, given sufficiently many instances, instead of the original bound of N1-is an element of. In addition, we also consider the effectiveness of the attacks when mounted against multi-prime RSA and Takagi's variant of RSA. For multi-prime RSA, we show three (or more) instances with a common modulus and private exponents smaller than N1/3-is an element of is unsafe. For Takagi's scheme, we show that three or more instances with a common modulus N = p(t)q is unsafe when all the private exponents are smaller than N2/(3(t+1))-is an element of. The results, for both variants, is obtained using Guo's method and are almost always successful with the inclusion of a small exhaustive search. When only two instances are available, Howgrave-Graham and Seifert's attack can be successfully mounted on multi prime RSA, with r primes in the modulus, when the private exponents are both smaller than N (3 +r)/7r-is an element of.