Converse results to the Wiener attack on RSA

A well-known attack on RSA with low secret-exponent d was given by Wiener about 15 years ago. Wiener showed that using continued fractions, one can efficiently recover the secret-exponent d from the public key (N, e) as long as d < N 1/4. Interestingly, Wiener stated that his attack may sometimes also work when d is slightly larger than N 1/4. This raises the question of how much larger d can be: could the attack work with non-negligible probability for d = N 1/4+ρ for some constant ρ> 0? We answer this question in the negative by proving a converse to Wiener’s result. Our result shows that, for any fixed ɛ> 0 and all sufficiently large modulus lengths, Wiener’s attack succeeds with negligible probability over a random choice of d < N δ (in an interval of size Ω(N δ)) as soon as δ> 1/4 + ɛ. Thus Wiener’s success bound d < N 1/4 for his algorithm is essentially tight. We also obtain a converse result for a natural class of extensions of the Wiener attack, which are guaranteed to succeed even when δ> 1/4. The known attacks in this class (by Verheul and Van Tilborg and Dujella) run in exponential time, so it is natural to ask whether there exists an attack in this class with subexponential run-time. Our second converse result answers this question also in the negative.