Paper 2025/1281

Improving RSA Cryptanalysis: Combining Continued Fractions and Coppersmith's Techniques

Abstract

In this paper, we present a new small private exponent attack on RSA by combining continued fractions and Coppersmith's techniques. Our results improve upon previous bounds, including Herrmann-May's attack, by leveraging a crucial relation derived from continued fraction. Additionally, we extend the range of vulnerable small private exponents by considering the partial leakage of prime factors or their sum. Our main result establishes an improved attack bound $ d < N^{1-\alpha/3-\gamma/2} $, where $ \alpha := \log_{N} e $ and $ \gamma := \log_{N} |p+q-S| $, with $ S $ being an approximation of the prime sum $ p+q $. Furthermore, we explore more applications of our main attack in scenarios where the primes share some most or least significant bits. The validity of our proposed main attack is confirmed through numerical experiments, demonstrating its improved performance over existing attacks.