New Attacks on RSA with Modulus N = p²q Using Continued Fractions

In this paper, we propose two new attacks on RSA with modulus N = p²q using continued fractions. Our first attack is based on the RSA key equation ed – ϕ(N)k = 1 where ϕ(N) = p(p – 1)(q – 1). Assuming that and , we show that can be recovered among the convergents of the continued fraction expansion of . Our second attack is based on the equation eX – (N – (ap² + bq²)) Y = Z where a,b are positive integers satisfying gcd(a,b) = 1, |ap² – bq²| < N^1/2 and ap² + bq² = N^2/3+α with 0 < α < 1/3. Given the conditions , we show that one can factor N = p²q in polynomial time.