New Finding on Factoring Prime Power RSA Modulus N = prq

This paper proposes three new attacks. In the first attack we consider the class of the public exponents satisfying an equation e X-N Y +（ap+ bq）Y = Z for suitably small positive integers a, b. Applying continued fractions we show thatY/Xcan be recovered among the convergents of the continued fraction expansion of e/N. Moreover, we show that the number of such exponents is at least Nwhere ε≥ 0 is arbitrarily small for large N. The second and third attacks works upon k RSA public keys（N, e） when there exist k relations of the form ex-Ny+（ap+ bq）y= zor of the form ex-Ny +（ap+ bq）y = zand the parameters x, x, y, y, zare suitably small in terms of the prime factors of the moduli. We apply the LLL algorithm, and show that our strategy enables us to simultaneously factor k prime power RSA moduli.