New cryptanalytic results upon prime power moduli N = prq

Abstract

In this paper we propose three attacks on the prime power modulus N = prq for r ≥ 2. The first attack is based on the equation eX − NY +(qr + pru)Y = Z for suitable positive integer u. Using continued fraction we show that YX can be recovered among the convergents of the continued fraction expansion of eN. Also we show that the number of such exponents is at least N5r−76(r+1)−e where e ≥ 0 is arbitrarily small for large N. Hence one can factor the prime power modulus N = prq in polynomial time. For i = 1,…,k, with k ≥ 2 and r ≥ 2 the second and third attacks works when attacks k RSA public keys (Ni, ei) are such that there exist k relations of the form eix−Niyi+(qir+piru)yi=zi or of the shape eixi−Niy+(qir+piru)y=zi where the parameters x, xi, y, yi, zi are suitably small in terms of the prime factors of the moduli. Based on LLL algorithm we show that our attack enable us to simultaneously factor the k prime power RSA moduli Ni.