The Discrete Logarithm Problem for LucasSequences and a New Class of Weak

Let n be an RSA modulus, that is, n = pq, where p, q are two large primes. We deene the discrete logarithm problem for Lucas sequences and show that solving the discrete logarithm problem for Lucas sequences modulo n gives a polynomial factorisation algorithm. Let d = jq?pj. We calculate m and Vd(m; 1) mod n in polynomial time and then solve for d in O(~ d 1 2 +) time where ~ d is an upper estimate for d such that ~ d d. Hence, we have found a factorisation algorithm that uses Lucas sequences and factorises n in O(~ d 1 2 +) time. This algorithm induces a new class of weak RSA moduli.