On Deterministic Polynomial-Time Equivalence of Computing the CRT-RSA Secret Keys and Factoring

Let N = pq be the product of two large primes. Consider CRT-RSA with the public encryption exponent e and private decryption exponents dp, dq. It is well known that given any one of dp or dq (or both) one can factorize N in probabilistic poly(logN) time with success probability almost equal to 1. Though this serves all the practical purposes, from theoretical point of view, this is not a deterministic polynomial time algorithm. In this paper, we present a lattice based deterministic poly(logN) time algorithm that uses both dp, dq (in addition to the public information e,N) to factorize N for certain ranges of dp, dq. We like to stress that proving the equivalence for all the values of dp, dq may be a nontrivial task.