Factoring multi power RSA moduli with a class of secret exponents

In this paper, we consider the RSA variant based on the key equation ed ≡ 1 (mod φ(N)) where N = p^rq, r ≥ 2. We show that if the secret exponent d is close to any multiple of the prime factor p or its powers, then it is possible to factor N in polynomial time in log N.