An application of low private exponent attack on RSA

RSA is one of the most popular and widely used public key cryptosystems. In this paper, an application of low private exponent attack on it is presented. More precisely, let (N, e) be an RSA public key with corresponding private key d, then N can be factored provided that there exists a proper integer k such that e^k is relatively small and d^k mod phi(N) is small (or large) enough. Experiment shows that our generalization of low private exponent attack can seek out some new weak keys of RSA.