In this paper, we analyze the security of the RSA public key cryptosystem where multiple encryption and decryption exponents are considered with the same RSA modulus N. We consider , where p, q are of the same bit size, i.e., . We show that if n many decryption exponents are used with the same N, then RSA is insecure when , for all i, and . Our result improves the bound of Howgrave-Graham and Seifert (CQRE 1999) for and also generalizes our recent work for (IPL 2010).