Consider RSA with N = pq, q< p < 2q, public encryption exponent e and private decryption exponent d. We study cryptanalysis of RSA when certain amount of the Most Significant Bits (MSBs) or Least Significant Bits (LSBs) of d is known. This problem has been well studied in literature as evident from the works of Boneh et. al. in Asiacrypt 1998, Blomer et. al. in Crypto 2003 and Ernst et. al. in Eurocrypt 2005. In this paper, we achieve significantly improved results by modifying the techniques presented by Ernst et. al. Our novel idea is to guess a few MSBs of the secret prime p (may be achieved by exhaustive search over those bits in certain cases) that substantially reduces the requirement of MSBs of d for the key exposure attack. © Springer-Verlag Berlin Heidelberg 2009.