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We prove that RSA least significant bit is  \tfrac12 + \tfrac1logc N\tfrac{1}{2} + \tfrac{1}{{log^c N}} secure, for any constant c (where N is the RSA modulus). This means that an adversary, given the ciphertext, cannot guess the least significant bit of the plaintext with probability better than  \tfrac12 + \tfrac1logc N\tfrac{1}{2} + \tfrac{1}{{log^c N}} , unless he can break RSA.  Our proof technique is strong enough to give, with slight modifications, the following related results:    (1)   The loglog N least significant bits are simultaneously  \tfrac12 + \tfrac1logc N\tfrac{1}{2} + \tfrac{1}{{log^c N}} secure.     (2)   The above also holds for Rabin’s encryption function.         Our results imply that Rabin/RSA encryption can be directly used for pseudo random bits generation, provided that factoring/inverting RSA is hard.

RSA/Rabin least significant bits are $$ \tfrac{1} {2} + \tfrac{1} {{poly \left( {\log N} \right)}} $$ secure (Extended Abstract)