Efficient and Secure Pseudo-Random Number Generation

Cryptographically secure pseudo-random number generators known so far suffer from the handicap of being inefficient; the most efficient ones can generate only one bit on each modular multiplication (n 2 steps). Blum, Blum and Shub ask the open problem of outputting even two bits securely. We state a simple condition, the XOR-Condition, and show that any generator satisfying this condition can output logn bits on each multiplication. We also show that the logn least significant bits of RSA, Rabin’s Scheme, and the x 2 mod N generator satisfy this condition. As a corollary, we prove that all boolean predicates of these bits are secure. Furthermore, we strengthen the security of the x 2 mod N generator, which being a Trapdoor Generator, has several applications, by proving it as hard as Factoring.