Recently, there has been a lot of interest in provably "good" pseudo-random number generators [lo, 4, 14, 31. These cryptographically secure generators are "good" in the sense that they pass all probabilistic polynomial time statistical tests. However, despite these nice properties, the secure generators known so far suffer from the han- cap of being inefiicient; the most efiicient of these take n2 steps (one modular multip- cation, n being the length of the seed) to generate one bit. Pseudc-random number g- erators that are currently used in practice output n bits per multiplication (n2 steps). An important open problem was to output even two bits on each multiplication in a cryptographically secure way. This problem was stated by Blum, Blum & Shub [3] in the context of their z2 mod N generator. They further ask: how many bits can be o- put per multiplication, maintaining cryptographic security? In this paper we state a simple condition, the XOR-Condition and show that any generator satisfying this condition can output logn bits on each multiplication. We show that the XOR-Condition is satisfied by the lop least significant bits of the z2-mod N generator. The security of the z2 mod N generator was based on Quadratic Residu- ity [3]. This generator is an example of a Trapdoor Generator [13], and its trapdoor properties have been used in protocol design. We strengthen the security of this gene- tor by proving it as hard as factoring.
