Cryptographically Secure Pseudorandom Number Generator

A cryptographically secure pseudorandom number generator (CSPRNG) is a deterministic algorithm that stretches a short random seed into a longer sequence of bits that is computationally indistinguishable from true randomness by any efficient adversary. The security requirement is strictly stronger than that of ordinary pseudorandom generators: a CSPRNG must resist not only statistical tests but deliberate cryptanalysis, meaning no polynomial-time algorithm can predict the next bit with probability significantly better than random guessing given any prefix of the output.

The formal definition, due to Yao and Blum-Micali, frames security as an indistinguishability game: an adversary who sees either the output of the CSPRNG or truly random bits should be unable to tell which is which. This definition bridges computational complexity and cryptography, and it distinguishes CSPRNGs from generators like the Mersenne Twister — which pass statistical tests but are trivially predictable given a small sequence of outputs. The Blum-Blum-Shub generator remains the canonical example of a CSPRNG with a reductionist security proof.