Blum–Blum–Shub Pseudorandom Bit Generator

Related Concepts

Pseudorandom Number Generator

Definition

The Blum–Blum–Shub (BBS) pseudorandom bit generator [1] is one of the most efficient pseudorandom number generators known that is provably secure under the assumption that factoring large composites is intractable (integer factoring).

Theory

The generator makes use of modular arithmetic and works as follows:

Setup. Given a security parameter $\tau \in \mathbb{Z}$ as input, generate two random $\tau$-bit primes p,  q where p = q = 3 mod 4. Set $N = {\it { pq}} \in Z$. Integers N of this type (where both prime factors are distinct and are 3 mod 4) are called Blum integers. Next pick a random y in the group ${\mathbb{Z}}_{N}^{{_\ast}}$ and set $s = {y}^{2} \in {\mathbb{Z}}_{N}^{{_\ast}}$. The secret seed is (N,  s). As we will see below, there is no need to keep the number N secret.

Generate. Given an input $\ell \in \mathbb{Z}$ and a seed (N,  s) we generate a pseudorandom sequence of length $\ell$. First, set \({x}_{1}...