Blum Blum Shub

This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these template messages) This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (September 2013) (Learn how and when to remove this message) This article relies excessively on references to primary sources. Please improve this article by adding secondary or tertiary sources. Find sources: "Blum Blum Shub" – news · newspapers · books · scholar · JSTOR (September 2013) (Learn how and when to remove this message) (Learn how and when to remove this message)

Blum Blum Shub (B.B.S.) is a pseudorandom number generator proposed in 1986 by Lenore Blum, Manuel Blum and Michael Shub^[1] that is derived from Michael O. Rabin's one-way function.

Blum Blum Shub takes the form

${\displaystyle x_{n+1}=x_{n}^{2}{\bmod {M}}}$,

where M = pq is the product of two large primes p and q. At each step of the algorithm, some output is derived from x_n+1; the output is commonly either the bit parity of x_n+1 or one or more of the least significant bits of x_n+1.

The seed x₀ should be an integer that is co-prime to M (i.e. p and q are not factors of x₀) and not 1 or 0.

The two primes, p and q, should both be congruent to 3 (mod 4) (this guarantees that each quadratic residue has one square root which is also a quadratic residue), and should be safe primes with a small gcd((p-3)/2, (q-3)/2) (this makes the cycle length large).

An interesting characteristic of the Blum Blum Shub generator is the possibility to calculate any x_i value directly (via Euler's theorem):

${\displaystyle x_{i}=\left(x_{0}^{2^{i}{\bmod {\lambda }}(M)}\right){\bmod {M}}}$,

where ${\displaystyle \lambda }$ is the Carmichael function. (Here we have ${\displaystyle \lambda (M)=\lambda (p\cdot q)=\operatorname {lcm} (p-1,q-1)}$).