Blum Blum Shub (B.B.S.) is a pseudorandom number generator proposed in 1986 by Lenore Blum, Manuel Blum and Michael Shub (Blum et al., 1986).
Blum Blum Shub takes the form:
where M=pq is the product of two large primes p and q. At each step of the algorithm, some output is derived from xn+1; the output is commonly either the bit parity of xn+1 or one or more of the least significant bits of xn+1.
The seed x0 should be an integer that's 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 gcd(φ(p-1), φ(q-1)) should be small (this makes the cycle length large).
An interesting characteristic of the Blum Blum Shub generator is the possibility to calculate any xi value directly (via Euler's Theorem):
where is the Carmichael function. (Here we have ).
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