Blum Blum Shub

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 oblivious transfer mapping.

Blum Blum Shub takes the form

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

where M = pq is the product of two large primes p and q. At each step of the algorithm, some output is derived from 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 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 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):

$x_i = \left( x_0^{2^i \bmod \lambda(M)} \right) \bmod M$,

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

Security

There is a proof reducing its security to the computational difficulty of solving the Quadratic residuosity problem.[1] When the primes are chosen appropriately, and O(log log M) lower-order bits of each xn are output, then in the limit as M grows large, distinguishing the output bits from random should be at least as difficult as solving the Quadratic residuosity problem modulo M.

Example

Let $p=11$, $q=19$ and $s=3$ (where $s$ is the seed). We can expect to get a large cycle length for those small numbers, because ${\rm gcd}(\varphi(p-1), \varphi(q-1))=2$. The generator starts to evaluate $x_0$ by using $x_{-1}=s$ and creates the sequence $x_0$, $x_1$, $x_2$, $\ldots$ $x_5$ = 9, 81, 82, 36, 42, 92. The following table shows the output (in bits) for the different bit selection methods used to determine the output.

Even parity bit Odd parity bit Least significant bit
0 1 1 0 1 0 1 0 0 1 0 1 1 1 0 0 0 0

References

1. ^ a b Blum, Lenore; Blum, Manuel; Shub, Mike (1 May 1986). "A Simple Unpredictable Pseudo-Random Number Generator". SIAM Journal on Computing 15 (2): 364–383. doi:10.1137/0215025.
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