Blum Blum Shub

Template:More footnotes Template:Primary sources Blum Blum Shub (B.B.S.) is a pseudorandom number generator proposed in 1986 by Lenore Blum, Manuel Blum and Michael Shub 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 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 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 x_i 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

The generator is very slow. However, there is a proof reducing its securityTemplate:Clarify to the computational difficulty of computing modular square roots, a problem whose difficulty is equivalent to factoring. When the primes are chosen appropriately, and O(log log M) lower-order bits of each x_n are output, then in the limit as M grows large, distinguishing the output bits from random should be at least as difficult as factoring 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

General

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External links

• GMPBBS , a GPL'ed GMP-based implementation of Blum Blum Shub in C by Maria Morisot with implementations in Java and PHP also.
• An implementation in Java
• Randomness tests