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The Blum-Micali algorithm is a cryptographically secure pseudorandom number generator. The algorithm gets its security from the difficulty of computing discrete logarithms.[1]

Let ${\displaystyle p}$ be an odd prime, and let ${\displaystyle g}$ be a primitive root modulo ${\displaystyle p}$. Let ${\displaystyle x_0}$ be a seed, and let

${\displaystyle x_{i+1} = g^{x_i}\ \bmod{\ p}}$.


The ${\displaystyle i}$th output of the algorithm is 1 if ${\displaystyle x_i < \frac{p-1}{2}}$. Otherwise the output is 0.

In order for this generator to be secure, the prime number ${\displaystyle p}$ needs to be large enough so that computing discrete logarithms modulo ${\displaystyle p}$ is infeasible.[1] To be more precise, if this generator is not secure then there is an algorithm that computes the discrete logarithm faster than is currently thought to be possible.[2]

References

1. Bruce Schneier, Applied Cryptography: Protocols, Algorithms, and Source Code in C, pages 416-417, Wiley; 2nd edition (October 18, 1996), ISBN 0471117099
2. Manuel Blum and Silvio Micali, How to Generate Cryptographically Strong Sequences of Pseudorandom Bits, SIAM Journal on Computing 13, no. 4 (1984): 850-864.