# Blum–Micali algorithm

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

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, any method that predicts the numbers generated will lead to an algorithm that solves the discrete logarithm problem for that prime.[2]

There is a paper discussing possible examples of the quantum permanent compromise attack to the Blum-Micali construction. This attacks illustrate how a previous attack to the Blum-Micali generator can be extended to the whole Blum-Micali construction, including the Blum Blum Shub and Kaliski generators.[3]

## 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. online (pdf)
3. Elloá B. Guedes, Francisco Marcos de Assis, Bernardo Lula Jr, Examples of the Generalized Quantum Permanent Compromise Attack to the Blum-Micali Construction http://arxiv.org/abs/1012.1776