# 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

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

The ${\displaystyle i}$th output of the algorithm is 1 if ${\displaystyle x_{i}\leq {\frac {p-1}{2}}}$. Otherwise the output is 0. This is equivalent to using one bit of ${\displaystyle x_{i}}$ as your random number. It has been shown that ${\displaystyle n-c-1}$ bits of ${\displaystyle x_{i}}$ can be used if solving the discrete log problem is infeasible even for exponents with as few as ${\displaystyle c}$ bits.[2]

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.[3]

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.[4]

## References

1. ^ a b Bruce Schneier, Applied Cryptography: Protocols, Algorithms, and Source Code in C, pages 416-417, Wiley; 2nd edition (October 18, 1996), ISBN 0471117099
2. ^ Gennaro, Rosario (2004). "An Improved Pseudo-Random Generator Based on the Discrete Logarithm Problem". Journal of Cryptology. 18 (2): 91–110. doi:10.1007/s00145-004-0215-y. ISSN 0933-2790. S2CID 18063426.
3. ^ Blum, Manuel; Micali, Silvio (1984). "How to Generate Cryptographically Strong Sequences of Pseudorandom Bits" (PDF). SIAM Journal on Computing. 13 (4): 850–864. doi:10.1137/0213053. S2CID 7008910. Archived from the original (PDF) on 2015-02-24.
4. ^ Guedes, Elloá B.; Francisco Marcos de Assis; Bernardo Lula Jr (2010). "Examples of the Generalized Quantum Permanent Compromise Attack to the Blum-Micali Construction". arXiv:1012.1776 [cs.IT].