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Blum–Micali algorithm

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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.

Let p {\displaystyle p} be an odd prime, and let g {\displaystyle g} be a primitive root modulo p {\displaystyle p} . Let x 0 {\displaystyle x_{0}} be a seed, and let

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

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

In order for this generator to be secure, the prime number p {\displaystyle p} needs to be large enough so that computing discrete logarithms modulo p {\displaystyle p} is infeasible. 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.

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.

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. 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 .

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