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.[1]

Let be an odd prime, and let be a primitive root modulo . Let be a seed, and let


The th output of the algorithm is 1 if . Otherwise the output is 0.

In order for this generator to be secure, the prime number needs to be large enough so that computing discrete logarithms modulo 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]


  1. 1.0 1.1 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

External links