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 p be an odd prime, and let g be a primitive root modulo p. Let x_0 be a seed, and let

x_{i%2B1} = g^{x_i}\ \bmod{\ p}.

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

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

External links