Complexity transitions in global algorithms for sparse linear systems over finite fields

Details Complexity transitions in global algorithms for sparse linear systems over finite fields Complexity transitions in global algorithms for sparse linear systems over finite fields

2002-03-29

arxiv.org/abs/cond-mat/0203613v2

J. Phys. A 35 (2002) 7559

Physics

Condensed Matter

Statistical Mechanics

23 pages, 8 figures

Scientific paper

10.1088/0305-4470/35/35/301

We study the computational complexity of a very basic problem, namely that of finding solutions to a very large set of random linear equations in a finite Galois Field modulo q. Using tools from statistical mechanics we are able to identify phase transitions in the structure of the solution space and to connect them to changes in performance of a global algorithm, namely Gaussian elimination. Crossing phase boundaries produces a dramatic increase in memory and CPU requirements necessary to the algorithms. In turn, this causes the saturation of the upper bounds for the running time. We illustrate the results on the specific problem of integer factorization, which is of central interest for deciphering messages encrypted with the RSA cryptosystem.

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