2002/03/31 by Alfredo Braunstein, A. Braunstein, Michele Leone +5 · 1 citation
Computer Science · Engineering · Physics and Astronomy · #Coding theory and cryptography #Complexity and Algorithms in Graphs #cond-mat.dis-nn #cond-mat.stat-mech #graph theory and CDMA systems
paper · pdf · doi:10.1088/0305-4470/35/35/301
published as J. Phys. A 35 (2002) 7559 · 23 pages, 8 figures
arxiv created 2002/07/08 · openalex publication_date 2002/08/20 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
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 the changes in the performance of a global algorithm, namely Gaussian elimination. Crossing phase boundaries produces a dramatic increase in memory and CPU requirements necessary for 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.