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

TL;DR

This paper investigates how the complexity of solving large random linear systems over finite fields changes at phase transitions, affecting the performance of Gaussian elimination and related algorithms, with implications for cryptography.

Contribution

It identifies phase transitions in the solution space of linear systems over finite fields and links these to drastic changes in algorithmic complexity and resource requirements.

Findings

Phase transitions cause sharp increases in memory and CPU usage.

Gaussian elimination performance drops dramatically at phase boundaries.

Results have implications for cryptographic algorithms like RSA.

Abstract

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.