Neural networks and the search for a quadratic residue detector

Abstract

This paper investigates the feasibility of employing artificial neural network techniques for solving fundamental cryptography problems, taking quadratic residue detection as an example. The problem of quadratic residue detection is one which is well known in both number theory and cryptography. While it garners less attention than problems such as factoring or discrete logarithms, it is similar in both difficulty and importance. No polynomial-time algorithm is currently known to the public by which the quadratic residue status of one number modulo another may be determined. This work leverages machine learning algorithms in an attempt to create a detector capable of solving instances of the problem more efficiently. A variety of neural networks, currently at the forefront of machine learning methodologies, were compared to see if any were capable of consistently outperforming random guessing as a mechanism for detection. Surprisingly, neural networks were repeatably able to achieve accuracies well in excess of random guessing on numbers up to 20 bits in length. Unfortunately, this performance was only achieved after a super-polynomial amount of network training, and therefore we do not believe that the system as implemented could scale to cryptographically relevant inputs of 500 to 1000 bits. This nonetheless suggests a new avenue of attack in the search for solutions to the quadratic residues problem, where future work focused on feature set refinement could potentially reveal the components necessary to construct a true closed-form solution.