Permutation error-correcting codes and their applications to public-key cryptography

Abstract

In this thesis, we study some computational problems in permutation group theory and their applications to public-key cryptography. The primary goal of this thesis is to come up with a cryptosystem similar to the McEliece cryptosystem using permutation groups instead of vector spaces over finite fields. Like vector spaces, permutation groups too have been explored as a setting for error-correcting codes. These objects are called permutation codes. We propose a framework for such a cryptosystem and also come up with several classical attacks on it. We prove that the cryptosystem using transitive permutation groups is quantum-secure. We also explore using the permutation codes proposed by Bailey and Cameron in our cryptosystem. Although our cryptosystem using permutation codes that exist currently is insecure, we hope that this work would encourage research on coming up with new classes of permutation codes with efficient decoding algorithms.