The ElGamal cryptosystem over circulant matrices

TL;DR

This paper investigates the discrete logarithm problem in the group of non-singular circulant matrices, aiming to identify secure parameters and compare its difficulty with finite fields and elliptic curve groups.

Contribution

It provides a detailed analysis of the discrete logarithm problem for circulant matrices and tabulates secure parameters for practical implementation.

Findings

Identified parameters for secure circulant matrix groups

Compared the discrete logarithm problem across different algebraic structures

Provided insights into the relative hardness of the problem in circulant matrices

Abstract

In this paper we study extensively the discrete logarithm problem in the group of non-singular circulant matrices. The emphasis of this study was to find the exact parameters for the group of circulant matrices for a secure implementation. We tabulate these parameters. We also compare the discrete logarithm problem in the group of circulant matrices with the discrete logarithm problem in finite fields and with the discrete logarithm problem in the group of rational points of an elliptic curve.