# A Public-Key Cryptosystem Using Cyclotomic Matrices

## TL;DR

This paper introduces a novel asymmetric cryptosystem based on cyclotomic matrices, leveraging their mathematical properties to create a secure one-way function suitable for encryption and decryption.

## Contribution

It presents the design of a new cryptosystem using cyclotomic matrices and analyzes its computational complexity, offering an alternative approach to asymmetric cryptography.

## Key findings

- Efficient encryption and decryption with complexity ~O(e^{2.373})
- Cryptosystem relies on the difficulty of inverting cyclotomic matrix-based functions
- Provides a new mathematical foundation for asymmetric cryptography

## Abstract

Confidentiality and Integrity are two paramount objectives in the evaluation of information and communication technology. In this paper, we propose an arithmetic approach for designing asymmetric key cryptography. Our method is based on the formulation of cyclotomic matrices correspond to the diophantine system. The proposed cyclotomic asymmetric cryptosystem (CAC) utilizes the cyclotomic matrices, whose entries are cyclotomic numbers of order $2l^{2}$, $l$ be prime over a finite field $\mathbb{F}_{p}$ of $p$ elements. The method utilize cyclotomic matrices to design a one-way function. The outcome of a one-way function that is efficient to compute however difficult to compute its inverse unless if secret data about the trapdoor is known. We demonstrate that the encryption and decryption can be efficiently performed with asymptotic complexity of $\mathcal{O}(e^{2.373})$. Besides, we study the computational complexity of the CAC.