A Public-Key Cryptosystem Using Cyclotomic Matrices

Confidentiality and Integrity are two paramount objectives of asymmetric key cryptography. Where two non-identical but mathematically related keys -- a public key and a private key effectuate the secure transmission of messages. Moreover, the private key is non-shareable and the public key has to be shared. The messages could be secured if the amount of computation rises to very high value. In this work we propose a public key cryptosystem using the cyclotomic numbers, where cyclotomic numbers are certain pair of solutions $(a,b)_{e}$ of order $e$ over a finite field $\mathbb{F}_{q}$ with characteristic $p$. The strategy employs cyclotomic matrices of order $2l^{2}$, whose entries are cyclotomic numbers of order $2l^{2}$, $l$ be prime. The public key is generated by choosing a particular generator $\gamma^{\prime}$ of $\mathbb{F}_{p}^{*}$. Secret key (private key) is accomplished by discrete logarithm problem (DLP) over a finite field $\mathbb{F}_{p}$.