  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.
