**Abstract**

Confidentiality and Integrity are two paramount objectives in the evaluation of information and communication technology. In this chapter, we propose an arithmetic approach for designing asymmetric key cryptography. Our method is based on the formulation of cyclotomic matrices correspond to a diophantine system. The strategy uses in cyclotomic matrices to design a one-way function. The result of a one-way function that is efficient to compute, however, is hard to process its inverse except if privileged information about the hidden entry is known. Also, we demonstrate that encryption and decryption can be efficiently performed with the asymptotic complexity of <sup>O</sup> *<sup>e</sup>*<sup>2</sup>*:*<sup>373</sup> ð Þ. Finally, we study the computational complexity of the cryptosystem.

**Keywords:** finite fields, discrete logarithm problem, cyclotomic numbers, cyclotomic matrix, public key, secret key
