**1.1 Outline of our scheme**

In this chapter, we consider two significant problems in the theory of cyclotomic numbers over **F***p*. The first one deals with an efficient algorithm for fast computation of all the cyclotomic numbers of order 2*l* 2 , where *l* is prime. The subsequent one deals with designing public key cryptosystem based on cyclotomic matrices of order 2*l* 2 . The strategy employs for designing public-key cryptosystem utilizing cyclotomic matrices of order 2*l* 2 , whose entries are cyclotomic numbers of order 2*l* 2 , *l* be prime, where cyclotomic numbers are certain pairs of solutions ð Þ *a*, *b* <sup>2</sup>*<sup>l</sup>* <sup>2</sup> of order 2*l* <sup>2</sup> over a finite field **F***<sup>p</sup>* with *p* elements.

In our approach, to designing cyclotomy asymmetric cryptosystem (CAC) based on trapdoor one-way function (OWF). The public key is obtained by choosing a non-trivial generator *γ* ∈**F**<sup>∗</sup> *<sup>p</sup>* . The chosen value of the generator constructs a cyclotomic matrix of order 2*l* 2 . It is believed that cyclotomic matrices of order 2*l* <sup>2</sup> is always non-singular if the value of *k*> 1. Since there are efficient algorithms for the construction of cyclotomic matrices. Consequently, the key setup time in our proposed cryptosystem is much shorter than previously designed cryptosystems.

In the scheme, the secret key is given by choosing a different non-trivial generator, which is accomplished by discrete logarithm problem (DLP) over a finite field **F**∗ *<sup>p</sup>* . A key-expansion algorithm is employed to expand the secret keys, which form a non-singular matrix of order 2*l* 2 . Here it is important to note that, if one can change the generators of **F**<sup>∗</sup> *<sup>p</sup>* , then entries of cyclotomic matrices get interchanged among themselves, however, the nature of the cyclotomic matrices remain as same. The decryption algorithm involves efficient algebraic operations of matrices. Hence the decryption in our proposed CAC is very efficient. In view of the perspective on the efficient encryption and decryption features, the polynomial time algorithm ensures that the proposed CAC makes it attractive in computationally restricted processors.

The chapter is organized as follows: Section 2 presents the definition and notations, including some well-known properties of cyclotomic numbers of order 2*l* 2 . Section 3 presents the construction of cyclotomic matrices of order 2*l* 2 . Section 4 contains encryption and decryption algorithms of CAC along with a numerical example. In addition, the computational complexity of the proposed CAC is discussed and in Section 5 presents the encryption & decryption can be efficiently perform with asymptotic complexity of <sup>O</sup> *<sup>e</sup>*<sup>2</sup>*:*<sup>373</sup> ð Þ. Finally, a brief conclusion is reflected in Section 6.
