Correspondences and cryptographic applications of certain type of irreducible polynomials over finite fields

Irreducibility of polynomials and construction of irreducible polynomials over finite fields have played an important role both in their theoretical importance and in their applications, especially in cryptography and coding theory. In the literature, in addition to some methods for constructing irreducible polynomials, there are also many encryption schemes using irreducible polynomials. The first goal to be achieved in this thesis is to produce irreducible polynomials defined on finite fields from certain types of irreducible polynomials defined on finite fields, called correspondence methods. The other is to propose a new encryption scheme based on the commutative law property of linear recurrence relations defined on finite fields, using the generated irreducible polynomials and the like. In this context, firstly, for an odd integer k, a k-th degree irreducible polynomial was produced on the finite field \F_{q^2} which corresponds to a k-th degree irreducible polynomial of a certain type on the finite field \F_q. In order to obtain this double-sided proposition, a system of linear equations on the finite field \F_q was obtained by using the roots of the irreducible polynomial over \F_{q^2} with some algebraic properties and the coefficients of the irreducible polynomial over \F_q were determined through the solutions of these systems. Additionally, it has been shown that this result covers similar studies in the literature. Later, a new encryption scheme based on the commutative law of the k-th order linear recurrence relation on the finite field \F_q for k > 2 was proposed as an application in cryptography, using the generated irreducible polynomials defined on finite fields and the like. This encryption scheme is an ephemeral-static encryption scheme where the recipient does not have to be online, which is useful in situations such as email. The security of the proposed encryption scheme is based on the difficulty of solving some LFSR problems. It has also the property of semantic security. Besides, the proposed encryption scheme does not have any multiplication cost on the finite field \F_q from the encryption and decryption procedures. The proposed encryption scheme was compared with algorithms of similar structure for k = 3. Finally, a similar encryption scheme has been proposed for k = 2 using the commutative law of Lucas and Fibonacci sequences, which has the same security as the proposed scheme for k > 2.