Constructions and Necessities of Some Permutation Polynomials over Finite Fields

Let F q denote the finite field with q elements. Permutation polynomials over finite fields have important applications in many areas of science and engineering such as coding theory, cryptography, and combinatorial design. The study of permutation polynomials has a long history, and many results are obtained in recent years. In this paper, we obtain some further results about the permutation properties of permutation polynomials. Some new classes of permutation polynomials are constructed, and the necessities of some permutation polynomials are studied.

Construction of irreducible polynomials over finite fields

Irreducible polynomials over finite fields and their applications have been quite well studied. Here, we discuss the construction of the irreducible polynomials of degree [Formula: see text] over the finite field [Formula: see text] for a given irreducible polynomial of degree [Formula: see text]. Furthermore, we construct the irreducible polynomials of degree [Formula: see text] over the finite field [Formula: see text] for a given irreducible polynomial of degree [Formula: see text] by using the method of composition of polynomials with some conditions on coefficients and degree of a given irreducible polynomial.

Linear permutations and their compositional inverses over 𝔽qn

The use of permutation polynomials over finite fields has appeared, along with their compositional inverses, as a good choice in the implementation of cryptographic systems. As a particular case, the construction of involutions is highly desired since their compositional inverses are themselves. In this work, we present an effective way of how to construct several linear permutation polynomials over [Formula: see text] as well as their compositional inverses using a decomposition of [Formula: see text] based on its primitive idempotents. As a consequence, involutions are also constructed.

Some Proposed Problems on Permutation Polynomials over Finite Fields

From the 19th century, the theory of permutation polynomial over finite fields, that are arose in the work of Hermite and Dickson, has drawn general attention. Permutation polynomials over finite fields are an active area of research due to their rising applications in mathematics and engineering. The last three decades has seen rapid progress on the research on permutation polynomials due to their diverse applications in cryptography, coding theory, finite geometry, combinatorics and many more areas of mathematics and engineering. For this reason, the study of permutation polynomials is important nowadays. In this chapter, we propose some new problems in connection to permutation polynomials over finite fields by the help of prime numbers.