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.

Permutation polynomials over Galois fields of characteristic

In this paper, a class of permutation polynomial known as o-polynomial over Galois fields of characteristic 2 was studied. A necessary and sufficients condition for a monomial 𝑥2k to be an o-polynomial over F2t is given and two results obtained by Gupta and Sharma (2016) were deduced.

A Class of New Permutation Polynomials over F 2 n

In this paper, according to the known results of some normalized permutation polynomials with degree 5 over F 2 n , we determine sufficient and necessary conditions on the coefficients b 1 , b 2 ∈ F 2 n 2 such that f x = x 3 x ¯ 2 + b 1 x 2 x ¯ + b 2 x permutes F 2 n . Meanwhile, we obtain a class of complete permutation binomials over F 2 n .

Polynomial functions on rings of dual numbers over residue class rings of the integers

The ring of dual numbers over a ring R is R[α] = R[x]/(x 2), where α denotes x + (x 2). For any finite commutative ring R, we characterize null polynomials and permutation polynomials on R[α] in terms of the functions induced by their coordinate polynomials (f 1, f 2 ∈ R[x], where f = f 1 + αf 2) and their formal derivatives on R. We derive explicit formulas for the number of polynomial functions and the number of polynomial permutations on ℤ p n [α] for n ≤ p (p prime).

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.

The Polypermutation Group of an Associative Ring

We study permutation polynomials through the device of the polypermutation group of an associative ring R, denoted by Pgr(R). We derive some basic properties and compute the cardinality of Pgr(Z/pk ) when p ≥ k. We use this computation to determine the structure of Pgr(Z/p2 ).