Abstract

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.

Highlights

  • Introduction to permutation polynomialswe collect some basic facts about permutation polynomials over a finite field that will be frequently used throught the chapter

  • We collect some basic facts about permutation polynomials over a finite field that will be frequently used throught the chapter

  • Given any arbitrary function φ : q ! q, the unique polynomial g ∈ q1⁄2xŠ with degðgÞ < q representing φ can be found by the formula gðxÞ 1⁄4 Pc ∈ q φðcÞ1 À ðx À cÞqÀ1, see ([1], Chapter 7)

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Summary

Introduction to permutation polynomials

We collect some basic facts about permutation polynomials over a finite field that will be frequently used throught the chapter. First it will be convenient to define permutation polynomial over a finite field. A polynomial f ðxÞ ∈ q1⁄2xŠ is said to be a permutation polynomial over q for which the associated polynomial function c↦f ðcÞ ia a permutation of q, that is, the mapping from q to q defined by f is one–one and onto. Finite fields are polynomially complete, that is, every mapping from q into q can be represented by a unique polynomial over q. Q, the unique polynomial g ∈ q1⁄2xŠ with degðgÞ < q representing φ can be found by the formula gðxÞ 1⁄4 Pc ∈ q φðcÞ1 À ðx À cÞqÀ1, see ([1], Chapter 7). Due to the finiteness of the field, the followings are the equivalent conditions for a polynomial to be a permutation polynomial. 1.1 Criteria for permutation polynomials Some well-known criteria for being permutation polynomials are the following

First criterion for permutation polynomials
Third criterion for permutation polynomials
Fourth criterion for permutation polynomials
Some well-known classes of permutation polynomials
Open problems on permutation polynomials
Applications of permutation polynomials
Coding theory
Cryptography
Finite geometry
Some proposed problems
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