Full classification of permutation rational functions and complete rational functions of degree three over finite fields

Autor: Andrea Ferraguti, Giacomo Micheli
Přispěvatelé: Ferraguti, Andrea, Micheli, Giacomo
Rok vydání: 2020
Předmět:
FOS: Computer and information sciences
Computer Science - Cryptography and Security
Discrete Mathematics (cs.DM)
Mathematics::Number Theory
Field (mathematics)
0102 computer and information sciences
02 engineering and technology
Rational function
01 natural sciences
Combinatorics
Densities
Finite fields
Permutation polynomials
Permutation polynomial
FOS: Mathematics
0202 electrical engineering
electronic engineering
information engineering
Number Theory (math.NT)
11T06
11R32
11R58
11R45
Prime power
Mathematics
Polynomial (hyperelastic model)
Mathematics - Number Theory
Degree (graph theory)
Applied Mathematics
Densitie
Finite field
Order (ring theory)
020206 networking & telecommunications
Computer Science Applications
Number theory
010201 computation theory & mathematics
Settore MAT/03 - Geometria
Cryptography and Security (cs.CR)
Computer Science - Discrete Mathematics
Zdroj: Designs, Codes and Cryptography
ISSN: 1573-7586
0925-1022
DOI: 10.1007/s10623-020-00715-0
Popis: Let $q$ be a prime power, $\mathbb F_q$ be the finite field of order $q$ and $\mathbb F_q(x)$ be the field of rational functions over $\mathbb F_q$. In this paper we classify all rational functions $\varphi\in \mathbb F_q(x)$ of degree 3 that induce a permutation of $\mathbb P^1(\mathbb F_q)$. Our methods are constructive and the classification is explicit: we provide equations for the coefficients of the rational functions using Galois theoretical methods and Chebotarev Density Theorem for global function fields. As a corollary, we obtain that a permutation rational function of degree 3 permutes $\mathbb F_q$ if and only if it permutes infinitely many of its extension fields. As another corollary, we derive the well-known classification of permutation polynomials of degree 3. As a consequence of our classification, we can also show that there is no complete permutation rational function of degree $3$ unless $3\mid q$ and $\varphi$ is a polynomial.
Comment: Using our results, we now also characterize complete permutation rational functions of degree 3
Databáze: OpenAIRE
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