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

Designs, Codes and Cryptography
Andrea Ferraguti, Giacomo Micheli
Wichtige Hinweise
Communicated by D. Panario.
The first author was partially supported by the Swiss National Science Foundation Grant Number 168459, and is grateful to the Max Planck Institute for Mathematics in Bonn for its hospitality and financial support. The second author was supported by the Swiss National Science Foundation Grant Number 171248.

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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 and count all rational functions \(\varphi \in \mathbb {F}_q(x)\) of degree 3 that induce a permutation of \(\mathbb {P}^1(\mathbb {F}_q)\). As a consequence of our classification, we can show that there is no complete permutation rational function of degree 3 unless \(3\mid q\) and \(\varphi \) is a polynomial.

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