Further Study of Planar Functions in Characteristic Two

Yubo Li, Kangquan Li, Longjiang Qu, Chao Li

Published 2020-02-19Version 1

Planar functions are of great importance in the constructions of DES-like iterated ciphers, error-correcting codes, signal sets and the area of mathematics. They are defined over finite fields of odd characteristic originally and generalized by Y. Zhou \cite{Zhou} in even characteristic. In 2016, L. Qu \cite{Q} proposed a new approach to constructing quadratic planar functions over $\F_{2^n}$. Very recently, D. Bartoli and M. Timpanella \cite{Bartoli} characterized the condition on coefficients $a,b$ such that the function $f_{a,b}(x)=ax^{2^{2m}+1}+bx^{2^m+1} \in\F_{2^{3m}}[x]$ is a planar function over $\F_{2^{3m}}$ by the Hasse-Weil Theorem. In this paper, together with the Hasse-Weil Theorem and the new approach introduced in \cite{Q}, we completely characterize the necessary and sufficient conditions on coefficients of four classes of planar functions over finite fields in characteristic two. The first and the last class of them are over $\F_{q^2}$ and $\F_{q^4}$ respectively, while the other two classes are over $\F_{q^3}$, where $q=2^m$. One class over $\F_{q^3}$ is an extension of $f_{a,b}(x)$ investigated in \cite{Bartoli}, while our proofs seem to be much simpler. In addition, although the planar binomial over $\F_{q^2}$ of our results is finally a known planar monomial, we also answer the necessity at the same time and solve partially an open problem for the binomial case proposed in \cite{Q}.