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- W3007396593 abstract "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 $f_{a,b}(x)=ax^{2^{2m}+1}+bx^{2^m+1} inF_{2^{3m}}[x]$ is a planar function over $F_{2^{3m}}$ by the Hasse-Weil bound. In this paper, using the Lang-Weil bound, a generalization of the Hasse-Weil bound, 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 $F_{q^k}$, where $q=2^m$ with $m$ sufficiently large (see Theorem ref{main}). The first and last classes of them are over $F_{q^2}$ and $F_{q^4}$ respectively, while the other two classes are over $F_{q^3}$. 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}." @default.
- W3007396593 created "2020-03-06" @default.
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- W3007396593 date "2020-02-19" @default.
- W3007396593 modified "2023-09-28" @default.
- W3007396593 title "Further Study of Planar Functions in Characteristic Two" @default.
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- W3007396593 doi "https://doi.org/10.48550/arxiv.2002.08149" @default.
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