2020/02/19 by Li, Yubo, Li, Kangquan, Qu, Longjiang +1
#94A60 11T06 11G20 12E05 #Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2002.08149
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 \citeZhou in even characteristic. In 2016, L. Qu \citeQ proposed a new approach to constructing quadratic planar functions over \F2n. Very recently, D. Bartoli and M. Timpanella \citeBartoli characterized the condition on coefficients a,b such that the function fa,b(x)=ax^22m+1+bx2m+1 ∈\F23m[x] is a planar function over \F23m 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 \citeQ, we completely characterize the necessary and sufficient conditions on coefficients of four classes of planar functions over \Fqk, where q=2m with m sufficiently large (see Theorem \refmain). The first and last classes of them are over \Fq2 and \Fq4 respectively, while the other two classes are over \Fq3. One class over \Fq3 is an extension of fa,b(x) investigated in \citeBartoli, while our proofs seem to be much simpler. In addition, although the planar binomial over \Fq2 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 \citeQ.