Further study of planar functions in characteristic two

Abstract

Planar functions are of great importance in the constructions of DES-like iterated ciphers, error-correcting codes, signal sets and mathematics. They are defined over finite fields of odd characteristic originally and generalized by Y. Zhou [28] in even characteristic. In 2016, L. Qu [23] proposed a new approach to constructing quadratic planar functions over F2n. Very recently, D. Bartoli and M. Timpanella [4] characterized the condition on coefficients a,b such that the function fa,b(x)=ax22m+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 [23], 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 1.1). 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 [4], 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 [23].