Hyper-bent functions and cyclic codes

The class of bent functions has strong properties and its elements are rare. It contains a subclass of functions which properties are still stronger and which elements are still rarer. Youssef and Gong have proved the existence of such hyper-bent functions in (A. M. Youssef et al. 2001), for every even n. We show that the hyper-bent functions they exhibit are exactly those elements of the well-known 𝒫𝒮_ap class, up to the linear transformations x → δx, δ ∈ F_2n*. We show that hyper-bent functions can all be obtained from some codewords of an extended cyclic code H_n with small dimension and we deduce from the study of the nonzeroes of H_n that the algebraic degree of hyper-bent functions is exactly n/2. We also prove that the functions of class 𝒫𝒮_ap are some codewords of weight 2^n-1 - 2ⁿ2-1/ of a subcode of H_n and we deduce that for some n, depending on the factorization of 2ⁿ - 1, the only hyper-bent functions on n variables are the elements of the class 𝒫𝒮_ap^#, obtained from 𝒫𝒮_ap by composing the functions by the transformations x → δx, δ≠0, and by adding constant functions.