On the Nonexistence of $q$ -Bent Boolean Functions
Andrew Klapper, Zhixiong Chen · IEEE Transactions on Information Theory · 2017
We continue the study of the properties of Boolean functions as reflected in The properties of a recently defined transform. For each non-constant Boolean function q, the q-transform of a Boolean function f is related to the Hamming distances from f to the functions obtainable from q by nonsingular linear change of basis. Many properties that can be characterized by the Walsh-Hadamard transform have (for each q) analogues that can be characterized by the q-transform. In this paper, we study one such property, bentness. We show that if q is balanced and not affine, then there is no function that is both bent and q-bent.