On $q$-nearly bent Boolean functions
This work solves a theoretical problem in cryptography and coding theory regarding the characterization of Boolean functions, but it is incremental as it builds on existing q-transform concepts.
The paper addresses the existence of q-nearly bent Boolean functions, proving that any balanced Boolean function is q-nearly bent when q has weight one, which resolves an open question by Klapper, and provides a necessary condition for when a function is not q-nearly bent.
For each non-constant Boolean function $q$, Klapper introduced the notion of $q$-transforms of Boolean functions. The {\em $q$-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. In this work we discuss the existence of $q$-nearly bent functions, a new family of Boolean functions characterized by the $q$-transform. Let $q$ be a non-affine Boolean function. We prove that any balanced Boolean functions (linear or non-linear) are $q$-nearly bent if $q$ has weight one, which gives a positive answer to an open question (whether there exist non-affine $q$-nearly bent functions) proposed by Klapper. We also prove a necessary condition for checking when a function isn't $q$-nearly bent.