On q-nearly bent Boolean functions
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For each non-constant Boolean function q, Klapper introduced the notion of q-transforms of Boolean functions. 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. 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.
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