3 years ago

# On the q -bentness of Boolean functions

Zhixiong Chen, Ting Gu, Andrew Klapper

### Abstract

For each non-constant q in the set of n-variable 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. Klapper conjectured that no Boolean function exists with its q-transform coefficients equal to $$\pm \, 2^{n/2}$$ (such function is called q-bent) when q is non-affine balanced. In our early work, we only gave partial results to confirm this conjecture for small n. Here we prove thoroughly that the conjecture is true for all n by investigating the nonexistence of the partial difference sets in abelian groups with special parameters. We also introduce a new family of functions called $$(\delta ,q)$$ -bent functions, which give a measurement of q-bentness.

Open URL: http://arxiv.org/pdf/1711.02917

DOI: 10.1007/s10623-018-0494-1

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