### Abstract

For c is an element of F(2)n, a c-bent4 function f from the finite field F(2)n to F-2 is a function with a fiat spectrum with respect to the unitary transform V-f(c), which is designed to describe the component functions of modified planar functions. For c = 0 the transform V-f(c) reduces to the conventional Walsh transform, and hence a 0-bent4 function is bent. In this article we generalize the concept of partially bent functions to the transforms V-f(c). We show that every quadratic function is partially bent, and hence it is plateaued with respect to any of the transforms V-f(c). In detail we analyse two quadratic monomials. The first has values as small as possible in its spectra with respect to all transforms V-f(c), and the second has a flat spectrum for a large number of c. Moreover, we show that every quadratic function is c-bent4 for at least three distinct c. In the last part we analyse a cubic monomial. We show that it is c-bent(4) only for c = 1, the function is then called negabent, which shows that non-quadratic functions exhibit a different behaviour. (C) 2017 Elsevier Inc. All rights reserved.

Original language | English |
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Journal | Finite Fields and Their Applications |

Volume | 46 |

Pages (from-to) | 163-178 |

ISSN | 1071-5797 |

DOIs | |

Publication status | Published - 2017 |

### Keywords

- Bent function
- Negabent function
- Bent4
- Boolean function
- Walsh transform
- Quadratic functions

## Cite this

Anbar Meidl, N., & Meidl, W. (2017). Bent and bent(4) spectra of Boolean functions over finite fields.

*Finite Fields and Their Applications*,*46*, 163-178. https://doi.org/10.1016/j.ffa.2017.03.008