## Abstract

There has been only limited information on the existence of p-ary bent functions. Recently there has been a result by the authors on finding necessary conditions for the existence of regular p-ary bent functions (from Zpn to Zp), where p is a prime. The general case of p-ary bent functions is, however, an open question for finding necessary conditions for their existence. In this paper we complete this open case. We state our main result in more detail. We find an explicit family of non-bent functions. We also show that there is no p-ary bent function f in n variables with w(Mf)>n2 if n is even (w(Mf)>n+32 if n is odd, respectively), and for a given nonnegative integer k there is no p-ary bent function f in n variables with w(Mf)=n2-k (w(Mf)=n+32-k, respectively) for an even n≥N_{p, k} (for an odd n≥N_{p, k}, respectively), where N_{p, k} is some positive integer which is explicitly determined and w(Mf) is some explicit value related to the power of each monomial of f. We point out that if f is not a p-ary bent function in n variables and g is any p-ary function in n variables such that L(Gf)=Gg for some CCZ-transformation L of f, then g cannot be bent either. This shows that our result produces a larger family of non-bent functions.

Original language | English |
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Article number | 19742 |

Pages (from-to) | 1177-1189 |

Number of pages | 13 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 433 |

Issue number | 2 |

DOIs | |

State | Published - 15 Jan 2016 |

### Bibliographical note

Publisher Copyright:© 2015 Elsevier Inc.

## Keywords

- Boolean function
- P-ary bent function
- P-ary function
- Weakly regular p-ary bent function