## CryptoDB

### Tor Helleseth

#### Affiliation: University of Bergen

#### Publications

**Year**

**Venue**

**Title**

2010

EPRINT

On isotopisms of commutative presemifields and CCZ-equivalence of functions
Abstract

A function $F$ from \textbf{F}$_{p^n}$ to itself is planar if for any $a\in$\textbf{F}$_{p^n}^*$ the function $F(x+a)-F(x)$ is a permutation. CCZ-equivalence is the most general known equivalence relation of functions preserving planar property. This paper considers two possible extensions of CCZ-equivalence for functions over fields of odd characteristics, one proposed by Coulter and Henderson and the other by Budaghyan and Carlet, and we show that they in fact coincide with CCZ-equivalence. We prove that two finite commutative presemifields of odd order are isotopic if and only if they are strongly isotopic. This result implies that two isotopic commutative presemifields always define CCZ-equivalent planar functions (this was unknown for the general case). Further we prove that, for any odd prime $p$ and any positive integers $n$ and $m$, the indicators of the graphs of functions $F$ and $F'$ from \textbf{F}$_{p^n}$ to \textbf{F}$_{p^m}$ are CCZ-equivalent if and only if $F$ and $F'$ are CCZ-equivalent.
We also prove that, for any odd prime $p$, CCZ-equivalence of functions from \textbf{F}$_{p^n}$ to \textbf{F}$_{p^m}$, is strictly more general than EA-equivalence when $n\ge3$ and $m$ is greater or equal to the smallest positive divisor of $n$ different from 1.

2009

EPRINT

New commutative semifields defined by PN multinomials
Abstract

We introduce infinite families of perfect nonlinear Dembowski-Ostrom multinomials over $F_{p^{2k}}$ where $p$ is any odd prime. We prove that for $k$ odd and $p\ne3$ these PN functions define new commutative semifields (in part by studying the nuclei of these semifields). This implies that these functions are CCZ-inequivalent to all previously known PN mappings.

2004

CRYPTO

#### Program Committees

- Eurocrypt 1998
- Eurocrypt 1993
- Eurocrypt 1992
- Eurocrypt 1988

#### Coauthors

- Navid Ghaedi Bardeh (1)
- Lilya Budaghyan (2)
- Thomas Johansson (1)
- Håvard Molland (1)
- Sondre Rønjom (1)