## CryptoDB

### Paper: On isotopisms of commutative presemifields and CCZ-equivalence of functions

Authors: Lilya Budaghyan Tor Helleseth URL: http://eprint.iacr.org/2010/320 Search ePrint Search Google 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.
##### BibTeX
@misc{eprint-2010-23221,
title={On isotopisms of commutative presemifields and CCZ-equivalence of functions},
booktitle={IACR Eprint archive},
keywords={foundations /},
url={http://eprint.iacr.org/2010/320},
note={Commutative semifield, CCZ-equivalence, EA-equivalence, isotopism of presemifields, Perfect nonlinear, Planar function. lilia.b@mail.ru 14768 received 27 May 2010, withdrawn 8 Jun 2010},
author={Lilya Budaghyan and Tor Helleseth},
year=2010
}