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

*Lilya Budaghyan and Tor Helleseth*

**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. We show that the second one in fact coincides with CCZ-equivalence, while using the first one we generalize one of the known families of PN functions. In particular, 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.

**Category / Keywords: **foundations / Commutative semifield, CCZ-equivalence, EA-equivalence, isotopism of presemifields, Perfect nonlinear, Planar function

**Date: **received 4 Oct 2010

**Contact author: **lilia b at mail ru

**Available format(s): **Postscript (PS) | Compressed Postscript (PS.GZ) | PDF | BibTeX Citation

**Note: **Section 4 is changed and corrected.

**Version: **20101005:143616 (All versions of this report)

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