Welcome to the resource topic for
**2010/507**

**Title:**

On isotopisms of commutative presemifields and CCZ-equivalence of functions

**Authors:**
Lilya Budaghyan, 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.

**ePrint:**
https://eprint.iacr.org/2010/507

See all topics related to this paper.

Feel free to post resources that are related to this paper below.

**Example resources include:**
implementations, explanation materials, talks, slides, links to previous discussions on other websites.

For more information, see the rules for Resource Topics .