**A Note on the Kasami Power Function**

*Doreen Hertel*

**Abstract: **This work is motivated by the observation that the function $\F{m}$ to $\F{m}$
defined by $x^d+(x+1)^d+a$ for some $a\in \F{m}$ can be used to construct difference sets.
A desired condition is, that the function $\varphi _d(x):=x^d+(x+1)^d$ is a $2^s$-to-1 mapping.
If $s=1$, then the function $x^d$ has to be APN.
If $s>1$, then there is up to equivalence only one function known:
The function $\varphi _d$ is a $2^s$-to-1 mapping if
$d$ is the Gold parameter $d=2^k+1$ with $\gcd (k,m)=s$.
We show in this paper, that $\varphi _d$ is also a $2^s$-to-1 mapping if
$d$ is the Kasami parameter $d=2^{2k}-2^k+1$ with $\gcd (k,m)=s$ and $m/s$ odd.
We hope, that this observation can be used to construct more difference sets.

**Category / Keywords: **foundations / number theory, finite field

**Publication Info: **submitted to IEEE Transactions on Information Theory

**Date: **received 29 Nov 2005

**Contact author: **doreen hertel at mathematik uni-magdeburg de

**Available format(s): **PDF | BibTeX Citation

**Version: **20051130:134219 (All versions of this report)

**Short URL: **ia.cr/2005/436

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