Cryptology ePrint Archive: Report 2005/436

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

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

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