# [Resource Topic] 2005/436: A Note on the Kasami Power Function

Welcome to the resource topic for 2005/436

Title:
A Note on the Kasami Power Function

Authors: 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.

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