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**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.

**ePrint:**
https://eprint.iacr.org/2005/436

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