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On EA-equivalence of certain permutations to power mappings. (English) Zbl 1216.94049
Summary: In this paper we investigate the existence of permutation polynomials of the form $$x^d+ L(x)$$ on $$\mathbb F_{2^n}$$, where $$L(x)\in\mathbb F_{2^n}[x]$$ is a linearized polynomial. It is shown that for some special $$d$$ with $$\gcd(d, 2^n-1)> 1$$, $$x^d+L(x)$$ is never a permutation on $$\mathbb F_{2^n}$$ for any linearized polynomial $$L(x)\in\mathbb F_{2^n}[x]$$. For the Gold functions $$x^{2^i+1}$$, it is shown that $$x^{2^i+1}+L(x)$$ is a permutation on $$\mathbb F_{2^n}$$ if and only if $$n$$ is odd and $$L(x)=\alpha^{2^i}x+\alpha x^{2^i}$$ for some $$\alpha\in\mathbb F_{2^n}^*$$. We also disprove a conjecture in [M. Macchetti, Addendum to “On the generalized linear equivalence of functions over finite fields”, Cryptology ePrint Archive, Report2004/347 (2004)] in a very simple way. At last some interesting results concerning permutation polynomials of the form $$x^{-1} + L(x)$$ are given.

##### MSC:
 94A60 Cryptography 11T06 Polynomials over finite fields
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##### References:
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