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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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