Abstract
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, 2n−1) > 1, x d + L(x) is nerve 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 (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.
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Communicated by Pascale Charpin.
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Li, Y., Wang, M. On EA-equivalence of certain permutations to power mappings. Des. Codes Cryptogr. 58, 259–269 (2011). https://doi.org/10.1007/s10623-010-9406-8
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DOI: https://doi.org/10.1007/s10623-010-9406-8