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Permutation polynomials EA-equivalent to the inverse function over \(\mathrm{GF}(2^n)\). (English) Zbl 1251.94032
Summary: In this paper, a proof is given that there does not exist a linearized polynomial \(L(x)\in \mathbb F_{2^{n}}[x]\) such that \(x^{-1}+L(x)\) is a permutation on \(\mathbb F_{2^{n}}\) when \(n \geq 5\), which was posed as a conjecture in [the authors, Des. Codes Cryptography 58, No. 3, 259–269 (2011; Zbl 1216.94049)]. As a consequence of this result, if a permutation is EA-equivalent to the inverse function over \(\mathbb F_{2^{n}}\), then it is affine equivalent to the inverse mapping when \(n\geq 5\).

MSC:
94A60 Cryptography
06E30 Boolean functions
11T06 Polynomials over finite fields
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