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A highly nonlinear differentially 4 uniform power mapping that permutes fields of even degree. (English) Zbl 1194.94182
Power functions on finite fields \(GF(2^{n})\) that permute the fields have wide cryptographic applications. H. Dobbertin [Appl. Algebra Eng. Commun. Comput. 9, No. 2, 139–152 (1998; Zbl 0924.94026)] gives a list of such mappings that meet the conjectured upper bound nonlinearity which in the case of even \(n\) is \(2^{n-1}-2^{n/2}\). One of the mappings on the list is \(f(x)=x^{2^{2k}+2^k+1}\) defined on \(GF(2^{4k})\) with \(k\) odd. The authors show that \(f(x)\) has differential uniformity of \(4\) (there is no need to assume that \(k\) is odd here; if \(k\) is even the function is not a permutation). They also give a slightly different proof (from that given by Dobbertin) of the fact that \(\text{NL}(f)=2^{n-1}-2^{n/2}\) (again, the proof covers also the case of even \(k\)).

MSC:
94A60 Cryptography
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
14G50 Applications to coding theory and cryptography of arithmetic geometry
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