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Some results concerning cryptographically significant mappings over \(\text{GF}(2^{n})\). (English) Zbl 1197.94201
Summary: We investigate the existence of permutation polynomials of the form \(F(x) = x ^{d } + L(x)\) over \(\text{GF}(2^{n})\), \(L\) being a linear polynomial. The results we derive have a certain impact on the long-term open problem on the nonexistence of APN permutations over \(\text{GF}(2^{n})\), when \(n\) is even. It is shown that certain choices of exponent \(d\) cannot yield APN permutations for even \(n\). When \(n\) is odd, an infinite class of APN permutations may be derived from Gold mapping \(x^{3}\) in a recursive manner, that is starting with a specific APN permutation on \(\text{GF}(2^{k})\), \(k\) odd, APN permutations are derived over \(\text{GF}(2^{k+2i })\) for any \(i \geq 1\). But it is demonstrated that these classes of functions are simply affine permutations of the inverse coset of the Gold mapping \(x^{3}\). This essentially excludes the possibility of deriving new EA-inequivalent classes of APN functions by applying the method of L. Berveglieri et al. [in: Advances in cryptology – ASIACRYPT 2004. 10th international conference on the theory and application of cryptology and information security, Jeju Island, Korea, December 5–9, 2004. Proceedings. Berlin: Springer. Lecture Notes in Computer Science 3329, 79–91 (2004; Zbl 1094.94028)] to arbitrary APN functions.

MSC:
94A60 Cryptography
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