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Christof Beierle, Nils-Gregor Leander

• We consider n-bit permutations with differential uniformity of 4 and null nonlinearity. We first show that the inverses of Gold functions have the interesting property that one component can be replaced by a linear function such that it still remains a permutation. This directly yields a construction of 4-uniform permutations with trivial nonlinearity in odd dimension. We further show their existence for all \(\it n\) = 3 and \(\it n\) \(\geq\) 5 based on a construction in Alsalami (Cryptogr. Commun. \(\bf 10\)(4): 611–628, \(\underline {2018}\)). In this context, we also show that 4-uniform 2-1 functions obtained from admissible sequences, as defined by Idrisova in (Cryptogr. Commun. \(\bf 11\)(1): 21–39, \(\underline {2019)}\), exist in every dimension \(\it n\) = 3 and \(\it n\) \(\geq\) 5. Such functions fulfill some necessary properties for being subfunctions of APN permutations. Finally, we use the 4-uniform permutations with null nonlinearity to construct some 4-uniform 2-1 functions from \(\mathbb{F}^{n}_{2}\) to \(\mathbb{F}^{n-1}_{2}\) which are not obtained from admissible sequences. This disproves a conjecture raised by Idrisova.