4-uniform permutations with null nonlinearity

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 n = 3 and n ≥ 5 based on a construction in Alsalami (Cryptogr. Commun. 10 (4): 611–628, 2018 ). In this context, we also show that 4-uniform 2-1 functions obtained from admissible sequences , as defined by Idrisova in (Cryptogr. Commun. 11 (1): 21–39, 2019 ), exist in every dimension n = 3 and n ≥ 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 F 2 n $\mathbb {F}_{2}^{n}$ to F 2 n − 1 $\mathbb {F}_{2}^{n-1}$ which are not obtained from admissible sequences. This disproves a conjecture raised by Idrisova.