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 \(\mathbb {F}_{2}^{n}\) to \(\mathbb {F}_{2}^{n-1}\) which are not obtained from admissible sequences. This disproves a conjecture raised by Idrisova.

我们考虑具有4的差分均匀性和零非线性的n位置换。我们首先表明，Gold函数的逆具有有趣的性质，即一个组件可以用线性函数代替，从而仍然保持排列。这直接产生具有奇数维琐碎非线性的4个均匀置换的构造。进一步的研究表明其存在于所有Ñ = 3和Ñ ≥5 Alsalami基于结构（Cryptogr COMMUN。10（4）：611-628，2018）。在这方面，我们还表明，从得到的4-均匀2-1功能受理序列，如通过在Idrisova（Cryptogr。COMMUN定义。11（1）：21-39，2019），存在于每个维度Ñ = 3和Ñ ≥5这样的功能实现为是APN排列的子功能，一些必要的属性。最后，我们使用具有零非线性的4均匀置换构造从\（\ mathbb {F} _ {2} ^ {n} \）到\（\ mathbb {F} _ { 2} ^ {n-1} \）不是从允许的序列中获得的。这证明了伊德里索娃提出的猜想。