Abstract
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.
Highlights
It is well known that an Almost Perfect Nonlinear (APN) function, i.e., a differentially 2-uniform function, must have non-trivial nonlinearity
We were interested in the following question: Can we find APN permutations for which one component can be replaced by a linear function such that it still remains a permutation?
By using the 4-uniform permutations with null nonlinearity constructed in the first part, we provide counterexamples to that conjecture in the final part of the paper
Summary
Cryptography and Communications (2020) 12:1133–1141 with higher nonlinearity. In particular, one can reduce any permutation with trivial nonlinearity to a 2-1 function of the same uniformity and extend it back to a permutation in many possible ways. In the first part of this work, we show that the inverses of Gold functions (see [7, 9]), i.e., the inverses of power permutations x → x2i+1 in F2n with gcd(i, n) = 1, have such a property They yield a construction of 4-uniform permutations with null nonlinearity. Since the Gold functions are permutations only in odd dimension, we further observe that the differentially 4-uniform 2-1 function constructed in [1], which is defined in even and odd dimension (except for n = 4), can be extended by a linear coordinate in order to obtain a 4-uniform permutation By showing that such a 2-1 function exists for all n = 3 and n ≥ 5, we conclude that 4-uniform permutations with trivial nonlinearity exist for all n = 3 and n ≥ 5. By using the 4-uniform permutations with null nonlinearity constructed in the first part, we provide counterexamples to that conjecture in the final part of the paper
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