New differentially 4-uniform permutations by modifying the inverse function on subfields.

*(English)*Zbl 1366.94526Summary: Permutations over \(\mathbb {F}_{2^{2k}}\) with low differential uniformity, high algebraic degree and high nonlinearity are of great cryptographic importance since they can be chosen as the substitution boxes (S-boxes) for many block ciphers with SPN (Substitution Permutation Network) structure. A well known example is that the S-box of the famous Advanced Encryption Standard (AES) is derived from the inverse function on \(\mathbb {F}_{2^{8}}\), which has been proved to be a differentially 4-uniform permutation with the optimal algebraic degree and known best nonlinearity. Recently, Z. Zha et al. [Finite Fields Appl. 25, 64–78 (2014; Zbl 1305.94084)] proposed two constructions of differentially 4-uniform permutations over \(\mathbb {F}_{2^{2k}}\), say \(G_t\) and \(G_{s, t}\) with \(\text{Tr}(s^{-1}) = 1\), by applying affine transformations to the inverse function on some subfields of \(\mathbb {F}_{2^{2k}}\) [Zha et al. (loc. cit)]. In this paper, we generalize their method by applying other types of EA (extended affine) equivalent transformations to the inverse function on some subfields of \(\mathbb {F}_{2^{2k}}\) and present two new constructions of differentially 4-uniform permutations, say \(F_{\alpha}\) and \(F_\beta\), \(\alpha\) with \(\text{Tr}(\beta^{-1}) = 1\). Furthermore, we prove that all the functions \(G_t\) with different \(t\) are CCZ (Carlet-Charpin-Zinoviev) equivalent to our subclass \(F_{0}\), while all the functions \(G_{s, t}\) with different \(t\) are CCZ-equivalent to our subclass \(F_{s,0}\). In addition, both our two constructions give many new CCZ-inequivalent classes of such functions, as checked by computer in small numbers of variables. Moreover, all these newly constructed permutations are proved to have the optimal algebraic degree and high nonlinearity.

##### Keywords:

S-box; differentially 4-uniform permutation; algebraic degree; nonlinearity; CCZ-equivalent
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\textit{J. Peng} and \textit{C. H. Tan}, Cryptogr. Commun. 9, No. 3, 363--378 (2017; Zbl 1366.94526)

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