New differentially 4-uniform permutations by modifying the inverse function on subfields
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- 作者：Jie Peng ; Chik How Tan
- 关键词：S ; box ; Differentially 4 ; uniform permutation ; Algebraic degree ; Nonlinearity ; CCZ ; equivalent
- 刊名：Cryptography and Communications
- 出版年：2017
- 出版时间：May 2017
- 年：2017
- 卷：9
- 期：3
- 页码：363-378
- 全文大小：
- 刊物类别：Computer Science
- 刊物主题：Data Structures, Cryptology and Information Theory; Coding and Information Theory; Communications Engineering, Networks; Information and Communication, Circuits; Mathematics of Computing;
- 出版者：Springer US
- ISSN：1936-2455
- 卷排序：9
文摘
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, Zha et al. proposed two constructions of differentially 4-uniform permutations over \(\mathbb {F}_{2^{2k}}\), say Gt and Gs, t with Tr(s−1) = 1, by applying affine transformations to the inverse function on some subfields of \(\mathbb {F}_{2^{2k}}\) (Zha et al. Finite Fields Appl. 25, 64–78, 2014). 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 Fα and Fβ, α with Tr(β−1) = 1. Furthermore, we prove that all the functions Gt with different t are CCZ (Carlet-Charpin-Zinoviev) equivalent to our subclass F0, while all the functions Gs, t with different t are CCZ-equivalent to our subclass Fs,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.