More New Classes of Differentially 4-Uniform Permutations with Good Cryptographic Properties

Jie PENG  Chik How TAN  Qichun WANG  Jianhua GAO  Haibin KAN  

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E101-A   No.6   pp.945-952
Publication Date: 2018/06/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E101.A.945
Type of Manuscript: PAPER
Category: Cryptography and Information Security
Keyword: 
permutation polynomial,  differentially 4-uniform,  algebraic degree,  nonlinearity,  CCZ-equivalence,  

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Summary: 
Research on permutation polynomials over the finite field F22k with significant cryptographical properties such as possibly low differential uniformity, possibly high nonlinearity and algebraic degree has attracted a lot of attention and made considerable progress in recent years. Once used as the substitution boxes (S-boxes) in the block ciphers with Substitution Permutation Network (SPN) structure, this kind of polynomials can have a good performance against the classical cryptographic analysis such as linear attacks, differential attacks and the higher order differential attacks. In this paper we put forward a new construction of differentially 4-uniformity permutations over F22k by modifying the inverse function on some specific subsets of the finite field. Compared with the previous similar works, there are several advantages of our new construction. One is that it can provide a very large number of Carlet-Charpin-Zinoviev equivalent classes of functions (increasing exponentially). Another advantage is that all the functions are explicitly constructed, and the polynomial forms are obtained for three subclasses. The third advantage is that the chosen subsets are very large, hence all the new functions are not close to the inverse function. Therefore, our construction may provide more choices for designing of S-boxes. Moreover, it has been checked by a software programm for k=3 that except for one special function, all the other functions in our construction are Carlet-Charpin-Zinoviev equivalent to the existing ones.