Abstract
The boomerang attack, introduced by Wagner in 1999, is a cryptanalysis technique against block ciphers based on differential cryptanalysis. In particular it takes into consideration two differentials, one for the upper part of the cipher and one for the lower part, and it exploits the dependency of these two differentials. At Eurocrypt’18, Cid et al. introduced a new tool, called the Boomerang Connectivity Table (BCT), that permits to simplify this analysis. Next, Boura and Canteaut introduced an important parameter for cryptographic S-boxes called boomerang uniformity, that is the maximum value in the BCT. Very recently, the boomerang uniformity of some classes of permutations (in particular quadratic functions) have been studied by Li, Qu, Sun and Li, and by Mesnager, Tang and Xiong. In this paper we further study the boomerang uniformity of some non-quadratic differentially 4-uniform functions. In particular, we consider the case of the Bracken-Leander cubic function and three classes of 4-uniform functions constructed by Li, Wang and Yu, obtained from modifying the inverse functions.
Highlights
IntroductionCryptography and Communications (2020) 12:1161–1178 such as block ciphers. In this context, a vectorial Boolean function is called an S-box
A vectorial Boolean function, or (n, m)-function, is a function F from the vector space Fn2 to Fm2
In this paper we studied the boomerang uniformity of some classes of differentially 4uniform permutations defined over F2n with n even
Summary
Cryptography and Communications (2020) 12:1161–1178 such as block ciphers. In this context, a vectorial Boolean function is called an S-box. In 1999, Wagner [22] introduced the boomerang attack, which is an important cryptanalysis technique against block ciphers This attack can be seen as an extension of classical differential attacks. Boura and Canteaut showed that the boomerang uniformity is invariant only with respect to affine equivalence and inverse transformation They gave the classification of all differentially 4-uniform permutations of 4 bits. Li et al [16] gave an equivalent definition to compute the BCT (and the boomerang uniformity) and provided a characterization by means of the Walsh transform of functions with a fixed boomerang uniformity They gave an upper bound for the boomerang uniformity of quadratic permutations, and provided a class of quadratic permutations (related to the Gold functions), defined for n even, with differential 4-uniformity and boomerang 4-uniformity. We compute the boomerang uniformities for three classes of differentially 4-uniform permutations of maximal algebraic degree n − 1, obtained in [17, 23] from modifying the inverse function
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