Many block ciphers use permutations defined on F(2 ^2k ) with low differential uniformity, high nonlinearity, and high algebraic degree as their S-boxes to provide confusion. It is well known that, for a function on F(2 ⁿ ), the lowest differential uniformity is 2 and the functions achieving this lower bound are called almost perfect nonlinear (APN) functions. However, due to the lack of knowledge on APN permutations on F(2 ^2k ), differentially 4-uniform permutations are usually chosen as S-boxes. For example, the currently endorsed Advanced Encryption Standard chooses one such function, the multiplicative inverse function, as its S-box. By a recent survey on differentially 4-uniform permutations over F(2 ^2k ), there are only five known infinite families of such functions, and most of them have small algebraic degrees. In this paper, we apply the powerful switching method to discover many CCZ-inequivalent infinite families of such functions on F(2 ^2k ) with optimal algebraic degree, where k is an arbitrary positive integer. This greatly expands the list of differentially 4-uniform permutations and hence provide more choices for the S-boxes. Furthermore, lower bounds for the nonlinearity of the functions obtained in this paper are presented and they imply that some infinite families have high nonlinearity.

In EUROCRYPT 2018, Cid et al. 16 introduced a new concept on the cryptographic property of Sboxes: Boomerang Connectivity Table (BCT for short) for evaluating the subtleties of boomerang-style attacks. Very recently, BCT and the boomerang uniformity, the maximum value in BCT, were further studied by Boura and Canteaut 4. Aiming at providing new insights, we show some new results about BCT and the boomerang uniformity of permutations in terms of theory and experiment in this paper. Firstly, we present an equivalent technique to compute BCT and the boomerang uniformity, which seems to be much simpler than the original definition from 16. Secondly, thanks to Carlet’s idea 15, we give a characterization of functions f from Fn 2 to itself with boomerang uniformity δf by means of the Walsh transform. Thirdly, by our method, we consider boomerang uniformities of some specific permutations, mainly the ones with low differential uniformity. Finally, we obtain another class of 4-uniform BCT permutation polynomials over F2n, which is the first binomial.

In this correspondence, we introduce a method to construct Boolean functions in any number of variables, with optimal algebraic immunity. Remarkably, all functions of this type with an odd number of variables can be obtained in this way. We study some cryptographic properties, such as balancedness, algebraic degree of the constructed functions. Moreover, a lower bound of the number of Boolean functions with optimal algebraic immunity is given.

Minimal linear codes are a special subclass of linear codes and have significant applications in secret sharing and secure two-party computation. In this paper, we focus on constructing minimal linear codes with wmin/wmax ≤ p–1/p for any odd prime p based on a generic construction of linear codes, where wmin and wmax denote the minimum and maximum nonzero weights in a code, respectively. First, we present two new infinite families of minimal linear codes with two or three weights by selecting suitable subcode of linear codes which are not minimal. Second, we also present an infinite family of minimal linear codes by employing partial spreads, which can be viewed as a generalization of the construction of Ding et al. In addition, we determine the weight distributions of all these minimal linear codes.