Abstract:In this paper, three constructions of balanced Boolean functions with optimum algebraic immunity are proposed. The cryptographical properties such as algebraic degree and nonlinearity of the constructed functions are also analyzed.

Based on a sufficient condition proposed by Hollmann and Xiang for constructing triple-error-correcting codes, the minimum distance of a binary cyclic code C 1,3,13 with three zeros α, α 3 , and α 13 of length 2 m −1 and the weight divisibility of its dual code are studied, where m ≥ 5 is odd and α is a primitive element of the finite field F 2 m . The code C 1,3,13 is proven to have the same weight distribution as the binary triple-error-correcting primitive BCH code C 1,3,5 of the same length.

Substitution boxes are used in many iterate block ciphers for the confusion part of round functions. In this paper, we present some constructions of differentially 6-uniform permutations over F 2 2m by modifying the image values of the Gold function. The resulted differentially 6-uniform permutations have high nonlinearities and algebraic degrees simultaneously, which provide more choices for the Substitution boxes.

In this paper, we present several new constructions of differentially 4-uniform permutations over F 2 2m by modifying the values of the inverse function on some subsets of F 2 2m . The resulted differentially 4-uniform permutations have high nonlinearities and algebraic degrees, which provide more choices for the design of crytographic substitution boxes.