Properties of a family of cryptographic boolean functions

Abstract

In 2008, Carlet and Feng studied a class of functions with good cryptographic properties. Based on that function, [18] proposed a family of cryptographically significant Boolean functions which contains the functions proposed by [28,30]. However, their study is not in-depth. In this paper, we investigate the properties of those functions further, and find that they can be divided into some affine equivalent classes. The bent functions proposed by [18] are in fact in the same class with the function proposed by [30]. We then prove that those functions have optimum algebraic immunity if and only if a combinatorial conjecture is correct, which gives a new direction to prove the conjecture. Furthermore, we improve upon the lower bound on the nonlinearity, and our bound is higher than all other similar bounds. Finally, we extend the construction to a balanced function, and give an example of a 12-variable function which has the best cryptographic properties among all currently known functions.