Constructing balanced 2p-variable rotation symmetric Boolean functions with optimal algebraic immunity, high nonlinearity and high algebraic degree

Abstract

How to design cryptographic Boolean functions is a challenge work in the design of stream and block ciphers. Cryptographic criteria of Boolean functions are connected with some known cryptanalytic attacks. To resist these known attacks, it is important to search Boolean functions with some properties, including balancedness, optimal algebraic immunity, high algebraic degree, good nonlinearity, high correlation immunity, etc. Rotation symmetric Boolean functions (RSBFs) can have these properties simultaneously. In this paper, we propose a new class of balanced 2p-variable RSBFs based on the compositions of an integer, where p is an odd prime. It is found that the functions of this class have optimal algebraic immunity, and their nonlinearity reaches $2^{2p-1}-\left(\begin{array}{c}2p-1\\ p\end{array}\right)+2{\sum }_{i=3}^{p-2}(i-1)(i-2)\left(\begin{array}{c}p-2\\ p-i-2\end{array}\right)+N_{\eta }+1$ (where $N_{\eta }=\frac{p-2-\left(p\mathrm{mod}4\right)}{2}$ and p is an odd prime), which is higher than the previously constructed balanced even-variable RSBFs with optimal algebraic immunity. At the same time, the algebraic degree of the constructed functions are studied, and the results show that they can be optimal under certain conditions.