As a part of the field of cryptography, rotation symmetric Boolean functions have rich cryptographic significance. In this paper, based on the knowledge of integer compositions, we present a new construction of odd-variable rotation symmetric Boolean functions with optimal algebraic immunity. The nonlinearity of the new rotation symmetric Boolean functions is much better than that of the previously ones with optimal algebraic immunity. And the algebraic degree of the function class is also much high. Moreover, it is shown that our new functions have almost optimal fast algebraic immunity within the range of variable numbers that ordinary computers can calculate.

Rotation symmetric Boolean functions (RSBFs) are nowadays studied a lot because of its easy operations and good performance in cryptosystem. This paper constructs a new class of odd-variable RSBFs with optimal algebraic immunity (AI). The nonlinearity of the new function, 2 n−1 − n−1 k +2 k−4 (k−3)(k−2), is the highest among all existing RSBFs with optimal AI and known nonlinearity, and its algebraic degree is also almost highest. Besides, the class of functions have almost optimal fast algebraic immunity (FAI) at least for n < 17, which is actually the highest possible value for the designated number of variables.