Lower Bounds on the Algebraic Immunity of Boolean Functions

From the motivation of algebraic attacks to stream and block ciphers([1,2,7,13,14,15]), the concept of {\em algebraic immunity} (AI) was introduced in [21] and studied in [3,5,10,11,17,18,19,20,21]. High algebraic immunity is a necessary condition for resisting algebraic attacks. In this paper, we give some lower bounds on algebraic immunity of Boolean functions. The results are applied to give lower bounds on AI of symmetric Boolean functions and rotation symmetric Boolean functions. Some balanced rotation symmetric Boolean functions with their AI near the maximum possible value $\lceil \frac{n}{2}\rceil$ are constructed.