  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.
