  To resist algebraic attack, a Boolean function should possess good algebraic
immunity (AI). Several papers constructed symmetric functions with the maximum
algebraic immunity $\lceil \frac{n}{2}\rceil $. In this correspondence we prove
that for each odd $n$, there is exactly one trivial balanced $n$-variable
symmetric Boolean function achieving the algebraic immunity $\lceil
\frac{n}{2}\rceil $. And we also obtain a necessary condition for the algebraic
normal form of a symmetric Boolean function with maximum algebraic immunity.
