  Constructing $2m$-variable Boolean functions with optimal algebraic immunity
based on decomposition of additive group of the finite field
$\mathbb{F}_{2^{2m}}$ seems to be a promising approach since Tu and Deng's
work. In this paper, we consider the same problem in a new way. Based on polar
decomposition of the multiplicative group of $\mathbb{F}_{2^{2m}}$, we propose
a new construction of Boolean functions with optimal algebraic immunity. By a
slight modification of it, we obtain a class of balanced Boolean functions
achieving optimal algebraic immunity, which also have optimal algebraic degree
and high nonlinearity. Computer investigations imply that this class of
functions also behave well against fast algebraic attacks.
