  We propose a general approach to construct cryptographic significant Boolean
functions of $(r+1)m$ variables based on the additive decomposition
$\mathbb{F}_{2^{rm}}\times\mathbb{F}_{2^m}$ of the finite field
$\mathbb{F}_{2^{(r+1)m}}$, where $r$ is odd and $m\geq3$. A class of unbalanced
functions are constructed first via this approach, which coincides with a
variant of the unbalanced class of generalized Tu-Deng functions in the case
$r=1$. This class of functions have high algebraic degree, but their algebraic
immunity does not exceeds $m$, which is impossible to be optimal when $r>1$. By
modifying these unbalanced functions, we obtain a class of balanced functions
which have optimal algebraic degree and high nonlinearity (shown by a lower
bound we prove). These functions have optimal algebraic immunity provided a
combinatorial conjecture on binary strings which generalizes the Tu-Deng
conjecture is true. Computer investigations show that, at least for small
values of number of variables, functions from this class also behave well
against fast algebraic attacks.
