  In a recent paper \cite{Zhang-Xiao}, Zhang and Xiao describe a technique on
constructing almost optimal resilient functions on even number of variables. In
this paper, we will present an extensive study of the constructions of almost
optimal resilient functions by using the generalized Maiorana-McFarland (GMM)
construction technique. It is shown that for any given $m$, it is possible to
construct infinitely many $n$-variable ($n$ even), $m$-resilient Boolean
functions with nonlinearity equal to $2^{n-1}-2^{n/2-1}-2^{k-1}$ where $k<n/2$.
A generalized version of GMM construction is further described to obtain almost
optimal resilient functions with higher nonlinearity. We then modify the GMM
construction slightly to make the constructed functions satisfying strict
avalanche criterion (SAC). Furthermore we can obtain infinitely many new
resilient functions with nonlinearity $>2^{n-2}-2^{(n-1)/2}$ ($n$ odd) by using
Patterson-Wiedemann functions or Kavut-Y$\ddot{u}$cel functions. Finally, we
provide a GMM construction technique for multiple-output almost optimal
$m$-resilient functions $F: \mathbb{F}_2^n\mapsto \mathbb{F}_2^r$ ($n$ even)
with nonlinearity $>2^{n-1}-2^{n/2}$. Using the methods proposed in this paper,
a large class of previously unknown cryptographic resilient functions are
obtained.
