Boolean functions are very important cryptographic primitives in stream or block
ciphers. In this context, these functions need to satisfy good properties
like high algebraic degree, nonlinearity and correlation immunity. We present
here an original and efficient method to enumerate all the correlation-immune
functions of a fixed Hamming weigth, in particular the class of $1$-resilient functions.
The key idea consists in defining equivalent classes to split boolean functions
along their distance from correlation-immune boolean functions.
These classes, called first-order correlation classes, are built using
a recursive decomposition of smaller classes. We derive from this method
several algorithms to enumerate their elements and to count their cardinality.
We first show that the exact number of $1$-resilient boolean functions with
$7$ variables is $23478015754788854439497622689296$
and we obtain a tight estimation of their number with $8$ variables,
between $4\;10^{67}$ and $5.6\;10^{68}$.
We then present a general lower bound for the number
of $1$-resilient boolean functions and improve Schneider's upper bound.
We also propose a general lower bound
for the number of $k$-resilient functions. Most of the bounds presented in
this paper, substantially improve the best known bounds in the literature.
We finally establish
that the probability of a Boolean function being
$1$-resilient is asymptotically between
$\frac{\left(n\pi\right)^{n/2}}{2^{n^2-\frac{3}{2}n-1}e^{n-1/2}}$ and
$2^{-\frac{n^2}{4}+\frac{n}{4}}$.