  Correlation-immune (CI) multi-output Boolean functions have the property of
keeping the same output distribution when some input variables are fixed.
Recently, a new application of CI functions has appeared in the system of
resisting side-channel attacks (SCA). In this paper, three new methods are
proposed to characterize the $t$ th-order CI multi-output Boolean functions
($n$-input and $m$-output). The first characterization is to regard the
multi-output Boolean functions as the corresponding generalized Boolean
functions. It is shown that a generalized Boolean functions $f_g$ is a $t$
th-order CI function if and only if the Walsh transform of $f_g$ defined here
vanishes at all points with Hamming weights between $1$ and $t$. Compared to
the previous Walsh transforms of component functions, our first method can
reduce the computational complexity from $(2^m-1)\sum^t_{j=1}\binom{n}{j}$ to
$m\sum^t_{j=1}\binom{n}{j}$. The last two methods are generalized from Fourier
spectral characterizations. Especially, Fourier spectral characterizations are
more efficient to characterize the symmetric multi-output CI Boolean functions.
