  The correlation-immune functions serve as an important metric for measuring
resistance of a cryptosystem against correlation attacks. Existing literature
emphasize on matrices, orthogonal arrays and Walsh-Hadamard spectra to
characterize the correlation-immune functions over $\mathbb{F}_p$ ($p \geq 2$
is a prime). %with prime $p$. Recently, Wang and Gong investigated the Fourier
spectral characterization over the complex field for correlation-immune Boolean
functions. In this paper, the discrete Fourier transform (DFT) of non-binary
functions was studied. It was shown that a function $f$ over $\mathbb{F}_p$ is
$m$th-order correlation-immune if and only if its Fourier spectrum vanishes at
a specific location under any permutation of variables. Moreover, if $f$ is a
symmetric function, $f$ is correlation-immune if and only if its Fourier
spectrum vanishes at only one location.
