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The correlation-immune functions serve as an important metric for measuringresistance of a cryptosystem against correlation attacks. Existing literatureemphasize on matrices, orthogonal arrays and Walsh-Hadamard spectra tocharacterize the correlation-immune functions over $\mathbb{F}_p$ ($p \geq 2$is a prime). %with prime $p$. Recently, Wang and Gong investigated the Fourierspectral characterization over the complex field for correlation-immune Booleanfunctions. In this paper, the discrete Fourier transform (DFT) of non-binaryfunctions 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 ata specific location under any permutation of variables. Moreover, if $f$ is asymmetric function, $f$ is correlation-immune if and only if its Fourierspectrum vanishes at only one location.

The Fourier Spectral Characterization for the Correlation-Immune Functions over Fp