## Cryptology ePrint Archive: Report 2012/283

Some properties of q-ary functions based on spectral analysis

Deep Singh and Maheshanand Bhaintwal

Abstract: In this paper, we generalize some existing results on Boolean functions to the $q$-ary functions defined over $\BBZ_q$, where $q\geq 2$ is an integer, and obtain some new characterization of $q$-ary functions based on spectral analysis. We provide a relationship between Walsh-Hadamard spectra of two $p$-ary functions $f$ and $g$ (for $p$ a prime) and their derivative $D_{f, g}$. We provide a relationship between the Walsh-Hadamard spectra and the decompositions of any two $p$-ary functions. Further, we investigate a relationship between the Walsh-Hadamard spectra and the autocorrelation of any two $q$-ary functions.

Category / Keywords: secret-key cryptography / Boolean functions; $q$-ary functions; Walsh-Hadamard spectrum; Crosscorrelation