A note on the Signal-to-noise ratio of $ (n, m) $-functions

The concept of the signal-to-noise ratio (SNR) as a useful measure indicator of the robustness of \begin{document}$ (n, m) $\end{document} -functions \begin{document}$ F = (f_1, \ldots, f_m) $\end{document} (cryptographic S-boxes) against differential power analysis (DPA), has received extensive attention during the previous decade. In this paper, we give an upper bound on the SNR of balanced \begin{document}$ (n, m) $\end{document} -functions, and a clear upper bound regarding unbalanced \begin{document}$ (n, m) $\end{document} -functions. Moreover, we derive some deep relationships between the SNR of \begin{document}$ (n, m) $\end{document} -functions and three other cryptographic parameters (the maximum value of the absolute value of the Walsh transform, the sum-of-squares indicator, and the nonlinearity of its coordinates), respectively. In particular, we give a trade-off between the SNR and the refined transparency order of \begin{document}$ (n, m) $\end{document} -functions. Finally, we prove that the SNR of \begin{document}$ (n, m) $\end{document} -functions is not affine invariant, and data experiments verify this result.