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

Abstract

<p style='text-indent:20px;'>The concept of the signal-to-noise ratio (SNR) as a useful measure indicator of the robustness of <inline-formula><tex-math id="M1">\begin{document}$ (n, m) $\end{document}</tex-math></inline-formula>-functions <inline-formula><tex-math id="M2">\begin{document}$ F = (f_1, \ldots, f_m) $\end{document}</tex-math></inline-formula> (cryptographic S-boxes) against differential power analysis (DPA), has received extensive attention during the previous decade. In this paper, we give an upper bound on the SNR of balanced <inline-formula><tex-math id="M3">\begin{document}$ (n, m) $\end{document}</tex-math></inline-formula>-functions, and a clear upper bound regarding unbalanced <inline-formula><tex-math id="M4">\begin{document}$ (n, m) $\end{document}</tex-math></inline-formula>-functions. Moreover, we derive some deep relationships between the SNR of <inline-formula><tex-math id="M5">\begin{document}$ (n, m) $\end{document}</tex-math></inline-formula>-functions and three other cryptographic parameters (the maximum value of the absolute value of the Walsh transform, the sum-of-squares indicator, and the nonlinearity of its coordinates), respectively. In particular, we give a trade-off between the SNR and the refined transparency order of <inline-formula><tex-math id="M6">\begin{document}$ (n, m) $\end{document}</tex-math></inline-formula>-functions. Finally, we prove that the SNR of <inline-formula><tex-math id="M7">\begin{document}$ (n, m) $\end{document}</tex-math></inline-formula>-functions is not affine invariant, and data experiments verify this result.</p>