{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,8]],"date-time":"2025-12-08T22:29:57Z","timestamp":1765232997713},"reference-count":27,"publisher":"American Institute of Mathematical Sciences (AIMS)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["AMC"],"published-print":{"date-parts":[[2022]]},"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><\/jats:p>","DOI":"10.3934\/amc.2020117","type":"journal-article","created":{"date-parts":[[2020,11,20]],"date-time":"2020-11-20T09:27:30Z","timestamp":1605864450000},"page":"413","source":"Crossref","is-referenced-by-count":3,"title":["A note on the Signal-to-noise ratio of $ (n, m) $-functions"],"prefix":"10.3934","volume":"16","author":[{"given":"Yu","family":"Zhou","sequence":"first","affiliation":[]},{"given":"Xinfeng","family":"Dong","sequence":"additional","affiliation":[]},{"given":"Yongzhuang","family":"Wei","sequence":"additional","affiliation":[]},{"given":"Fengrong","family":"Zhang","sequence":"additional","affiliation":[]}],"member":"2321","reference":[{"key":"key-10.3934\/amc.2020117-1","doi-asserted-by":"publisher","unstructured":"S. Banik, A. 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