Abstract
The transparency order (denoted by \(\mathcal {TO}\)) is a useful measure of the robustness of (n, m)-functions (cryptographic S-boxes as mappings from \(GF(2)^n\) to \(GF(2)^m\)) to multi-bits Differential Power Analysis (DPA). An improved version of transparency order (denoted by \(\mathcal {RTO}\)), based on the use of cross-correlation coefficients, was also introduced recently. For the first time, we resolve this open problem which (n, m)-functions reach the upper bound on \(\mathcal {TO}\) for odd n (m is a power of 2). We also investigate the tightness of upper and lower bounds related to \(\mathcal {RTO}\) and derive its relationship to main cryptographic characterizations of (n, m)-functions (such as nonlinearity, the sum-of-square indicator and algebraic immunity). Finally, concerning S-boxes of size \(4\times 4\), the distributions of \(\mathcal {RTO}\) for all 302 balanced S-boxes (up to affine equivalence) and 16 equivalence classes of optimal S-boxes are given.
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Acknowledgments
Yu Zhou is supported in part by the Sichuan Science and Technology Program (2020JDJQ0076). Yongzhuang Wei is supported by the National Natural Science Foundation of China (61872103), the Guangxi Science and Technology Foundation (Guike AB18281019) and the Guangxi Natural Science Foundation (2019GXNSFGA245004). Hailong Zhang is supported by the National Natural Science Foundation of China (61872040). Enes Pasalic is supported in part by the Slovenian Research Agency (research program P1-0404 and research projects J1-9108, J1-1694, N1-0159, J1-2451). Luyang Li is supported by the Natural Science Foundation of Shaanxi Provincial Department of Education (20JK0911).
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Zhou, Y., Wei, Y., Zhang, H., Li, L., Pasalic, E., Wu, W. (2021). On Characterization of Transparency Order for (n, m)-functions. In: Yu, Y., Yung, M. (eds) Information Security and Cryptology. Inscrypt 2021. Lecture Notes in Computer Science(), vol 13007. Springer, Cham. https://doi.org/10.1007/978-3-030-88323-2_19
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