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Threshold implementations of small S-boxes

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Abstract

Threshold implementation (TI) is a masking method that provides security against first-order DPA with minimal assumptions on the hardware. It is based on multi-party computation and secret sharing. In this paper, we provide an efficient technique to find TIs for all 3 and 4-bit permutations which also covers the set of 3×3 and 4×4 invertible S-boxes. We also discuss alternative methods to construct shared functions by changing the number of variables or shares. Moreover, we further consider the TI of 5-bit almost bent and 6-bit almost perfect nonlinear permutations. Finally, we compare the areas of these various TIs.

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Notes

  1. The component function defined for shared functions in this paper is different than the definition provided in [16]

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Acknowledgments

We would like to thank the reviewers for their detailed comments, Christophe De Cannière for the fruitful discussions and for sharing part of his toolkit for affine equivalent classes with us, Georg Stütz for contributing to the proof of Theorem 1 and Anastasiya Gorodilova for kind assistance with APN permutations.

This work has been supported in part by the Research Council of KU Leuven (OT/13/071), B. Bilgin was partially supported by the FWO project G0B4213N, V. Nikov was supported by the European Commission (FP7) within the Tamper Resistant Sensor Node (TAMPRES) project with contract number 258754 and N. Tokareva and V. Vitkup were supported by the Russian Foundation for Basic Research (project 120131097) and by Grant NSh1939.2014.1 of President of Russia for Leading Scientific Schools.

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Authors and Affiliations

  1. Katholieke Universiteit Leuven, ESAT-COSIC and iMinds, Leuven, Belgium

    Begül Bilgin, Svetla Nikova & Vincent Rijmen

  2. NXP Semiconductors, Leuven, Belgium

    Ventzislav Nikov

  3. Sobolev Institute of Mathematics and Novosibirsk State University, Novosibirsk, Russia

    Natalia Tokareva & Valeriya Vitkup

Authors
  1. Begül Bilgin
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  2. Svetla Nikova
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  3. Ventzislav Nikov
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  4. Vincent Rijmen
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  5. Natalia Tokareva
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  6. Valeriya Vitkup
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Corresponding author

Correspondence to Begül Bilgin.

Appendix Tables

Appendix Tables

Table 9 The 4 classes of 3-bit permutations
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Table 10 The 302 classes of 4-bit permutations
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Table 11 The 302 classes of 4-bit permutations
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Table 12 The 302 classes of 4-bit permutations
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Table 13 The 302 classes of 4-bit permutations
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Table 14 Known S-boxes and their classes
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Table 15 Quadratic decomposition length 2
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Bilgin, B., Nikova, S., Nikov, V. et al. Threshold implementations of small S-boxes. Cryptogr. Commun. 7, 3–33 (2015). https://doi.org/10.1007/s12095-014-0104-7

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