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Categorizing all linear codes of IPM over \({\mathbb {F}}_{2^{8}}\)

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Abstract

Inner Product Masking (IPM) is a generalization of several masking schemes including the Boolean one to protect cryptographic implementation against side-channel analysis. The core competitiveness of IPM is that it provides higher side-channel resistance than Boolean masking with the same number of shares. In this paper, we follow a coding theoretic approach and categorize all linear codes of IPM with n = 2 shares over the finite field \({\mathbb {F}}_{2^{8}}\) in terms of side-channel resistance. We focus on 2-share masking schemes, as they provide, at bit-level, as high as 3rd-order security (much higher than the 1st-order security of Boolean masking). We present the optimal codes for IPM in the sense of side-channel resistance assessed by the signal-to-noise ratio (SNR) and the mutual information (MI). We also show that IPM with equivalent linear codes have comparable level of side-channel resistance. Furthermore, we take the Best Known Linear Codes into consideration for comparison. The numerical results of SNR and MI confirm the effectiveness of our proposal for categorizing.

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Notes

  1. The only criteria of BKLC is the highest minimum Hamming distance [18].

  2. As an example, this irreducible polynomial is used to construct the optimal codes in [10].

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Acknowledgements

This work has been partly financed via the project TeamPlay (https://teamplay-h2020.eu/), a project from European Union’s Horizon2020 research and innovation program, under grant agreement N∘ 779882, and also supported by SECODE project (https://secode.telecom-paristech.fr/) under grant N∘ ANR-15-CHR2-0007 funded by the CHIST-ERA programme and coordinated by ANR.

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Authors and Affiliations

  1. LTCI, Télécom Paris, Institut Polytechnique de Paris, Palaiseau, France

    Wei Cheng, Sylvain Guilley & Jean-Luc Danger

  2. Secure-IC S.A.S., Cesson-Sévigné, France

    Sylvain Guilley & Jean-Luc Danger

  3. Département d’informatique de l’ENS (DIENS), ENS, CNRS, PSL University, Paris, France

    Sylvain Guilley

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  1. Wei Cheng
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  2. Sylvain Guilley
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Correspondence to Wei Cheng.

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Appendix: Generating matrices for the best IPM codes identified on \({\mathbb {F}}_{256}\) by Alg. 1

Appendix: Generating matrices for the best IPM codes identified on \({\mathbb {F}}_{256}\) by Alg. 1

This appendix provides the details about the three non-equivalent optimal codes identified by Alg. 1 and reported in the last line of Table 1.

Extension of the first optimal code from \({\mathbb {F}}_{256}\) to \({\mathbb {F}}_{2}\)

The generating matrix for the expanded code spanned by \(\begin {pmatrix} 1 & \alpha ^{8} \end {pmatrix}\) from \({\mathbb {F}}_{256}\) on the base field \({\mathbb {F}}_{2}\) is:

$$ \mathbf{H}^{\perp}_{1}= \left( \begin{array}{lllllllllllllllll} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & & 0 & 1 & 0 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & & 0 & 0 & 1 & 0 & 1 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 \end{array}\right) \in{\mathbb{F}}_{2}^{8\times 16}. $$

Extension of the second optimal code from \({\mathbb {F}}_{256}\) to \({\mathbb {F}}_{2}\)

The generating matrix for the expanded code spanned by \(\begin {pmatrix} 1 & \alpha ^{126} \end {pmatrix}\) from \({\mathbb {F}}_{256}\) on the base field \({\mathbb {F}}_{2}\) is:

$$ \mathbf{H}^{\perp}_{2}= \left( \begin{array}{lllllllllllllllll} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & & 0 & 1 & 0 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & & 0 & 0 & 1 & 0 & 1 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 \end{array}\right) \in{\mathbb{F}}_{2}^{8\times 16}. $$

Extension of the third optimal code from \({\mathbb {F}}_{256}\) to \({\mathbb {F}}_{2}\)

The generating matrix for the expanded code spanned by \(\begin {pmatrix} 1 & \alpha ^{127} \end {pmatrix}\) from \({\mathbb {F}}_{256}\) on the base field \({\mathbb {F}}_{2}\) is:

$$ \mathbf{H}^{\perp}_{3}= \left( \begin{array}{lllllllllllllllll} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & & 0 & 1 & 1 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & & 0 & 0 & 1 & 1 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 \end{array}\right) \in{\mathbb{F}}_{2}^{8\times 16}. $$

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Cheng, W., Guilley, S. & Danger, JL. Categorizing all linear codes of IPM over \({\mathbb {F}}_{2^{8}}\). Cryptogr. Commun. 13, 527–542 (2021). https://doi.org/10.1007/s12095-021-00483-1

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