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Arithmetic Circuit Implementations of S-boxes for SKINNY and PHOTON in MPC

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Computer Security – ESORICS 2023 (ESORICS 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14344))

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Abstract

Secure multi-party computation (MPC) enables multiple distrusting parties to compute a function while keeping their respective inputs private. In a threshold implementation of a symmetric primitive, e.g., of a block cipher, each party holds a share of the secret key or of the input block. The output block is computed without reconstructing the secret key. This enables the construction of distributed TPMs or transciphering for secure data transmission in/out of the MPC context.

This paper investigates implementation approaches for the lightweight primitives SKINNY and PHOTON in arithmetic circuits. For these primitives, we identify arithmetic expressions for the S-box that result in smaller arithmetic circuits compared to the Boolean expressions from the literature. We validate the optimization using a generic actively secure MPC protocol and obtain 18% faster execution time with 49% less communication data for SKINNY-64-128 and 27% to 74% faster execution time with 49% to 81% less data for PHOTON \(P_{100}\) and \(P_{288}\). Furthermore, we find a new set of parameters for the heuristic method of polynomial decomposition, introduced by Coron, Roy and Vivek, specialized for SKINNY’s 8-bit S-box. We reduce the multiplicative depth from 9 to 5.

This work is supported by the Flemish Government through FWO SBO project MOZAIK S003321N.

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Notes

  1. 1.

    https://csrc.nist.gov/Projects/Lightweight-Cryptography.

  2. 2.

    However, the players must know an upper bound on the number of required multiplication triples resp. random bits.

  3. 3.

    Since the SKINNY reference does not specify operations in a field, we are free to pick a suitable one.

  4. 4.

    The parameters cannot be used to decompose the AES S-box, for instance.

  5. 5.

    Each party runs on a separate machine with 4 cores and 16 GB RAM connected with a bandwidth of 10 Gbit/sec and <1 ms latency.

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Authors and Affiliations

  1. COSIC, KU Leuven, Leuven, Belgium

    Aysajan Abidin, Erik Pohle & Bart Preneel

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  1. Aysajan Abidin
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  2. Erik Pohle
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  3. Bart Preneel
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Corresponding author

Correspondence to Erik Pohle .

Editor information

Editors and Affiliations

  1. University of California, Irvine, CA, USA

    Gene Tsudik

  2. University of Padua, Padua, Italy

    Mauro Conti

  3. Delft University of Technology, Delft, The Netherlands

    Kaitai Liang

  4. Delft University of Technology, Delft, The Netherlands

    Georgios Smaragdakis

A Appendix

A Appendix

We detail the used (inverse) embeddings in Table 7. The inversion of the embedding of \(\mathbb {F}_{2^{4}}\) and \(\mathbb {F}_{2^{8}}\) only costs 4 and 8 random bits from \(\mathcal {F}_{\textsf {Bit}}\), respectively.

Table 7. The used embeddings from \(\mathbb {F}_{2^{4}}\) and \(\mathbb {F}_{2^{8}}\) into \(\mathbb {F}_{2^{40}}\) on a bit level. Let \(b_3 X^3 + b_2 X^2 + b_1 X + b_0\) be an element in \(\mathbb {F}_{2^{4}}\) and \(b_7 X^7 + b_6 X^6 + b_5 X^5 + b_4 X^4 + b_3 X^3 + b_2 X^2 + b_1 X + b_0\) be an element in \(\mathbb {F}_{2^{8}}\). An element in \(\mathbb {F}_{2^{40}}\) is \(\sum _{i=0}^{39} b_i'Y^i\). Bits \(b_i'\) that are not set below are 0.
Full size table
Fig. 5.
figure 5

Additional figures for shortest addition chain and the trade-off between multiplication and free squares.

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Abidin, A., Pohle, E., Preneel, B. (2024). Arithmetic Circuit Implementations of S-boxes for SKINNY and PHOTON in MPC. In: Tsudik, G., Conti, M., Liang, K., Smaragdakis, G. (eds) Computer Security – ESORICS 2023. ESORICS 2023. Lecture Notes in Computer Science, vol 14344. Springer, Cham. https://doi.org/10.1007/978-3-031-50594-2_5

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