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Sparse polynomial multiplication for lattice-based cryptography with small complexity

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Abstract

In this paper, we propose efficient modular polynomial multiplication methods with applications in lattice-based cryptography. We provide a sparse polynomial multiplication to be used in the quotient ring \(({\mathbb {Z}}/ p{\mathbb {Z}}) [x] / (x^{n}+1)\). Then, we modify this algorithm with sliding window method for sparse polynomial multiplication. Moreover, the proposed methods are independent of the choice of reduction polynomial. We also implement the proposed algorithms on the Core i5-3210M CPU platform and compare them with number theoretic transform multiplication. According to the experimental results, we speed up the multiplication operation in \(({\mathbb {Z}}/ p{\mathbb {Z}}) [x] / (x^{n}+1)\) at least \(80~\%\) and improve the performance of the signature generation and verification process of GLP scheme significantly.

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Acknowledgments

Sedat Akleylek is partially supported by TÜBITAK under 2219-Postdoctoral Research Program Grant. Erdem Alkım is partially supported by TÜBITAK under 2214-A Doctoral Research Program Grant.

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Authors and Affiliations

  1. Cryptography and Computer Algebra Group, TU Darmstadt, Darmstadt, Germany

    Sedat Akleylek

  2. Department of Computer Engineering, Ondokuz Mayıs University, Samsun, Turkey

    Sedat Akleylek

  3. Department of Mathematics, Ege University, Izmir, Turkey

    Erdem Alkım

  4. Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey

    Zaliha Yüce Tok

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  1. Sedat Akleylek
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  2. Erdem Alkım
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  3. Zaliha Yüce Tok
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Correspondence to Sedat Akleylek.

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Akleylek, S., Alkım, E. & Tok, Z.Y. Sparse polynomial multiplication for lattice-based cryptography with small complexity. J Supercomput 72, 438–450 (2016). https://doi.org/10.1007/s11227-015-1570-1

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  • DOI: https://doi.org/10.1007/s11227-015-1570-1

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