The Journal of Supercomputing

, Volume 72, Issue 2, pp 438–450 | Cite as

Sparse polynomial multiplication for lattice-based cryptography with small complexity

  • Sedat Akleylek
  • Erdem Alkım
  • Zaliha Yüce Tok


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.


Polynomial multiplication Lattice-based cryptography  Sparse polynomial Sliding window method Software implementation 



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.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Sedat Akleylek
    • 1
    • 2
  • Erdem Alkım
    • 3
  • Zaliha Yüce Tok
    • 4
  1. 1.Cryptography and Computer Algebra GroupTU DarmstadtDarmstadtGermany
  2. 2.Department of Computer EngineeringOndokuz Mayıs UniversitySamsunTurkey
  3. 3.Department of MathematicsEge UniversityIzmirTurkey
  4. 4.Institute of Applied MathematicsMiddle East Technical UniversityAnkaraTurkey

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