Advertisement

Efficient Modular Squaring in Binary Fields on CPU Supporting AVX and GPU

  • Paweł AugustynowiczEmail author
  • Andrzej Paszkiewicz
Conference paper
  • 39 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12043)

Abstract

This paper deals with the acceleration of modular squaring operation in binary fields on both modern CPUs and GPUs. The key idea is based on applying bit-slicing methodology with a view to maximizing the advantage of Single Instruction Multiple Data (SIMD) and Single Instruction Multiple Threads (SIMT) execution patterns. The developed implementation of modular squaring was adjusted to testing for the irreducibility of binary polynomials of some particular forms.

Keywords

GPU SIMD Parallel algorithms 

References

  1. 1.
    Augustynowicz, P., Paszkiewicz, A.: Empirical verification of a hypothesis on the inner degree of sedimentary irreducible polynomials over \(GF(2)\). Przegląd Telekomunikacyjny + Wiadomości Telekomunikacyjne 8–9, 799–802 (2017).  https://doi.org/10.15199/59.2017.8-9.33
  2. 2.
    Augustynowicz, P., Paszkiewicz, A.: On trinomials irreducible over \(GF(2)\) accompanying to the polynomials of the form \(x^{2\cdot 3^l}+x^{3^l}+1\). Przegląd Telekomunikacyjny + Wiadomości Telekomunikacyjne 8–9 (2018).  https://doi.org/10.15199/59.2018.8-9.48
  3. 3.
    Ben-Or, M.: Probabilistic algorithms in finite fields. In: 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981), pp. 394–398, October 1981.  https://doi.org/10.1109/SFCS.1981.37
  4. 4.
    Ben-Sasson, E., Hamilis, M., Silberstein, M., Tromer, E.: Fast multiplication in binary fields on GPUs via register cache. In: Proceedings of the 2016 International Conference on Supercomputing, ICS 2016, pp. 35:1–35:12, Istanbul, Turkey (2016).  https://doi.org/10.1145/2925426.2926259, ISBN 978-1-4503-4361-9
  5. 5.
    Cohen, H., et al.: Handbook of Elliptic and Hyperelliptic Curve Cryptography, 2nd edn. Chapman & Hall/CRC (2012). ISBN 9781439840009Google Scholar
  6. 6.
    Feng, K., Ma, W., Huang, W., Zhang, Q., Gong, Y.: Speeding up Galois field arithmetic on Intel MIC architecture. In: Hsu, C.-H., Li, X., Shi, X., Zheng, R. (eds.) NPC 2013. LNCS, vol. 8147, pp. 143–154. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-40820-5_13CrossRefGoogle Scholar
  7. 7.
    Flynn, M.J.: Some computer organizations and their effectiveness. IEEE Trans. Comput. 21(9), 948–960 (1972).  https://doi.org/10.1109/TC.1972.5009071, ISSN 0018–9340
  8. 8.
    Kalcher, S., Lindenstruth, V.: Accelerating Galois field arithmetic for Reed-Solomon erasure codes in storage applications. In: 2011 IEEE International Conference on Cluster Computing, pp. 290–298, September 2011.  https://doi.org/10.1109/CLUSTER.2011.40
  9. 9.
    Lidl, R., Niederreiter, H.: Introduction to Finite Fields and Their Applications. Cambridge University Press, New York (1986)zbMATHGoogle Scholar
  10. 10.
    Lidl, R., et al.: Finite Fields. EBL-Schweitzer t. 20, pkt 1. Cambridge University Press (1997). https://books.google.pl/books?id=xqMqxQTFUkMC, ISBN 9780521392310
  11. 11.
    Plank, J.S., Greenan, K.M., Miller, E.L.: Screaming fast Galois field arithmetic using intel SIMD instructions. In: Proceedings of the 11th USENIX Conference on File and Storage Technologies, FAST 2013, San Jose, CA, USA, 12–15 February 2013, pp. 299–306 (2013). https://www.usenix.org/conference/fast13/technicalsessions/presentation/plank%5C_james%5C_simd
  12. 12.
    Rabin, M.O.: Probabilistic algorithms in finite fields. SIAM J. Comput. 9, 273–280 (1979)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Shoup,V.: NTL: A library for doing number theory (2003). http://www.shoup.net/ntl

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Faculty of CyberneticsMilitary University of TechnologyWarsawPoland

Personalised recommendations