Efficient Multiplication in Finite Field Extensions of Degree 5

  • Nadia El Mrabet
  • Aurore Guillevic
  • Sorina Ionica
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6737)


Small degree extensions of finite fields are commonly used for cryptographic purposes. For extension fields of degree 2 and 3, the Karatsuba and Toom Cook formulæ perform a multiplication in the extension field using 3 and 5 multiplications in the base field, respectively. For degree 5 extensions, Montgomery has given a method to multiply two elements in the extension field with 13 base field multiplications. We propose a faster algorithm, which requires only 9 base field multiplications. Our method, based on Newton’s interpolation, uses a larger number of additions than Montgomery’s one but our implementation of the two methods shows that for cryptographic sizes, our algorithm is much faster.


finite field arithmetic implementation interpolation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Recommendations for Key Management, Special Publication 800-57 Part 1 (2007)Google Scholar
  2. 2.
    Avanzi, R., Cesena, E.: Trace Zero Varieties over Fields of Characteristic 2 for Cryptographic Applications. In: Hromkovič, J., Královič, R., Nunkesser, M., Widmayer, P. (eds.) SAGA 2007. LNCS, vol. 4665, Springer, Heidelberg (2007)Google Scholar
  3. 3.
    Bajard, J.C., Imbert, L., Negre, C.: Arithmetic operations in finite fields of medium prime characteristic using the Lagrange representation. IEEE Transactions on Computers 55(9), 1167–1177 (2006)CrossRefGoogle Scholar
  4. 4.
    Bodrato, M.: Towards Optimal Toom-Cook Multiplication for Univariate and Multivariate Polynomials in Characteristic 2 and 0. In: Carlet, C., Sunar, B. (eds.) WAIFI 2007. LNCS, vol. 4547, pp. 116–133. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  5. 5.
    Devegili, A.J., Ó hÉigeartaigh, C., Scott, M., Dahab, R.: Multiplication and squaring on pairing-friendly fields. Cryptology ePrint Archive, Report 2006/471 (2006),
  6. 6.
    van Dijk, M., Granger, R., Page, D., Rubin, K., Silverberg, A., Stam, M., Woodruff, D.: Practical cryptography in high dimensional tori. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 234–250. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  7. 7.
    Freeman, D.: Constructing pairing-friendly elliptic curves with embedding degree 10. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 452–465. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  8. 8.
    Freeman, D., Scott, M., Teske, E.: A taxonomy of pairing-friendly elliptic curves. Journal of Cryptology 23, 224–280 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Granger, R., Page, D., Smart, N.: On small characteristic algebraic tori in pairing based cryptography. LMS Journal of Computation and Mathematics (9), 64–85 (2006)Google Scholar
  10. 10.
    Itoh, T., Tsujii, S.: A Fast Algorithm for Computing Multiplicative Inverses in GF(2m) Using Normal Bases. Info. and Comp. 78(3), 171–177 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Koblitz, N., Menezes, A.: Pairing-based cryptography at high security levels. In: Smart, N.P. (ed.) Cryptography and Coding 2005. LNCS, vol. 3796, pp. 13–36. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  12. 12.
    Lidl, R., Niederreiter, H.: Finite Fields, 2nd edn. Cambridge University Press, Cambridge (1997)zbMATHGoogle Scholar
  13. 13.
    Menezes, A., van Oorschot, P., Vanstone, S.: Handbook of Applied Cryptology. CRC Press, Boca Raton (2001)zbMATHGoogle Scholar
  14. 14.
    Montgomery, P.L.: Five, six, and seven-term Karatsuba-like formulae. IEEE Transactions on Computers 54(3), 362–369 (2005)CrossRefzbMATHGoogle Scholar
  15. 15.
    Naehrig, M., Barreto, P., Schwabe, P.: On compressible pairings and their computation. In: Vaudenay, S. (ed.) AFRICACRYPT 2008. LNCS, vol. 5023, pp. 371–388. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  16. 16.
    Rubin, K., Silverberg, A.: Torus-based cryptography. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 349–365. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  17. 17.
    Thales Communications. LibCryptoLCH Librairie cryptographique du Laboratoire Chiffre (2011)Google Scholar
  18. 18.
    Von ZurGathen, J., Gerhard, J.: Modern Computer Algebra. Cambridge University Press, New York (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Nadia El Mrabet
    • 1
  • Aurore Guillevic
    • 2
    • 3
  • Sorina Ionica
    • 4
  1. 1.LIASD - Université Paris 8France
  2. 2.Laboratoire ChiffreThales Communications S.A.Colombes CedexFrance
  3. 3.Équipe crypto DI/LIENS, École Normale SupérieureFrance
  4. 4.TANC, Inria Saclay and LIX, École PolytechniqueFrance

Personalised recommendations