Optimal Extension Field Inversion in the Frequency Domain

  • Selçuk Baktır
  • Berk Sunar
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5130)


In this paper, we propose an adaptation of the Itoh-Tsujii algorithm to the frequency domain for efficient inversion in a class of Optimal Extension Fields. To the best of our knowledge, this is the first time a frequency domain finite field inversion algorithm is proposed for elliptic curve cryptography. We believe the proposed algorithm would be well suited especially for efficient low-power hardware implementation of elliptic curve cryptography using affine coordinates in constrained small devices such as smart cards and wireless sensor network nodes.


Elliptic curve cryptography finite fields inversion discrete Fourier transform number theoretic transform 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bailey, D.V., Paar, C.: Optimal Extension Fields for Fast Arithmetic in Public-Key Algorithms. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 472–485. Springer, Heidelberg (1998)Google Scholar
  2. 2.
    Bailey, D.V., Paar, C.: Efficient Arithmetic in Finite Field Extensions with Application in Elliptic Curve Cryptography. Journal of Cryptology 14(3), 153–176 (2001)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Baktır, S.: Efficient Algorithms for Finite Fields, with Applications in Elliptic Curve Cryptography. Master’s thesis, Electrical and Computer Engineering Department, Worcester Polytechnic Institute, Worcester, MA, USA (April 2003)Google Scholar
  4. 4.
    Baktır, S., Sunar, B.: Achieving Efficient Polynomial Multiplication in Fermat Fields Using the Fast Fourier Transform. In: Proceedings of the 44th ACM Southeast Conference (ACMSE 2006), March 2006, pp. 549–554. ACM Press, New York (2006)CrossRefGoogle Scholar
  5. 5.
    Baktır, S., Sunar, B.: Finite Field Polynomial Multiplication in the Frequency Domain with Application to Elliptic Curve Cryptography. In: Levi, A., Savaş, E., Yenigün, H., Balcısoy, S., Saygın, Y. (eds.) ISCIS 2006. LNCS, vol. 4263, pp. 991–1001. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  6. 6.
    Baktır, S., Sunar, B.: Frequency Domain Finite Field Arithmetic for Elliptic Curve Cryptography (preprint, 2007),
  7. 7.
    Baktır, S., Kumar, S., Paar, C., Sunar, B.: A State-of-the-art Elliptic Curve Cryptographic Processor Operating in the Frequency Domain. Mobile Networks and Applications (MONET) 12(4), 259–270 (2007)CrossRefGoogle Scholar
  8. 8.
    Burrus, C.S., Parks, T.W.: DFT/FFT and Convolution Algorithms. John Wiley & Sons, Chichester (1985)Google Scholar
  9. 9.
    Cooley, J., Tukey, J.: An Algorithm for the Machine Calculation of Complex Fourier Series. Mathematics of Computation 19, 297–301 (1965)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Guajardo, J., Paar, C.: Itoh-Tsujii Inversion in Standard Basis and Its Application in Cryptography. Design, Codes, and Cryptography (25), 207–216 (2002)Google Scholar
  11. 11.
    Hinton, G., Sager, D., Upton, M., Boggs, D., Carmean, D., Kyker, A., Roussel, P.: The Microarchitecture of the Pentium 4 Processor. Intel Technology Journal Q1 (2001)Google Scholar
  12. 12.
    Itoh, T., Tsujii, S.: A Fast Algorithm for Computing Multiplicative Inverses in GF(2m) Using Normal Bases. Information and Computation 78, 171–177 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Lidl, R., Niederreiter, H.: Finite Fields. Encyclopedia of Mathematics and its Applications, vol. 20. Addison-Wesley, Reading (1983)zbMATHGoogle Scholar
  14. 14.
    Pollard, J.M.: The Fast Fourier Transform in a Finite Field. Mathematics of Computation 25, 365–374 (1971)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Rivest, R.L., Shamir, A., Adleman, L.: A Method for Obtaining Digital Signatures and Public-Key Cryptosystems. Communications of the ACM 21(2), 120–126 (1978)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Saldamlı, G., Koç, Ç.K.: Spectral Modular Exponentiation. In: Proceedings of the 18th IEEE Symposium on Computer Arithmetic (2007)Google Scholar
  17. 17.
    Woodbury, A., Bailey, D.V., Paar, C.: Elliptic Curve Cryptography on Smart Cards without Coprocessors. In: IFIP CARDIS 2000, Fourth Smart Card Research and Advanced Application Conference, Bristol, UK, September 20–22, 2000. Kluwer, Dordrecht (2000)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Selçuk Baktır
    • 1
  • Berk Sunar
    • 1
  1. 1.WPI, Cryptography & Information Security LaboratoryWorcesterUSA

Personalised recommendations