Dual-form Elliptic Curves Simple Hardware Implementation

  • Jianxin Wang
  • Xingjun Wang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7389)


Performing standard Weierstrass-form curves’ operations based on Edwards-form curves’ addition law, the overall security of elliptic curves can be strengthened while remain compatible with existing ECC system. We present a simplified algorithm for finding such dual-form elliptic curves over prime field F p with p ≡ 3 mod 4. Using the generated curves, algorithms for implementing dual-form operations on affine, projective and twisted coordinates are further discussed and optimized for the case of Weierstrass-form operations. The algorithms are implemented on FPGA and show competitive time and area performance both in Edwards form and Weierstrass form.


Elliptic curve Edwards curve birational equivalence FPGA 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Koblitz, N.: Elliptic Curve Cryptosystems. Mathematics of Computation 48, 203–209 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Miller, V.S.: Use of Elliptic Curves in Cryptography. In: Williams, H.C. (ed.) CRYPTO 1985. LNCS, vol. 218, pp. 417–426. Springer, Heidelberg (1986)Google Scholar
  3. 3.
    Lenstra, A.K., Verhul, E.R.: Selecting Cryptographic Key Sizes. J. Cryptol. 14, 255–293 (2001)zbMATHGoogle Scholar
  4. 4.
    Kocher, P.C.: Timing Attacks on Implementations of Diffie-Hellman, RSA, DSS, and Other Systems. In: Koblitz, N. (ed.) CRYPTO 1996. LNCS, vol. 1109, pp. 104–113. Springer, Heidelberg (1996)Google Scholar
  5. 5.
    Biehl, I., Meyer, B., Müller, V.: Differential Fault Attacks on Elliptic Curve Cryptosystems. In: Bellare, M. (ed.) CRYPTO 2000. LNCS, vol. 1880, pp. 131–146. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  6. 6.
    Edwards, H.M.: A Normal Form for Elliptic Curves. Bulletin of the American Mathematical Society 44(3), 393–422 (2007)zbMATHCrossRefGoogle Scholar
  7. 7.
    Bernstein, D.J., Birkner, P., Joye, M., Lange, T., Peters, C.: Twisted Edwards Curves. In: Vaudenay, S. (ed.) AFRICACRYPT 2008. LNCS, vol. 5023, pp. 389–405. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  8. 8.
    Bernstein, D.J., Lange, T.: Faster Addition and Doubling on Elliptic Curves. In: Kurosawa, K. (ed.) ASIACRYPT 2007. LNCS, vol. 4833, pp. 29–50. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  9. 9.
    Verneuil, V.: Elliptic Curve Cryptography on Standard Curves Using the Edwards Addition Law (2011) (not published yet)Google Scholar
  10. 10.
    Hankerson, D.R., Vanstone, S.A., Menezes, A.J.: Guide to Elliptic Curve Cryptography. Springer (2004)Google Scholar
  11. 11.
    Montgomery, P.L.: Speeding the Pollard and Elliptic Curve Methods of Factorizations. Math. Comp. 48, 243–264 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Okeya, K., Kurumatani, H., Sakurai, K.: Elliptic Curves with the Montgomery-Form and Their Cryptographic Applications. In: Imai, H., Zheng, Y. (eds.) PKC 2000. LNCS, vol. 1751, pp. 238–257. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  13. 13.
    Koc, C.K., Acar, T., Kaliski, B.S.: Analyzing and Comparing Montgomery Multiplication Algorithms. IEEE Micro 16(3), 26–33 (1996)CrossRefGoogle Scholar
  14. 14.
    Sakiyama, K., Mentens, N., Batina, L., Preneel, B., Verbauwhede, I.: Reconfigurable Modular Arithmetic Logic Unit Supporting High-performance RSA and ECC over GF(p). International Journal of Electronics 94(5), 501–514 (2007)CrossRefGoogle Scholar
  15. 15.
    Kocabas, U., Fan, J., Verbauwhede, I.: Implementation of Binary Edwards Curves for very-Constrained Devices. In: 21st IEEE International Conference on Application-specific Systems Architectures and Processors, ASAP 2010, pp. 185–191 (2010)Google Scholar
  16. 16.
    McIvor, C., McLoone, M., McCanny, J.: Hardware Elliptic Curve Cryptographic Processor over GF(p). IEEE Trans. Circuits and Systems I 53(9), 1946–1957 (2006)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Chatterjee, A., Gupta, I.S.: FPGA Implementation of Extended Reconfigurable Binary Edwards Curve Based Processor. In: 2012 International Conference on Computing, Networking and Communications (ICNC), pp. 211–215 (2012)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Jianxin Wang
    • 1
  • Xingjun Wang
    • 1
  1. 1.Department of Electronic and EngineeringBeijing Tsinghua UniversityBeijingChina

Personalised recommendations