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Efficient Simultaneous Inversion in Parallel and Application to Point Multiplication in ECC

  • Pradeep Kumar Mishra
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3822)

Abstract

Inversion is the costliest of all finite field operations. Some algorithms require computation of several finite field elements simultaneously (elliptic curve factorization for example). Montgomery’s trick is a well known technique for performing the same in a sequential set up with little scope for parallelization. In the current work we propose an algorithm which needs almost same computational resources as Montgomery’s trick, but can be easily parallelized. Our algorithm uses binary tree structures for computation and using 2 r − 1 multipliers, it can simultaneously invert 2 r elements in 2r multiplication rounds and one inversion round. We also describe how the algorithm can be used when 2, 4, ... number of multipliers are available. To exhibit the utility of the method, we apply it to obtain a parallel algorithm for elliptic curve point multiplication. The proposed method is immune to side-channel attacks and compares favourably to many parallel algorithms existing in literature.

Keywords

Elliptic Curve Cryptosystems Scalar Multiplication parallel algorithm Montgomery ladder simultaneous inversion 

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References

  1. 1.
    Bertoni, G., Breveglieri, L., Wollinger, T.J., Paar, C.: Finding Optimum Parallel Coprocessor Design for Genus 2 Hyperelliptic Curve Cryptosystems. ITCC (2), 538–546 (2004)Google Scholar
  2. 2.
    Brier, E., Dechene, I., Joye, M.: Unified point addition formulae for elliptic curve cryptosystems. In: Nedjah, N., de Macedo, L. (eds.) Embedded Cryptographic Hardware: Methodolgies & Architectures, Nova Science Publishers, Bombay (2004)Google Scholar
  3. 3.
    Brier, É., Joye, M.: Weierstraß elliptic curves and side-channel attacks. In: Naccache, D., Paillier, P. (eds.) PKC 2002. LNCS, vol. 2274, pp. 335–345. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  4. 4.
    Coron, J.-S.: Resistance against differential power analysis for elliptic curve cryptosystems. In: Koç, Ç.K., Paar, C. (eds.) CHES 1999. LNCS, vol. 1717, pp. 292–302. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  5. 5.
    Fischer, W., Giraud, C., Knudsen, E.W., Seifert, J.-P.: Parallel Scalar Multiplication on General Elliptic Curves over F p hedged against Non-Differential Side-Channel Attacks. In: Available at IACR eprint Archive, Technical Report No 2002/007, http://www.eprint.iacr.org
  6. 6.
    Fong, K., Hankerson, D., Lòpez, J., Menezes, A.: Field inversion and point halving revisited. IEEE Transactions on Computers 53(8), 1047–1059 (2004)CrossRefGoogle Scholar
  7. 7.
    Garcia, J.M.G., Garcia, R.M.: Parallel Algorithm for Multiplication on Elliptic Curves. Cryptology ePrint Archive, Report 2002/179 (2002), Available at http://eprint.iacr.org
  8. 8.
    Hankerson, D., Menezes, A., Vanstone, S.: Guide to Elliptic Curve Cryptography. Springer, Heidelberg (2004)zbMATHGoogle Scholar
  9. 9.
    Izu, T., Moller, B., Takagi, T.: Improved elliptic curve multiplication methods resistant against side channel attacks. In: Menezes, A., Sarkar, P. (eds.) INDOCRYPT 2002. LNCS, vol. 2551, pp. 296–313. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  10. 10.
    Izu, T., Takagi, T.: A fast parallel elliptic curve multiplication resistant against side channel attacks. In: Naccache, D., Paillier, P. (eds.) PKC 2002. LNCS, vol. 2274, pp. 280–296. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  11. 11.
    Joye, M., Tymen, C.: Protections against differential analysis for elliptic curve cryptography. In: Koç, Ç.K., Naccache, D., Paar, C. (eds.) CHES 2001. LNCS, vol. 2162, pp. 377–390. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  12. 12.
    Koblitz, N.: Elliptic curve cryptosystems. Mathematics of Computation 48(177), 203–209 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Kocher, P.: 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
  14. 14.
    Kocher, P., Jaffe, J., Jun, B.: Differential power analysis. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 388–397. Springer, Heidelberg (1999)Google Scholar
  15. 15.
    Lenstra, H.W.: Factoring Integers with Elliptic Curves. Ann. of Math. 126, 649–673 (1987)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Menezes, A., Van Oorschot, P.C., Vanstone, S.A.: Handbook of applied cryptography. CRC Press, Boca Raton (1997)zbMATHGoogle Scholar
  17. 17.
    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
  18. 18.
    Möller, B.: Personal CommunicationGoogle Scholar
  19. 19.
    Montgomery, P.L.: Speeding The Pollard and Elliptic Curve methods of Factorization. Math. Comp. 48, 243–264 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Oswald, E.: On Side-Channel Attacks and Application of Algorithmic Countermeasures. Ph.D. Thesis, Graz University of Technology, Austria (2003)Google Scholar
  21. 21.
    Shacham, H., Boneh, D.: Improving SSL handshake performance via batching. In: Naccache, D. (ed.) CT-RSA 2001. LNCS, vol. 2020, p. 28. Springer, Heidelberg (2001)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Pradeep Kumar Mishra
    • 1
  1. 1.Centre for Information Security and Cryptography (CISaC)University of CalgaryCalgaryCanada

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