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Elliptic Curve Arithmetic Using SIMD

  • Kazumaro Aoki
  • Fumitaka Hoshino
  • Tetsutaro Kobayashi
  • Hiroaki Oguro
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2200)

Abstract

Focusing on servers that process many signatures or ciphertexts, this paper proposes two techniques for parallel computing with SIMD, which significantly enhances the speed of elliptic curve scalar multiplication. We also evaluate one of them based on a real implementation on a Pentium III, which incorporates the SIMD architecture. The results show that the proposed method is about 4.4 times faster than the conventional method.

Keywords

Elliptic Curve Smart Card Elliptic Curf Single Instruction Multiple Data Window Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Lenstra, A.K., Verheu, E.R.: Selecting cryptographic key sizes. In Imai, H., Zheng, Y., eds.: Public Key Cryptography-Third International Workshop on Practice and Theory in Public Key Cryptosystems, PKC 2000. Volume 1751 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, Heidelberg, New York (2000) 446–465Google Scholar
  2. 2.
    Miller, V.S.: Use of elliptic curves in cryptography. In Williams, H.C., ed.: Advances in Cryptology — CRYPTO’85. Volume 218 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, Heidelberg, New York (1986) 417–426Google Scholar
  3. 3.
    Koblitz, N.: Elliptic curve cryptosystems. Mathematics of Computation 48 (1987) 203–209zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Cohen, H., Miyaji, A., Ono, T.: Efficient elliptic curve exponentiation using mixed coordinates. In Ohta, K., Pei, D., eds.: Advances in Cryptology — ASIACRYPT’98. Volume 1514 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, Heidelberg, New York (1998) 51–65CrossRefGoogle Scholar
  5. 5.
    Lopez, J., Dahab, R.: Improved algorithms for elliptic curve arithmetic in gf(2n). In Tavares, S., Meijer, H., eds.: Selected Areas in Cryptography — 5th Annual International Workshop, SAC’98. Volume 1556 of Lecture Notes in Computer Science., Berlin, Heidelberg, New York, Springer-Verlag (1999) 210–212Google Scholar
  6. 6.
    Woodbury, A., Bailey, D., Paar, C.: Elliptic curve cryptography on smart cards without coprocessors. In: the Fourth Smart Card Research and Advanced Applications (CARDIS 2000) Conference. CADIS’2000, Bristol, UK (2000)Google Scholar
  7. 7.
    Koyama, K., Tsuruoka, Y.: Speeding up elliptic curve cryptosystems by using a signed binary windows method. In Brickell, E.F., ed.: Advances in Cryptology — CRYPTO’92. Volume 740 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, Heidelberg, New York (1993) 345–357Google Scholar
  8. 8.
    Smart, N.P.: The hessian form of an elliptic curve. In: Preproceedings of Cryptographic Hardware and Embedded Systems. CHES2001 (2001) 121–128Google Scholar
  9. 9.
    Lipmaa, H.: Idea: A cipher for multimedia architectures? In Tavares, S., Meijer, H., eds.: Selected Areas in Cryptography-5th Annual International Workshop, SAC’98. Volume 1556 of Lecture Notes in Computer Science., Berlin, Heidelberg, New York, Springer-Verlag (1999) 248–263Google Scholar
  10. 10.
    Dixon, B., Lenstra, A.K.: Factoring integers using simd sieves. In Helleseth, T., ed.: Advances in Cryptology — EUROCRYPT’93. Volume 765 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, Heidelberg, New York (1993) 28–39Google Scholar
  11. 11.
    Menezes, A.J., van Oorschot, P.C., Vanstone, S.A.: Handbook of applied cryptography. CRC Press (1997)Google Scholar
  12. 12.
    Biham, E.: A fast new des implementation in sofware. In Biham, E., ed.: Fast Software Encryption-4th International Workshop, FSE’97. Volume 1267 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, Heidelberg, New York (1997) 260–272Google Scholar
  13. 13.
    Nakajima, J., Matsui, M.: Fast software implementations of misty1 on alpha processors. IEICE Transactions Fundamentals of Electronics, Communications and Computer Sciences (Japan) E82-A (1999) 107–116Google Scholar
  14. 14.
    NIST: Recommended elliptic curves for federal government use (1999) (available at http://csrc.nist.gov/csrc/fedstandards.html).
  15. 15.
    Knuth, D.E.: Seminumerical Algorithms. Third edn. Volume 2 of The Art of Computer Programming. Addison Wesley (1997)Google Scholar
  16. 16.
    Itoh, T., Tsujii, S.: A fast algorithm for computing multiplicative inverses in gf(2m) using normal bases. In: Information and Computation. Volume 78. (1988) 171–177zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Schroeppel, R., Orman, H., O’Malley, S., Sparscheck, O.: Fast key exchange with ellipic curve systems. In Coppersmith, D., ed.: Advances in Cryptology — CRYPTO’95. Volume 963 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, Heidelberg, New York (1995) 43–56Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Kazumaro Aoki
  • Fumitaka Hoshino
  • Tetsutaro Kobayashi
  • Hiroaki Oguro

There are no affiliations available

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