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Mathematics of Computation

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Evaluation of discrete logarithms
in a group of $p$-torsion points
of an elliptic curve in characteristic $p$

Author: I. A. Semaev
Journal: Math. Comp. 67 (1998), 353-356
MSC (1991): Primary 94A60
MathSciNet review: 1432133
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Abstract | References | Similar Articles | Additional Information

Abstract: We show that to solve the discrete log problem in a subgroup of order $p$ of an elliptic curve over the finite field of characteristic $p$ one needs $O(\ln p)$ operations in this field.

References [Enhancements On Off] (What's this?)

  • 1. V. Miller, Use of elliptic curves in cryptography, Advances in Cryptology-Crypto '85, Springer-Verlag, New York, 1986, 417-426. MR 88b:68040
  • 2. N. Koblitz, Elliptic curve cryptosystems, Math. Comp. 48 (1987), 203-209. MR 88b:94017
  • 3. A. Menezes, S. Vanstone, and O. Tatsuaki, Reducing elliptic curve logarithms to logarithms in a finite field, Proc. 23rd ACM Sympos. Theory of Computing, 1991, pp. 80-89.
  • 4. $XXXX$ $XXXX$, International Conference ``Modern Problems in Number Theory'', Russia, Tula, Sept. 20-25, 1993, Abstracts of papers.
  • 5. G. Frey and H.-G. Ruck, A remark concerning $m$-divisibility and the discrete logarithm in the divisor class group of curves, Math. Comp. 62 (1994), 865-874. MR 94h:11056
  • 6. S. Pohlig and M. Hellman, An improved algorithm for computing logarithms over $GF(p)$ and its cryptographic significance, IEEE Trans. Inform. Theory, IT-24 (1978), 106-110. MR 58:4617
  • 7. J. M. Pollard, Monte-Carlo methods for index computation $(\operatorname{mod}p)$, Math. Comp. 32 (1978), 918-924. MR 58:10684
  • 8. J. H. Silverman, The arithmetic of elliptic curves, Springer-Verlag, 1986. MR 87g:11070
  • 9. J. P. Serre, Sur la topologie des variétés algébriques en caractéristique $p$, Sympos. Internac. Topologia Algebraica, Mexico City, 1956, 24-53. MR 20:4559
  • 10. H.-G. Ruck, A remark on the paper ``Evaluation of discrete logarithms on some elliptic curves, by I. A. Semaev'', communication to ``Mathematics of Computation''.

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Additional Information

I. A. Semaev
Affiliation: 43-2 Profsoyusnaya ul., Apt. 723, 117420 Moscow, Russia

Keywords: Cryptography, elliptic curves, discrete logarithms
Received by editor(s): August 30, 1995
Received by editor(s) in revised form: September 11, 1996
Article copyright: © Copyright 1998 American Mathematical Society

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