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Global Duality, Signature Calculus and the Discrete Logarithm Problem

  • Ming-Deh Huang (a1) and Wayne Raskind (a2)
Abstract
Abstract

We develop a formalism for studying the discrete logarithm problem for the multiplicative group and for elliptic curves over finite fields by lifting the respective group to an algebraic number field and using global duality. One of our main tools is the signature of a Dirichlet character (in the multiplicative group case) or principal homogeneous space (in the elliptic curve case), which is a measure of its ramification at certain places. We then develop signature calculus, which generalizes and refines the index calculus method. Finally, using some heuristics, we show the random polynomial time equivalence for these two cases between the problem of computing signatures and the discrete logarithm problem. This relates the discrete logarithm problem to some very well-known problems in algebraic number theory and arithmetic geometry.

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

5. H. Cohen and H.W. Lenstra Jr., ‘Heuristics on class groups of number fields’, Number theory, Noordwijkerhout 1983, 3362, Lecture Notes in Math., 1068 (Springer, Berlin, 1984).

12. D. Goldfeld , ‘Conjectures on elliptic curves over quadratic fields’, in Number Theory (Carbondale, Ill., 1979), Lecture Notes in Math. 751 (Springer, Berlin, 1979) 108118.

13. R. Hartshorne , Algebraic Geometry, Graduate Texts in Mathematics, Volume 52 (Springer-Verlag, New York, Heidelberg, Berlin 1977).

22. K. McCurley , ‘The discrete logarithm problem’, Cryptology and Computational Number Theory, ed. C. Pomerance , Proceedings of Symposia in Applied Mathematics, 42 (1990) 4974.

36. J.H. Silverman , The Arithmetic of Elliptic Curves (Graduate Texts in Mathematics, Volume 106, Springer Verlag, 1986).

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LMS Journal of Computation and Mathematics
  • ISSN: -
  • EISSN: 1461-1570
  • URL: /core/journals/lms-journal-of-computation-and-mathematics
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