Abstract
The discrete logarithm problem is analyzed from the perspective of Tate local duality. Local duality in the multiplicative case and the case of Jacobians of curves over p-adic local fields are considered. When the local field contains the necessary roots of unity, the case of curves over local fields is polynomial time reducible to the multiplicative case, and the multiplicative case is polynomial time equivalent to computing discrete logarithm in finite fields. When the local field does not contains the necessary roots of unity, similar results can be obtained at the cost of going to an extension that contains these roots of unity. There was evidence in the analysis that suggests that the minimal extension where the local duality can be rationally and algorithmically defined must contain the roots of unity. Therefore, the discrete logarithm problem appears to be well protected against an attack using local duality. These results are also of independent interest for algorithmic study of arithmetic duality as they explicitly relate local duality in the case of curves over local fields to the multiplicative case and Tate-Lichtenbaum pairing (over finite fields).
Similar content being viewed by others
References
Cassels J W S, Fröhlich A. Algebraic Number Theory. New York: Academic Press, 1967
Frey G. On Bilinear Structures on Divisor Class Groups. Ann Math Blaise Pascal, 2009, 16: 1–26
Frey G, Müller M, Rück H G. The Tate pairing and the discrete logarithm applied to elliptic curve cryptosystems. IEEE Trans Inform Theory, 1999, 45: 1717–1719
Frey G, Rück H G. A remark concerning m-divisibility and the discrete logarithm in the divisor class group of curves. Math Comput, 1994, 62: 865–874
Huang M D. Local duality and the discrete logarithm problem. In: Lecture Notes in Computer Science, vol. 6639. Berlin: Springer, 2011, 213–222
Joux A. The Weil and Tate Pairings as Building Blocks for Public Key Cryptosystems. In: Lecture Notes in Computer Science, vol. 2369. Berlin: Springer, 2002, 20–32
Lichtenbaum S. Duality theorems for curves over p-adic fields. Invent Math, 1969, 7: 120–136
Menezes A, Okamoto S, Vanstone T. Reducing elliptic curve logarithms to logarithms in a finite field. IEEE Trans Infor Theory, 1993, 39: 1639–1646
Milne J S. Arithmetic Duality Theorems. New York: Academic Press, 1986
Nguyen K. Explicit Arithmetic of Brauer Groups-Ray Class Fields and Index Calculus. PhD Thesis. Essen: University of Essen, 2001
Serre J P. Local Fields. New York: Springer-Verlag, 1979
Tate J. WC-groups over p-adic fields. Sem Bourbaki, 1957, 4: 265–277
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Huang, M. The discrete logarithm problem from a local duality perspective. Sci. China Math. 56, 1421–1427 (2013). https://doi.org/10.1007/s11425-013-4674-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-013-4674-1