Abstract
Using a recent idea of Gaudry and exploiting rational representations of algebraic tori, we present an index calculus type algorithm for solving the discrete logarithm problem that works directly in these groups. Using a prototype implementation, we obtain practical upper bounds for the difficulty of solving the DLP in the tori \(T_2(\mathbb{F}_{p^m})\) and \(T_6(\mathbb{F}_{p^m})\) for various p and m. Our results do not affect the security of the cryptosystems LUC, XTR, or CEILIDH over prime fields. However, the practical efficiency of our method against other methods needs further examining, for certain choices of p and m in regions of cryptographic interest.
Keywords
- Discrete Logarithm
- Discrete Logarithm Problem
- Compression Factor
- Cyclotomic Polynomial
- Cryptology ePrint Archive
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.
The work described in this paper has been supported in part by the European Commission through the IST Programme under Contract IST-2002-507932 ECRYPT. The information in this document reflects only the authors’ views, is provided as is and no guarantee or warranty is given that the information is fit for any particular purpose. The user thereof uses the information at its sole risk and liability.
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Granger, R., Vercauteren, F. (2005). On the Discrete Logarithm Problem on Algebraic Tori. In: Shoup, V. (eds) Advances in Cryptology – CRYPTO 2005. CRYPTO 2005. Lecture Notes in Computer Science, vol 3621. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11535218_5
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