Abstract
In this paper, we study the application of the function field sieve algorithm for computing discrete logarithms over finite fields of the form \({\mathbb {F}}_{q^n}\) when q is a medium-sized prime power. This approach is an alternative to a recent paper of Granger and Vercauteren for computing discrete logarithms in tori, using efficient torus representations. We show that when q is not too large, a very efficient L(1/3) variation of the function field sieve can be used. Surprisingly, using this algorithm, discrete logarithms computations over some of these fields are even easier than computations in the prime field and characteristic two field cases. We also show that this new algorithm has security implications on some existing cryptosystems, such as torus based cryptography in T 30, short signature schemes in characteristic 3 and cryptosystems based on supersingular abelian varieties. On the other hand, cryptosystems involving larger basefields and smaller extension degrees, typically of degree at most 6, such as LUC, XTR or T 6 torus cryptography, are not affected.
The original version of this chapter was revised: The copyright line was incorrect. This has been corrected. The Erratum to this chapter is available at DOI: 10.1007/978-3-540-34547-3_36
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Keywords
- Discrete Logarithm
- Discrete Logarithm Problem
- Multiplicative Identity
- Asymptotic Complexity
- Extension Degree
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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Joux, A., Lercier, R. (2006). The Function Field Sieve in the Medium Prime Case. In: Vaudenay, S. (eds) Advances in Cryptology - EUROCRYPT 2006. EUROCRYPT 2006. Lecture Notes in Computer Science, vol 4004. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11761679_16
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