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The index calculus method using non-smooth polynomials


Authors: Theodoulos Garefalakis and Daniel Panario
Journal: Math. Comp. 70 (2001), 1253-1264
MSC (2000): Primary 11Y16, 12E05; Secondary 11T71, 68P25, 68Q25, 94A60
DOI: https://doi.org/10.1090/S0025-5718-01-01298-4
Published electronically: March 7, 2001
MathSciNet review: 1826581
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Abstract:

We study a generalized version of the index calculus method for the discrete logarithm problem in ${\mathbb F}_{ q}$, when $q=p^n$, $p$ is a small prime and $n\rightarrow \infty$. The database consists of the logarithms of all irreducible polynomials of degree between given bounds; the original version of the algorithm uses lower bound equal to one. We show theoretically that the algorithm has the same asymptotic running time as the original version. The analysis shows that the best upper limit for the interval coincides with the one for the original version. The lower limit for the interval remains a free variable of the process. We provide experimental results that indicate practical values for that bound. We also give heuristic arguments for the running time of the Waterloo variant and of the Coppersmith method with our generalized database.


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Additional Information

Theodoulos Garefalakis
Affiliation: Department of Computer Science, University of Toronto, Toronto, M5S 3G4, Canada
Email: theo@cs.toronto.edu

Daniel Panario
Affiliation: School of Mathematics and Statistics, Carleton University, Ottawa, K1S 5B6, Canada
Email: daniel@math.carleton.ca

DOI: https://doi.org/10.1090/S0025-5718-01-01298-4
Keywords: Finite fields, discrete logarithm problem, cryptography, smooth polynomials
Received by editor(s): May 24, 1999
Published electronically: March 7, 2001
Additional Notes: Work done while the second author was with the Department of Computer Science, University of Toronto.
Article copyright: © Copyright 2001 American Mathematical Society

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