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Autour de la cryptographie à base de tores algébriques

Abstract : Discrete logarithm-based cryptography has sustained many studies in the last decade. Lenstra and Verheul have especially proposed to make use of algebraic tori. We focus here on the computational aspect of these ideas and on the parametrization of such structures. Van Dijk and Woodruff have recently given an explicit way of compactly representing a large set of points on an algebraic torus. The computational cost of this algorithm can be improved thanks to several tools. First we use a new class of bases for finite field extensions, namely elliptic normal bases due to Couveignes and Lercier. Besides we notice that the size of the groups involved is given in terms of cyclotomic polynomials and their modular inverses. The magnitude of their coefficients plays a dramatic role in the complexity study. In the case of indices dividing the product of two distinct primes, we manage to find bounds or explicit expressions of these coefficients. This allows us to compute the communication cost of protocoles such as Diffie-Hellman multiple key exchange.
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Contributor : Clément Dunand <>
Submitted on : Friday, February 25, 2011 - 10:28:38 AM
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  • HAL Id : tel-00569448, version 1


Clément Dunand. Autour de la cryptographie à base de tores algébriques. Mathématiques [math]. Université Rennes 1, 2010. Français. ⟨NNT : 2010REN1S112⟩. ⟨tel-00569448⟩



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