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Resolution of Linear Algebra for the Discrete Logarithm Problem using GPU and Multi-core Architectures

Hamza Jeljeli
CARAMEL project-team, LORIA, INRIA / CNRS / Universite de Lorraine, Campus Scientique, BP 239, 54506 Vanduvre-les-Nancy Cedex, France
hal-00946895, (14 February 2014)

@unpublished{jeljeli:hal-00946895,

   hal_id={hal-00946895},

   url={http://hal.inria.fr/hal-00946895},

   title={Resolution of Linear Algebra for the Discrete Logarithm Problem using GPU and Multi-core Architectures},

   author={Jeljeli, Hamza},

   keywords={Discrete logarithm problem; sparse linear algebra; parallel computing; GPU acceleration; multi-core processors; InfiniBand},

   language={Anglais},

   affiliation={CARAMEL – INRIA Nancy – Grand Est / LORIA},

   year={2014},

   month={Feb},

   pdf={http://hal.inria.fr/hal-00946895/PDF/linalg.pdf}

}

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In cryptanalysis, solving the discrete logarithm problem (DLP) is key to assessing the security of many public-key cryptosystems. The index-calculus methods, that attack the DLP in multiplicative subgroups of finite fields, require solving large sparse systems of linear equations modulo large primes. This article deals with how we can run this computation on GPU- and multi-core-based clusters, featuring InfiniBand networking. More specifically, we present the sparse linear algebra algorithms that are proposed in the literature, in particular the block Wiedemann algorithm. We discuss the parallelization of the central matrix–vector product operation from both algorithmic and practical points of view, and illustrate how our approach has contributed to the recent record-sized DLP computation in GF($2^{809}$).
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