Euclidean lattices: algorithms and cryptography

Abstract : Euclidean lattices are a rich algebraic object that occurs in a wide variety of contexts in mathematics and in computer science. The present thesis considers several algorithmic aspects of lattices. The concept of lattice basis reduction is thoroughly investigated: in particular, we cover the full range of time-quality trade-offs of reduction algorithms. On the first hand, we describe and analyse fast algorithms for finding a relatively short basis (LLL-reduced basis) of an arbitrary given lattice. On the second hand, we propose novel analyses for (slower) algorithms that compute very short bases (HKZ-reduced and BKZ-reduced bases). This study on how to efficiently solve algorithmic problems on lattices is completed by a constructive application exploiting their apparent hardness. We propose and analyze cryptographic schemes, including the NTRU encryption function, and prove them at least as secure as well-specified worst-case problems on lattices.
Document type :
Habilitation à diriger des recherches
Cryptography and Security. Ecole normale supérieure de lyon - ENS LYON, 2011


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Contributor : Damien Stehle <>
Submitted on : Tuesday, November 29, 2011 - 12:07:06 PM
Last modification on : Tuesday, November 29, 2011 - 1:53:44 PM

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  • HAL Id : tel-00645387, version 1

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Damien Stehlé. Euclidean lattices: algorithms and cryptography. Cryptography and Security. Ecole normale supérieure de lyon - ENS LYON, 2011. <tel-00645387>

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