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Euclidean lattices: algorithms and cryptography

Damien Stehlé 1
1 ARENAIRE - Computer arithmetic
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme
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.
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Contributor : Damien Stehle <>
Submitted on : Tuesday, November 29, 2011 - 12:07:06 PM
Last modification on : Thursday, November 21, 2019 - 2:41:11 AM
Long-term archiving on: : Sunday, December 4, 2016 - 11:44:03 PM


  • HAL Id : tel-00645387, version 1



Damien Stehlé. Euclidean lattices: algorithms and cryptography. Cryptography and Security [cs.CR]. Ecole normale supérieure de lyon - ENS LYON, 2011. ⟨tel-00645387⟩



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