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Contributions to the hardness foundations of lattice-based cryptography

Weiqiang Wen 1, 2
2 ARIC - Arithmetic and Computing
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme
Abstract : Lattice-based cryptography is one of the most competitive candidates for protecting privacy, both in current applications and post quantum period. The central problem that serves as the hardness foundation of lattice-based cryptography is called the Learning with Errors (LWE). It asks to solve a noisy equation system, which is linear and over-determined modulo q. Normally, we call LWE problem as an average-case problem as all the coefficients in the equation system are randomly chosen modulo q. The LWE problem is conjectured to be hard even wtih a large scale quantum computer. It is at least as hard as standard problems defined in the lattices, such as Bounded Distance Decoding (BDD) and unique Shortest Vector Problem (uSVP). Finally, the best known algorithm for solving these problems is BKZ, which is very expensive. In this thesis, we study the quantum hardness of LWE, the hardness relations between the underlying problems BDD and uSVP, and the practical performance of the BKZ algorithm. First, we give a strong evidence of quantum hardness of LWE. Concretely, we consider a relaxed version of the quantum version of dihedral coset problem and show an computational equivalence between LWE and this problem. Second, we tighten the hardness relation between BDD and uSVP. More precisely, We improve the reduction from BDD to uSVP by a factor √2, compared to the one by Lyubashevsky and Micciancio. Third, we propose a more precise simulator for BKZ. In the last work, we propose the first probabilistic simulotor for BKZ, which can pridict the practical behavior of BKZ very precisely.
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  • HAL Id : tel-01949339, version 2


Weiqiang Wen. Contributions to the hardness foundations of lattice-based cryptography. Computational Complexity [cs.CC]. Université de Lyon, 2018. English. ⟨NNT : 2018LYSEN070⟩. ⟨tel-01949339v2⟩



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