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Table of contents
Abstract
(French) Résumé
(French) Introduction
1 Motivation
2 Organisation de la thèse et résultats
Introduction
1 Motivation
2 Contributions & Organization
I Preliminaries
1 Algebraic Number Theory Tools
1.1 Linear algebra & Lattices
1.2 Number fields & Polynomials
1.3 Orders & Ideals
1.4 Norms & Smoothness
1.5 Ideal classes & Units
2 Previous work on class group computations and related problems
2.1 Exponential strategies for quadratic number fields
2.2 Class group generation
2.3 Subexponential complexity, using index calculus method
2.4 Algorithms related to number fields
II Reducing the complexity of Class Group Computation
3 Reduction of the defining polynomial
3.1 Motivations and link with class group computation
3.2 An optimal algorithm for NF defining polynomial reduction
3.3 Complexity analysis
3.4 Application to class group computation
4 Refinements for complexities appearing in the literature for the general case
4.1 The classification defined by classes D is sufficient
4.2 The relation collection
4.3 Complexity analyses
4.4 Using HNF to get an even smaller complexity
5 Reducing the complexity using good defining polynomials
5.1 Motivation
5.2 Deriving relations by sieving
5.3 Complexity analyses
5.4 Conclusion on sieving strategy
5.5 Application to Principal Ideal Problem
III Applications to Cryptology
6 PIP solution in cyclotomic fields and cryptanalysis of an FHE scheme
6.1 Situation of the problem and cryptosystems that rely on SPIP
6.2 Solving the PIP or how to performa full key recovery?
6.3 Description of the algorithm
6.4 Complexity analysis
6.5 Implementation results
Bibliography
