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Calcul De Groupes De Classes D'un Corps De Nombres Et Applications THÈSE DE DOCTORAT DE L’UNIVERSITÉ PIERRE ET MARIE CURIE Spécialité Informatique École Doctorale Informatique, Télécommunications et Électronique (Paris) Présentée par Alexandre GÉLIN Pour obtenir le grade de DOCTEUR de l’UNIVERSITÉ PIERRE ET MARIE CURIE Calcul de Groupes de Classes d’un Corps de Nombres et Applications à la Cryptologie Thèse dirigée par Antoine JOUX et Arjen LENSTRA soutenue le vendredi 22 septembre 2017 après avis des rapporteurs : M. Andreas ENGE Directeur de Recherche, Inria Bordeaux-Sud-Ouest & IMB M. Claus FIEKER Professeur, Université de Kaiserslautern devant le jury composé de : M. Karim BELABAS Professeur, Université de Bordeaux M. Andreas ENGE Directeur de Recherche, Inria Bordeaux-Sud-Ouest & IMB M. Claus FIEKER Professeur, Université de Kaiserslautern M. Louis GOUBIN Professeur, Université de Versailles-St-Quentin-en-Yvelines M. Antoine JOUX Chaire de Cryptologie à la Fondation UPMC, Paris M. Arjen LENSTRA Professeur, École Polytechnique Fédérale de Lausanne Mme Ariane MÉZARD Professeur, Université Pierre et Marie Curie, Paris M. Nigel SMART Professeur, Université de Bristol Class Group Computations in Number Fields and Applications to Cryptology Alexandre GÉLIN Team ALMASTY, Laboratoire d’Informatique de Paris 6 UPMC Sorbonne Universités Laboratoire d’Informatique de Paris 6 Université Pierre et Marie Curie Équipe ALMASTY École doctorale EDITE – ED 130 UPMC - Sorbonne Universités Faculté d’ingénierie – UFR 919 4, place Jussieu 4, place Jussieu 75005 Paris, FRANCE 75005 Paris, FRANCE Mathematics may be defined as the subject where we never know what we are talking about, nor whether what we are saying is true. — Bertrand Russell À tous ceux qui ne pourront lire ces mots. Abstract A common way to construct public-key cryptosystems is to use the discrete exponentiation in a finite group where the inverse problem, the discrete logarithm problem, is considered difficult. Several groups are commonly used: multiplicative group of finite fields or Jacobians of algebraic curves defined over finite fields. The class group of a number field is another candidate which was sometimes proposed. To study its viability, it is important to identify the difficulty for this kind of group to determine its cardinality, its structure and the possible hardness of the discrete logarithm problem. Irrespective of cryptologic applications, methods to determine the properties of these groups is of independent mathematical interest. In this thesis, we focus on class group computations in number fields. We start by de- scribing an algorithm for reducing the size of a defining polynomial of a number field. There exist infinitely many polynomials that define a specific number field, with arbitrarily large coefficients, but our algorithm constructs the one that has the absolutely smallest coefficients. The advantage of knowing such a “small” defining polynomial is that it makes calculations in the number field easier because smaller values are involved. In addition, thanks to such a small polynomial, one can use specific algorithms that are more efficient than the general ones for class group computations. The generic algorithm to determine the structure of a class group is based on ideal reduc- tion, where ideals are viewed as lattices. We describe and simplify the algorithm presented by Biasse and Fieker in 2014 at ANTS and provide a more thorough complexity analysis for it. We also examine carefully the case of number fields defined by a polynomial with small coeffi- cients. We describe an algorithm similar to the Number Field Sieve, which, depending on the ¡ 1 ¢ field parameters, may reach the hope for complexity L 3 . Finally, our results can be adapted to solve an associated problem: the Principal Ideal Problem. Given any basis of a principal ideal (generated by a unique element), we are able to find such a generator. As this problem, known to be hard, is the key-point in several homomorphic cryptosystems, the slight modifications of our algorithms provide efficient attacks against these cryptographic schemes. i Résumé L’une des voies privilégiées pour la construction de cryptosystèmes à clé publique est l’uti- lisation dans des groupes finis de l’exponentiation discrète, dont le problème inverse, celui du logarithme discret, est réputé difficile. Plusieurs groupes sont classiquement utilisés : le groupe multiplicatif d’un corps fini ou la Jacobienne d’une courbe algébrique définie sur un corps fini. Le groupe de classes d’un corps de nombres est un autre candidat qui a parfois été proposé. Pour étudier sa viabilité, il est important de bien cerner la difficulté de déterminer la cardinalité et la structure du groupe, ainsi que la difficulté éventuelle du problème du logarithme discret. Indépendamment de ces applications cryptographiques, les méthodes utilisées pour résoudre ces problèmes ont aussi un intérêt mathématique. Dans cette thèse, nous nous intéressons au calcul du groupe de classes d’un corps de nombres. Nous débutons par décrire un algorithme de réduction du polynôme de définition d’un corps de nombres. Il existe une infinité de polynômes qui définissent un corps de nombres fixé, avec des coefficients arbitrairement gros. Notre algorithme calcule celui qui a les plus petits coefficients. L’avantage de connaître un petit polynôme de définition est qu’il simplifie les calculs entre éléments de ce corps de nombres, en impliquant des quantités plus petites. En outre, la connaissance d’un tel polynôme permet l’utilisation d’algorithmes plus efficaces que dans le cas général pour calculer le groupe de classes. L’algorithme général pour calculer la structure du groupe de classes repose sur la réduction d’idéaux, vus comme des réseaux. Nous décrivons et simplifions l’algorithme présenté par Biasse et Fieker en 2014 à ANTS et approfondissons l’analyse de complexité. Nous nous sommes aussi intéressés au cas des corps de nombres définis par un polynôme à petits coefficients. Nous décrivons un algorithme similaire au crible par corps de nombres (NFS) ¡ 1 ¢ dont la complexité en fonction des paramètres du corps de nombres peut atteindre L 3 . Enfin, nos algorithmes peuvent être adaptés pour résoudre un problème lié : le Problème de l’Idéal Principal. Étant donné n’importe quelle base d’un idéal principal (généré par un seul élément), nous sommes capables de retrouver ce générateur. Cette application de nos algorithmes fournit une attaque efficace contre certains schémas de chiffrement homomorphe basés sur ce problème. iii Contents Abstract i (French) Résumé iii (French) Introduction1 1 Motivation.........................................1 2 Organisation de la thèse et résultats..........................6 Introduction 13 1 Motivation......................................... 13 2 Contributions & Organization.............................. 17 I Preliminaries 23 1 Algebraic Number Theory Tools 25 1.1 Linear algebra & Lattices................................. 26 1.2 Number fields & Polynomials.............................. 31 1.3 Orders & Ideals...................................... 38 1.4 Norms & Smoothness.................................. 41 1.5 Ideal classes & Units................................... 48 2 Previous work on class group computations and related problems 53 2.1 Exponential strategies for quadratic number fields................. 54 2.2 Class group generation.................................. 60 2.3 Subexponential complexity, using index calculus method............. 61 2.4 Algorithms related to number fields.......................... 69 v Contents II Reducing the complexity of Class Group Computation 73 3 Reduction of the defining polynomial 75 3.1 Motivations and link with class group computation................. 76 3.2 An optimal algorithm for NF defining polynomial reduction............ 79 3.3 Complexity analysis................................... 84 3.4 Application to class group computation........................ 87 4 Refinements for complexities appearing in the literature for the general case 93 4.1 The classification defined by classes D is sufficient................. 94 4.2 The relation collection.................................. 97 4.3 Complexity analyses................................... 102 4.4 Using HNF to get an even smaller complexity.................... 104 5 Reducing the complexity using good defining polynomials 109 5.1 Motivation......................................... 110 5.2 Deriving relations by sieving.............................. 111 5.3 Complexity analyses................................... 114 5.4 Conclusion on sieving strategy............................. 118 5.5 Application to Principal Ideal Problem........................ 119 III Applications to Cryptology 129 6 PIP solution in cyclotomic fields and cryptanalysis of an FHE scheme 131 6.1 Situation of the problem and cryptosystems that rely on SPIP........... 132 6.2 Solving the PIP or how to perform a full key recovery?............... 134 6.3 Description of the algorithm.............................. 135 6.4 Complexity analysis................................... 145 6.5 Implementation results................................. 146 Bibliography 151 vi Introduction 1 Motivation Cryptographie. La cryptographie moderne se divise en deux grandes sous-familles. La plus ancienne est la cryptographie symétrique. L’idée principale est que l’émetteur et le desti- nataire d’un message partagent un secret commun, appelé clé secrète. En 1976, Diffie et Hellman introduisent la cryptographie asymétrique — aussi appelée cryptographie à clé pu- blique — dans leur article intitulé New Directions in Cryptography
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