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Linear Algebra Algorithms for Cryptography Claire Delaplace

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Linear Algebra Algorithms for Cryptography Claire Delaplace Linear algebra algorithms for cryptography Claire Delaplace To cite this version: Claire Delaplace. Linear algebra algorithms for cryptography. Cryptography and Security [cs.CR]. Université Rennes 1, 2018. English. NNT : 2018REN1S045. tel-02000721 HAL Id: tel-02000721 https://tel.archives-ouvertes.fr/tel-02000721 Submitted on 1 Feb 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THÈSE DE DOCTORAT DE L’UNIVERSITE DE RENNES 1 COMUE UNIVERSITE BRETAGNE LOIRE Ecole Doctorale N°601 Mathématiques et Sciences et Technologies de l’Information et de la Communication Spécialité : Informatique Par Claire Delaplace Algorithmes d’algèbre linéaire pour la cryptographie Thèse présentée et soutenue à RENNES , le 21 Novembre 2018 Unité de recherche : Institut de Recherche en Informatique et Système Aléatoire (IRISA), UMR 6074 Rapporteurs avant soutenance : Guénaël Renault : Expert Cryptographie. Damien Vergnaud : Professeur Université de Paris 6. Composition du jury : Examinateurs : Aurélie Bauer : Expert Cryptographie. Jean-Guillaume Dumas : Professeur Université Grenoble Alpes. María Naya-Plasencia : Directrice de Recherche Inria. Marion Videau : Maître de Conférences Université de Lorraine. Rapporteurs : Guénaël Renault : Expert Cryptographie. Damien Vergnaud : Professeur Université de Paris 6. Dir. de thèse : Pierre-Alain Fouque : Professeur Université de Rennes 1. Co-dir. de thèse : Charles Bouillaguet : Maître de Conférence Université de Lille. Remerciements Je tiens à remercier tous ceux qui m’ont aidée à réaliser cette thèse, et tout d’abord l’ANR Brutus pour l’avoir financée. Un grand merci à mes deux directeurs, Pierre-Alain Fouque et Charles Bouillaguet, qui m’ont soutenue et encadrée tout au long de ces trois années. Plus particulièrement je remercie Pierre- Alain pour sa patience et son calme à toute épreuve, et pour avoir toujours pris le temps de suivre mes travaux, malgré son emploi du temps chargé. Je remercie également Charles, pour son enthousiasme, sa bonne humeur, et pour m’avoir conseillée et guidée dans mes recherches, et cela malgré la distance entre Lille et Rennes. Je remercie ensuite les membres du jury : Aurélie Bauer, Jean-Guillaume Dumas, Maria Naya- Plasencia, Guénaël Renault, Damien Vergnaud et Marion Videau. Je remercie d’autant plus Guén- aël Renault et Damien Vergnaud, pour avoir accepté d’être les rapporteurs, et pour leur remarques pertinentes qui ont amélioré la qualité de ce manuscrit. Durant ces trois ans j’ai travaillé dans deux équipes où j’ai pu bénéficier d’un excellent environ- nement. Tout d’abord à Lille, où je remercie les membres (et anciens membres) de l’équipe CFPH, pour leur accueil chaleureux: François Boulier (pour ses conseils et ses remarques), Yacine Bouz- idi, François Lemaire (pour les discussions que nous avons eu), Adrien Poteaux, Alban Quadrat, Alexandre Sedoglavic, Alexandre Temperville (qui entamait sa dernière année de thèse quand je débutais et qui a ainsi joué un rôle de « grand frère » tout au long de cette première année) et Marie-Emilie Voge (avec laquelle j’ai également eu le plaisir de co-écrire un article). Une petite pensée également pour Eric Wegrzynowski, Léopold Weinberg et tous ceux avec lesquels j’avais l’habitude de manger le midi. Pendant les deux années suivantes, j’ai eu le plaisir de cotoyer les membres de l’équipe EMSEC de l’IRISA de Rennes. Merci tout d’abord aux permanents : Patrick Derbez et Adeline Roux- Langlois pour leurs remarques et conseils, à Benoît Gérard, Clémentine Maurice et Mohamed Sabt pour leur enthousiasme et leur bonne humeur, merci aussi à Gildas Avoine, Stéphanie Delaune, et Barbara Kordy. Ces deux années n’auraient pas été les mêmes sans les thésards/thésardes et postdocs EMSEC, qui ont grandement contribué à la bonne ambiance de l’équipe. Merci à vous pour tous les bons moments passés ensemble, que ce soit lors des pauses cafés ou des soirées bowling. Plus particulièrement merci à Florent Tardif pour son humour sarcastique, Alban Siffer parce qu’il ne « boit pas l’eau des pâtes », Wojciech Widel for making the effort of learning a bit of french, Alexandre Debant pour son accent toulousain, Angèle Bossuat pour avoir enrichi mon vocabulaire d’onomatopées variées et d’expressions peu communes, Merci également aux « exilés » de l’étage vert avec lesquels j’ai passé le plus de temps dans nos bureaux mitoyens : Pauline Bert pour dire régulièrement des phrases étranges et amusantes sorties de leur contexte, Chen Qian parce qu’il est bizarre et qu’il voit des girafes là où le reste du monde ne voit qu’un trait, Guillaume Kaim pour l’ajout de « petit pain » à l’éternel débat « pain au chocolat » ou « chocolatine », et merci à Baptiste Lambin pour m’avoir supportée comme co-bureau pendant deux ans, merci pour les discussions improbables sur la grammaire française, anglaise et elfique, pour les débats finalement peu constructifs sur l’enseignement, pour tes « là haut » utilisés à tord et à travers... Je pense que ces deux ans auraient été plus longs, si je n’avais pas eu l’occasion de t’embêter régulièrement. Merci aussi à Adina Nedelcu et Céline Duguey, qui sont venues un jour par semaine égayer le bureau de leurs sourires respectifs. J’ai également une pensée pour les anciens thésards/postdocs: Raphael Bost, Pierre Karpman, Pierre Lestringant, Vincent Migliore, Brice Minaud, Cristina Onete (pour les conversations sur la musique ou la littérature), Benjamin Richard (le chauffeur de taxi le moins ponctuel de Rennes mais qui a le mérite d’être gratuit), et Cyrille Wiedling (pour ses gâteaux). i Enfin, j’en profite pour saluer les nouveaux: Adeline Bailly, Katharina Boudgoust, Xavier Bultel, Victor Mollimard, Solène Moreau, Sam Thomas et Weiqiang Wen. Je remercie aussi Sylvain Duquesne pour avoir fait partie de mon CSID, avec Patrick Derbez. Je tiens également à saluer mes autres co-auteurs: merci à Jonathan Bootle, Thomas Espitau, Mehdi Tibouchi, et Paul Kirchner pour leurs contributions à la réussite de cette thèse ; ainsi que Bonan Cuan, Aliénor Damien et Mathieu Valois, avec lesquels j’ai de plus passé une semaine à la fois fort sympathique et productive lors de l’évènement REDOCS’17. Ce fut un véritable plaisir de travailler avec vous. J’en profite pour remercier Pascal Lafourcade, pour ses encouragements et ses conseils valables en toutes circonstances. Merci également Jean-Guillaume Dumas, Damien Stehlé, et Morgan Barbier pour m’avoir per- mis de présenter mes travaux respectivement sur l’Elimination Gaussienne creuse, SPRING, et le problème 3XOR dans leurs séminaires respectifs. Je remercie aussi la LinBox Team pour m’avoir permis de faire partie de ce projet. Je n’ai malheureusement pas eu (pris) le temps de m’y in- vestir autant que je l’aurais souhaité. Merci également à Gaëtan Leurent pour avoir mis à notre disposition les codes source de SPRING-BCH et SPRING-CRT, qui m’ont été d’une grande aide. J’en profite pour saluer le corps enseignant de l’Université de Versailles-Saint-Quentin en Yve- lines, où j’ai effectué mes études de la L1 jusqu’au Master. Plus particulièrement merci à Luca Defeo pour avoir soutenu ma candidature de stage de M2 à l’Université de Lille, m’aidant ainsi à mettre un pied dans le monde de la recherche. J’ai aussi une pensée (sans vous nommer car vous êtes nombreux) pour toutes les personnes avec lesquelles j’ai sympathisé lors de conférences ou d’autres déplacements. Merci pour les bons moments passés ensemble. Pour finir, je voudrais remercier ma famille. En particulier, je remercie ma tante Agnès, pour m’avoir laissée utiliser sa chambre d’amis pendant le temps que j’ai passé à Lille. Enfin, je remercie mes parents, pour m’avoir toujours encouragée, et en particulier pendant ces derniers mois de thèse qui ont été assez éprouvants. Merci à mon père pour m’avoir donné le goût des mathématiques. Merci à ma mère, qui a du supporter régulièrement des discussions mathématiques pendant les repas, et pour avoir pris le temps de relire ces deux-cents pages de blabla technique afin de repérer un maximum de typos. Je ne serais jamais arrivée jusque là sans votre soutien. ii Résumé (en français) Contexte et organisation de cette thèse Un algorithme est une suite d’instructions qui doivent être suivies afin de retrouver une solution à un problème donné. La complexité en temps d’un algorithme est donnée par le nombre d’étapes nécessaires pour retrouver une solution. La complexité en mémoire est la quantité d’espace néces- saire au déroulement de l’algorithme. Pour un problème donné, il existe autant d’algorithmes qu’il y a de façons de résoudre ce problème, ce qui fait beaucoup, en général. Tous ces algorithmes ne sont pas équivalents en ce sens qu’ils n’ont pas tous la même complexité en temps et en mémoire. Généralement, nous considérons que le « meilleur » algorithme est celui avec la plus petite com- plexité en temps. Ce n’est pourtant pas toujours le cas, selon ce que nous recherchons. En effet, il existe des algorithmes qui ont la meilleure complexité en temps en théorie, mais qui sont cependant totalement inutilisables en pratique (par exemple [LG14]). Cette thèse traite principalement de plusieurs problèmes rencontrés en cryptologie et en calcul formel. En m’intéressant aux aspects algorithmiques de ces problèmes, j’ai souvent été confrontée à cet écart entre la théorie et la réalité.
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