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DESIGN and CRYPTANALYSIS of POST QUANTUM CRYPTOSYSTEMS Olive Chakraborty

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DESIGN and CRYPTANALYSIS of POST QUANTUM CRYPTOSYSTEMS Olive Chakraborty DESIGN AND CRYPTANALYSIS OF POST QUANTUM CRYPTOSYSTEMS Olive Chakraborty To cite this version: Olive Chakraborty. DESIGN AND CRYPTANALYSIS OF POST QUANTUM CRYPTOSYSTEMS. Cryptography and Security [cs.CR]. Sorbonne Université, 2020. English. tel-03135217 HAL Id: tel-03135217 https://tel.archives-ouvertes.fr/tel-03135217 Submitted on 12 Feb 2021 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 DOCTORANT DE SORBONNE UNIVERSITÉ Spécialité Informatique École Doctorale Informatique, Télécommunications et Électronique (Paris) Présentée par OLIVE CHAKRABORTY Pur obtenir le grade de DOCTEUR DE SORBONNE UNIVERSITÈ DESIGN AND CRYPTANALYSIS OF POST QUANTUM CRYPTOSYSTEMS Thèse dirigée par JEAN-CHARLES FAUGÈRE et LUDOVIC PERRET après avis des rapporteurs: Mme. Delaram KAHROBAEI Professeur, University of York, U.K M. Jacques PATARIN Professeur, Université de Versailles devant le jury composé de : M. Jean-Charles FAUGÈRE Directeur de recherche, INRIA Paris M. Stef GRAILLAT Professeur, Sorbonne Université, LIP6 Mme. Delaram KAHROBAEI Professeur, University of York, U.K M. Jacques PATARIN Professeur, Université de Versailles M. Ludovic PERRET Maître de Conférences, Sorbonne Université, LIP6 M. Mohab SAFEY EL DIN Professeur, Sorbonne Université, LIP6 Date de soutenance : 16-12-2020 Résumé La résolution de systèmes polynomiaux est l’un des problèmes les plus anciens et des plus importants en Calcul Formel et a de nombreuses applications. C’est un problème intrinsèquement difficile avec une complexité, en générale, au moins exponentielle en le nombre de variables. Dans cette thèse, nous nous concen- trons sur des schémas cryptographiques basés sur la difficulté de ce problème. Cependant, les systèmes polynomiaux provenant d’applications telles que la cryp- tographie multivariée, ont souvent une structure additionnelle cachée. En parti- culier, nous donnons la première cryptanalyse connue du crypto-système « Exten- sion Field Cancellation ». Nous travaillons sur le schéma à partir de deux aspects, d’abord nous montrons que les paramètres de challenge ne satisfont pas les 80 bits de sécurité revendiqués en utilisant les techniques de base Gröbner pour résoudre le système algébrique sous-jacent. Deuxièmement, en utilisant la struc- ture des clés publiques, nous développons une nouvelle technique pour montrer que même en modifiant les paramètres du schéma, le schéma reste vulnérable aux attaques permettant de retrouver le secret. Nous montrons que la variante avec erreurs du problème de résolution d’un système d’équations est encore dif- ficile à résoudre. Enfin, en utilisant ce nouveau problème pour concevoir un nouveau schéma multivarié d’échange de clés nous présentons un candidat qui a été soumis à la compétition Post-Quantique du NIST. Mots clés : Cryptographie, Post-quantique, Multivariée, cryptage à clé publique, base de Gröbner, cryptanalyse algébrique, système polynomial avec erreurs, NIST. Abstract Polynomial system solving is one of the oldest and most important problems in computational mathematics and has many applications in computer science. It is intrinsically a hard problem with complexity at least single exponential in the number of variables. In this thesis, we focus on cryptographic schemes based on the hardness of this problem. In particular, we give the first known cryptanalysis of the Extension Field Cancellation cryptosystem. We work on the scheme from two aspects, first we show that the challenge parameters don’t satisfy the 80 bits of security claimed by using Gröbner basis techniques to solve the underlying algebraic system. Secondly, using the structure of the public keys, we develop a new technique to show that even altering the parameters of the scheme still keeps the scheme vulnerable to attacks for recovering the hidden secret. We show that noisy variant of the problem of solving a system of equations is still hard to solve. Finally, using this new problem to design a new multivariate key- exchange scheme as a candidate for NIST Post Quantum Cryptographic Stan- dards. Keywords: Post-quantum Cryptography, Multivariate, Extension Field Cancel- lation, Gröbner basis, Algebraic Cryptanalysis, Polynomial systems with Errors, NIST. To my dearest mother Moushumi and heavenly father Haridash Acknowledgements My thesis has only been possible because of a lot of effort, help and support of the people that I came across during this process. First and foremost, I thank my mother and my heavenly father, it is because of them I am where I am. Without their thankless efforts for all these years noth- ing of this would have been possible. I am in your debt for my entire life. I thank my advisors Jean-Charles Faugère and Ludovic Perret for their guid- ance throughout this journey. I learned an incredible amount of things from them, but in particular how to do research and, more importantly how to deal with roller coaster of emotions that is associated with PhD. They inspired my love for the subjects on which I worked and my decision to pursue an academic career. They are the role models for the scientists that I would like to become. I would like to thank Jacques Patarin and Deleram Kahrobaei for reviewing this manuscript and for their comments that helped me to improve it. I thank Stef Graillant and Mohan Safey El Din for accepting to be part of the jury of my thesis. Additionally I thank Stef and Jacques again for being a part of my mid PhD evaluation committees and their advice on many topics. I thank the members of the PolSys, both present and past, for their compan- ionship all these years. In particular, my heartiest thanks to Mohab Safey El Din for his invaluable advice every time I went to him, whether it be academic, ad- ministrative or personal. To Jérémy Berthomieu for his unconditional help with every possible thing I can think of (especially teaching me French). I thank my fellow PhD mates, Huu-Phuoc, Xuan, Jocelyn, Eliane, Solane, Nagarjun, Andrew, Jorge and Hieu, for their time shared. I thank the secretaries of our team, lab and école doctoral, for their help all these years. I would like to thank the CROUS and its staffs who took care of our health providing delicious and healthy food, which I consider is one of the crucial things that allowed me to carry on with my work without worrying about food. iii I thank the people that this work gave me who now I proudly call as friends. To Matias, Rachel, James, and Kaie for making my time at work and after it mem- orable. To Elias Tsigaridas, who I can’t thank enough for everything he has done for me during this time and treated me like his own. To Mme. Corado, Rahma, Maurice, Alice, Andrina, Rafa, George, Steph for being the best flatmates ever and making this quarantine a little fun for me. In most of the manuscripts that I have read, this section ends with some words about the author’s significant other, and this shall not be an exception, I want to express my deepest gratitude to Saptaparni for being a constant by my side and supporting me all this time. I especially thank you for proof reading my thesis. The amount of love and understanding you have shown for me and my work during all this time cannot be repaid by any means. I cannot thank you enough for lifting up my mood whenever I was down and constantly supporting every decision I took. This work would not have been possible without you. Now that this chapter of my life is reaching its conclusion, I am looking forward to the next one. And I cannot wait to write it together with you. Contents List of Figures ix List of Tables xi 1 Introduction 1 1.1 Organization and Contributions of the thesis ............ 5 1.2 Publications .............................. 8 I Preliminaries 9 2 Polynomial System Solving 11 2.1 General Framework ......................... 11 2.2 Combinatorial Methods ....................... 12 2.2.1 Classical Setting ....................... 12 2.2.2 Quantum Setting ....................... 14 2.3 Gröbner Basis ............................. 15 2.3.1 Preliminary Definitions and Properties ........... 15 2.3.2 Gröbner Basis Algorithms .................. 22 2.3.3 Complexity of Gröbner Basis Computation ......... 32 2.4 Hybrid Combinatorial-Algebraic methods .............. 37 2.4.1 Classical Hybrid Algorithms ................. 37 2.4.2 Quantum Hybrid Approach ................. 40 2.5 Conclusion .............................. 41 3 Quantum-Safe Public-key Cryptography 43 3.1 Multivariate Public-Key Cryptography ................ 43 3.1.1 General Structure ....................... 44 3.1.2 Historical Cryptosystems ................... 46 3.1.3 Generic Modifications on MQ-schemes ............ 48 3.1.4 EFC Scheme .......................... 50 3.2 Standard attacks on MPKCs ..................... 53 3.2.1 Key Recovery Attacks ..................... 53 v 3.2.2 Message Recovery Attacks .................. 57 3.3 Lattice Based Cryptosystems ..................... 59 3.3.1 Frodo Key Exchange ..................... 63 II Contribution 67 4 Cryptanalysis of EFC Cryptosystem 69 4.1 Introduction .............................. 69 4.1.1 Main Results and Organization ............... 69 4.2 Algebraic Cryptanalysis of EFC ................... 71 4.2.1 A Key Recovery Attack .................... 71 4.2.2 A Message Recovery Attack ................. 72 4.2.3 Lower Degree of Regularity ................. 74 F 4.2.4 Analysis of the EFCq(0) and EFCq (0) instances ....... 74 − 4.2.5 Extending to EFCq (a) ..................... 78 − 4.2.6 Analysis on the case EFC2 (1) ................. 83 − 4.2.7 Analysis on the case EFC2 (2) ................. 84 − − 4.2.8 Analysis on the case EFC3 (1) and EFC3 (2) ......... 85 4.3 A Method to Find Degree Fall Equations .............
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