[{"command":"insert","method":null,"selector":".course-modal-ajax-placeholder-957876","data":"\u003Cdiv class=\u0022container content-rte\u0022\u003E\n  \u003Cdiv class=\u0022row\u0022\u003E\n    \u003Cdiv class=\u0022box col-md-6 bg-primary text-white\u0022\u003E\n      \u003Ch2 class=\u0022h1 text-left\u0022\u003E  Algorithmes en th\u00e9orie des nombres 2025-2026\n\u003C\/h2\u003E\n                  \u003Cdiv class=\u0022field-spacing\u0022\u003E\n        \u003Cdiv class=\u0022field__label h3 dash\u0022\u003EECTS :\n          6\n        \u003C\/div\u003E\n      \u003C\/div\u003E\n                  \u003Cdiv class=\u0022field-spacing\u0022\u003E\n        \u003Cdiv class=\u0022field__label h3 dash\u0022\u003ESemestre calendaire :\n          Semester 2\n        \u003C\/div\u003E\n      \u003C\/div\u003E\n            \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003ED\u00e9tail du volume horaire\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                          \u003Cspan class=\u0022field__label h4 mt-5 \u0022\u003ECours magistraux :\u003C\/span\u003E\n                                      \u003Cspan\u003E20\u003C\/span\u003E\n                      \u003C\/div\u003E\n                                \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                          \u003Cspan class=\u0022field__label h4 mt-5 \u0022\u003ETravaux pratiques :\u003C\/span\u003E\n                                      \u003Cspan\u003E10\u003C\/span\u003E\n                      \u003C\/div\u003E\n                                                                          \u003C\/div\u003E\n        \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003ELangue d\u0026#039;enseignement\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                                      \u003Cspan\u003EAnglais\u003C\/span\u003E\n                      \u003C\/div\u003E\n                  \u003C\/div\u003E\n        \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003EEnseignement \u00e0 distance\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                                      \u003Cspan\u003Enon\u003C\/span\u003E\n                      \u003C\/div\u003E\n                  \u003C\/div\u003E\n      \u003C\/div\u003E\n    \u003Cdiv class=\u0022box col-md-6\u0022\u003E\n            \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003EPr\u00e9requis\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                                      \u003Cspan\u003E\u003Cp\u003EM1 de Math\u00e9matique Fondamentale\u003C\/p\u003E\u003C\/span\u003E\n                      \u003C\/div\u003E\n                  \u003C\/div\u003E\n        \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003EProgramme \/ plan \/ contenus\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                                      \u003Cspan\u003E\u003Cp\u003EL\u0027objectif de ce cours est d\u0027aborder les fondements th\u00e9oriques d\u0027algorithmes actuels utilis\u00e9s en cryptographie. Les principaux outils viennent de la th\u00e9orie analytique des nombres et des courbes elliptiques. Des preuves math\u00e9matiques compl\u00e8tes des algorithmes seront donn\u00e9es, ainsi qu\u0027une \u00e9tude d\u00e9taill\u00e9e de leur complexit\u00e9 et une impl\u00e9mentation lors des s\u00e9ances de TP. Contenu : 1. Tests de primalit\u00e9 : tests probabilistes (Solovay-Strassen ; Miller-Rabin), d\u00e9terministes (AKS, Agrawal-Kayal-Saxena ; ECPP, Elliptic Curve Primality Proving), certificats (Pratt ; Atkin-Goldwasser-Kilian-Morain). 2. Factorisation de grands entiers : algorithmes Rho et p-1 de Pollard, courbes elliptiques (ECM), crible quadratique (QS), crible de corps de nombres g\u00e9n\u00e9ralis\u00e9 (GNFS). 3. Le probl\u00e8me du logarithme discret : algorithmes Baby step Giant step, Rho de Pollard, Pohlig-Hellman, calcul d\u0027indice.\u003C\/p\u003E\n\u003Cp\u003EThe aim of this course is to tackle the theoretical background of some modern algorithms currently used in cryptography. The main tools stem from analytic number theory and the theory of elliptic curves. Thorough mathematical proofs of the algorithms will be provided, along with a detailed study of their complexity and an implementation during hands-on sessions on computers. 1. Primality tests: probabilistic (Solovay-Strassen; Miller-Rabin), deterministic (AKS, Agrawal-Kayal-Saxena; ECPP, Elliptic Curve Primality Proving), certificates (Pratt; Atkin-Goldwasser-Kilian-Morain). 2. Large integer factorization: Pollard\u0027s Rho and p-1 algorithms, elliptic curves (ECM), quadratic sieve (QS), general number field sieve (GNFS). 3. Discrete logarithm problem: Baby step Giant step, Pollard\u0027s Rho, Pohlig-Hellman, index calculus algorithms.\u003C\/p\u003E\u003C\/span\u003E\n                      \u003C\/div\u003E\n                  \u003C\/div\u003E\n                                        \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003EBibliographie\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                                      \u003Cspan\u003E\u003Cp\u003E[1] H. Cohen et G. Frey, Handbook of elliptic and hyperelliptic curve cryptography, Chapman \u0026amp; Hall, 2006. [2] R. Crandall et C. Pomerance, Prime numbers: a computational perspective, 2nd ed., Springer, 2005.\u003C\/p\u003E\u003C\/span\u003E\n                      \u003C\/div\u003E\n                  \u003C\/div\u003E\n        \u003Cdiv class=\u0022field-spacing\u0022\u003E\n      \u003Cdiv class=\u0022field__label h3 dash\u0022\u003ENature de l\u0026#039;\u00e9valuation\u003C\/div\u003E\n                        \u003Cdiv class=\u0022mt-5 text-justify\u0022\u003E\n                                      \u003Cspan\u003EEvaluation Terminale\u003C\/span\u003E\n                      \u003C\/div\u003E\n                  \u003C\/div\u003E\n      \u003C\/div\u003E\n  \u003C\/div\u003E\n\u003C\/div\u003E\n","settings":null}]