From 087dde15580db025dfd260c67c2f9d5b478590c6 Mon Sep 17 00:00:00 2001 From: Unknown Date: Mon, 2 Jul 2018 18:22:03 +0200 Subject: [PATCH] Cours 6 crypto el gamal, diffie et tout l'tralala --- .../Boite à outils arithmétique.md | 9 +- .../Ressources/Cryptographie-cours1.pdf | Bin 3587209 -> 0 bytes ... DIFFIE HELLMAN.pdf => Diffie_Helmann.pdf} | Bin ...Master Eii - EL GAMAL.pdf => El_Gamal.pdf} | Bin .../Ressources/{ => Notes}/Cours 3.docx | Bin .../Exercice-Type-Cryptographie_v1.1.docx | Bin .../Prise de note Arithmétique modulaire.docx | Bin ... de note Histoire de la cryptographie.docx | Bin .../{Master Eii - RSA.pdf => RSA.pdf} | Bin Cours Cryptographie/Ressources/Sujet2014.pdf | Bin 0 -> 758858 bytes Cours Cryptographie/Ressources/TD.pdf | Bin 0 -> 586020 bytes Cours Cryptographie/À voir.md | 6 +- .../Éléments d'arithmétique.md | 135 +++++++++++++----- 13 files changed, 107 insertions(+), 43 deletions(-) delete mode 100644 Cours Cryptographie/Ressources/Cryptographie-cours1.pdf rename Cours Cryptographie/Ressources/{Master Eii - DIFFIE HELLMAN.pdf => Diffie_Helmann.pdf} (100%) rename Cours Cryptographie/Ressources/{Master Eii - EL GAMAL.pdf => El_Gamal.pdf} (100%) rename Cours Cryptographie/Ressources/{ => Notes}/Cours 3.docx (100%) rename Cours Cryptographie/Ressources/{ => Notes}/Exercice-Type-Cryptographie_v1.1.docx (100%) rename Cours Cryptographie/Ressources/{ => Notes}/Prise de note Arithmétique modulaire.docx (100%) rename Cours Cryptographie/Ressources/{ => Notes}/Prise de note Histoire de la cryptographie.docx (100%) rename Cours Cryptographie/Ressources/{Master Eii - RSA.pdf => RSA.pdf} (100%) create mode 100644 Cours Cryptographie/Ressources/Sujet2014.pdf create mode 100644 Cours Cryptographie/Ressources/TD.pdf diff --git a/Cours Cryptographie/Boite à outils arithmétique.md b/Cours Cryptographie/Boite à outils arithmétique.md index 4b20ac9..0e3703c 100644 --- a/Cours Cryptographie/Boite à outils arithmétique.md +++ b/Cours Cryptographie/Boite à outils arithmétique.md @@ -1,10 +1,13 @@ Boite à outils arithmétique -======================== +=== + Liste outils vus en cours et bien utiles -- division euclidienne +- Division euclidienne - Congruence modulo - Univers Zn (+;x) - Exponentiation modulaire - PGCD avec algo d'Euclide -- Bachet Bezoud \ No 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parler du Z*n* : on se base dans le modulo de *n* dnas les entiers, ce sont des univers. On compte en modulo. Exmeple avc Z6 : +On va commencer à parler du Z*n* : on se base dans le modulo de *n* dans les entiers, ce sont des univers. On compte en modulo. Exemple avec Z6 : -| + | 1 | 2 | 3 | 4 | 5 | -| 1 | 2 | 3 | 4 | 5 | 0 | -| 2 | 3 | 4 | 5 | 0 | 1 | -| 3 | 4 | 5 | 0 | 1 | 2 | -| 4 | 5 | 0 | 1 | 2 | 3 | -| 5 | 0 | 1 | 2 | 3 | 4 | +| + | 1 | 2 | 3 | 4 | 5 | +| --- | --- | --- | --- | --- | --- | +| 1 | 2 | 3 | 4 | 5 | 0 | +| 2 | 3 | 4 | 5 | 0 | 1 | +| 3 | 4 | 5 | 0 | 1 | 2 | +| 4 | 5 | 0 | 1 | 2 | 3 | +| 5 | 0 | 1 | 2 | 3 | 4 | + +Du coup l'opposé d'un chiffre est celui qui correspond au croisement avec 0 dans le tableau. -Du coup l'opposé d'un chiffre est celui qui correspond au croisement avec 0 dans le tableau. La loi universelle qu'on ressort est : soit *n* un chiffre et *o* son opposé, dans l'univers *U*, o = U - n . Voirs cours pour la suite/règles. @@ -88,24 +88,24 @@ L'univers Z2 est le concept du OU exclusif : on a 0 si valeurs identiques, 1 si ### Multiplication -Voir cours. Il faudrait faire toutes les tabls de Z5, Z6, Z7, Z9, Z11 +Voir cours. Il faudrait faire toutes les tables de Z5, Z6, Z7, Z9, Z11 Z6 à nouveau : -| x | 1 | 2 | 3 | 4 | 5 | -| 1 | 1 | 2 | 3 | 4 | 5 | -| 2 | 2 | 4 | 0 | 2 | 4 | -| 3 | 3 | 0 | 3 | 0 | 3 | -| 4 | 4 | 2 | 0 | 4 | 2 | -| 5 | 5 | 4 | 3 | 2 | 1 | +| $$\times$$ | 1 | 2 | 3 | 4 | 5 | +| ---------- | --- | --- | --- | --- | --- | +| 1 | 1 | 2 | 3 | 4 | 5 | +| 2 | 2 | 4 | 0 | 2 | 4 | +| 3 | 3 | 0 | 3 | 0 | 3 | +| 4 | 4 | 2 | 0 | 4 | 2 | +| 5 | 5 | 4 | 3 | 2 | 1 | -Attention : on oublie le "si a * b = 0, a = 0 et/ou b = 0" est faux dans cet univers. (Ici 2 * 3=0) +Attention : on oublie le "si a * b = 0, a = 0 et/ou b = 0" : c'est faux dans cet univers. (Ici 2 * 3=0) Si on prend 2, il a pas d'inverse (on cherche 2 * x = 1, ça n'existe pas dans cette table). En fait on a pas d'inverse à part 1 et n-1, c'est valable pour tous ls univers. Pour 2, 3, 4 y a pas d'inverse et c'est pas bijectif (il y a des répétitions, pas des valeurs uniques) On se rend compte que ça "marche" partout pour chiffres premiers, mais que quand pas premier, les lignes des diviseurs et de leurs multiples sont foireux. Dans ces cas seules les lignes restantes auront des inverses et seront bijectives. - ### Exercices A) Les seuls éléments inversibles dans Z10 sont 1, 3, 7 et 9 @@ -114,11 +114,9 @@ Z15* = {1,2,4,7,8,11,13,14} (8 éléments) Z12* = {1,5,7,11} (4 éléments) Z17* = {TOUS} (16 éléments) - - Prochain cour son va voir Bachet bézout >> à utiliser dans grands univers, dans petits autant faire la table. -b - +b - x = 7 (9) insolvable (9) x = {0;3;6} (9) @@ -162,13 +160,12 @@ Plusieurs formules permettent de savoir si il y a un nombre permier dans un éca on peut utiliser le chapeau ( ^ ) pour l'indiquer. Il y a une technique . Si PGCD est 1, les nombres sont étrangers. Pour trouver on décompose en nombre premier et on voit. Mais on va préférer l'algorithme d'euclide : on divise a par b et on note le reste. Puis b devient a et r devient b et on recommence jusqu'à trouver 0 ou 1 -### PPCM +### PPCM Noté a v b a v b * a ^ b = ab; donc si a ^ b = 1, a v b = ab; - ### Bachet Bezoud **Ultra important : outil principal** @@ -187,11 +184,12 @@ donc on a au + bv = 1 si les nombres sont étrangers. À partir de ça, on se pl Modulo b : bv congruent 0 dont au = 1 (b) donc u = a^-1 (b) Modulo a : au congruent 0 donc bv = -Cette démonstration permet de : -- déterminer u et v dans une équation diophantiene -- trouver facilement les éléments inversibles ! +Cette démonstration permet de : -### Indicatif d'Euler +- déterminer u et v dans une équation diophantiene +- trouver facilement les éléments inversibles ! + +### Indicatrice d'Euler Z9* = {1,2,4,5,7,8} (6 éléments) Z15* = {1,2,4,7,8,11,13,14} (8 éléments) @@ -201,17 +199,25 @@ Z17* = {TOUS} (16 éléments) => Dans Z*p, il y a p-1 éléments inversibles => Dans Z*n... on va utiliser la formule d'Eulère -Elle permet de trouver nombre d'élémnts inversibles à partir de la factorisation d'un nombre. +Elle permet de trouver nombre d'élémnts inversibles à partir de la factorisation d'un nombre. -On va relier ça à** Fermat :** +On va relier ça à **Fermat :** On note phi (n) le nombre d'éléments inversibles dans n +Définir Indicatrice d'Euler et justifier son utilisation dans un cryptosystème : + +- ça sert à déterminer clef privée + +- donne l'interface de bouclage + +- nombre d'éléments inversibles dans modulo *n* + ### Théorème d'Euler il nous dit que si a^n, alors a^phi(n) congruent 1(n) -devient ensuite +devient ensuite ### Petit théorème de Fermat @@ -219,11 +225,66 @@ Dans un univers Z*p, n^p-1 congruent à 1 (p) Dans un univers Z*n, la puissance correspond à l'indicatif d'Euler - ### Théorème des restes chinois On l'utilise pas le théorème chinois en tant que tel, on va utiliser son corollaire : théorème des restes chinois ou lemme chinois. +On a vu théorème avec un système de deux, maintenant on voit la règle générale (avec plus de conditions dans le système) + +$$ +\begin{cases} +x_1 \equiv a \pmod p \\ +x_2 \equiv b \pmod q \\ +x_3 \equiv c \pmod r +\end{cases} +$$ + +On obtient + +$$ +x = a \times y_1 \times M_1 + b \times y_2 \times M_2 + a \times y_3 \times M_3 \pmod M +$$ + +Où + +$$ +M = p \times q \times r \\ +M_i = \frac MM_i \\ +y_1 = M_1^{-1} \pmod p +$$ + ### Exercices Sur feuille + +## Exam + +On aura 6 questions : + +Q1 : Éléments d'arithmétique - 2pts - Sera dur, le prof pense que beaucoup vont échouer, faire preuve de **logique** + +Q2: Problème 1 - 6pts - C'est plutôt des maths + +Q3: Histoire - 3pts - Des connaissances du cours + +Q4: Éléments d'arithmétique - 3pts - Basé sur méthodologie (fortes chances pour restes chinois) + +Q5: Problème 2 - 6pts - Basé sur la compréhension, faisable + +## Algorithme de Diffie-Hellman + +Protocole de génération de clé par canal public. Il ne s'agit pas d'un système de cryptage. + +Repose sur le principe : + +$$ +g^{a^b} = g^{ab} = g^{b^a} +$$ + +Il est décidé de deux nombres g et n transmis publiquement. De chaque côté, quelqu'un fait l'opération à partir de g modulo n avec un a ou b tenu secret et transmet le résultat r. Ensuite chacun élève ce résultat r à la puissance de son a ou b, modulo n et obtient la même clé de chaque côté. + +## El-Gamal + +Il s'agit ici non plus d'une simple transmission de clef, mais carrément d'un message chiffré, avec le concept de clé publique qui apparait. Assez proche de Diffie-Hellman dans l'idée, puisqu'on utilise encore une fois le délire des puissances et la transmission de chiffres publics, sauf que cette fois-ci l'expéditeur utilise le bouzin pour définir une clé, puis utilise cette clef pour chiffrer le message , et renvoie le tout. + +Exercice sur feuille

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