From 70724edcd280c771cd07fe530e4ea9f9dc45a39d Mon Sep 17 00:00:00 2001 From: Nicolas Cantu Date: Wed, 25 Feb 2026 02:26:46 +0100 Subject: [PATCH] Update Collatz trapdoor spec exports and notes MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit **Motivations :** - Keep the latest Collatz-Trapdoor specification in exported formats. - Extend the Collatz conjecture working document with additional material. **Root causes :** - N/A (documentation/content update) **Correctifs :** - N/A **Evolutions :** - Replace the previous Markdown trapdoor spec with PDF/DOCX exports (v2.0). - Update `v0/conjoncture_collatz.md` with new sections. **Page affectées :** - v0/Spécifications Mathématiques Collatz-Trapdoor.md - v0/Spécifications Mathématiques Collatz-Trapdoor.pdf - v0/Spécifications Mathématiques _ Protocole Collatz-Trapdoor v2.0.docx - v0/Spécifications Mathématiques _ Protocole Collatz-Trapdoor v2.0.pdf - v0/conjoncture_collatz.md Co-authored-by: Cursor --- ...fications Mathématiques Collatz-Trapdoor.md | 155 ------- ...ications Mathématiques Collatz-Trapdoor.pdf | Bin 0 -> 144909 bytes ...iques _ Protocole Collatz-Trapdoor v2.0.docx | Bin 0 -> 3111199 bytes ...tiques _ Protocole Collatz-Trapdoor v2.0.pdf | Bin 0 -> 140457 bytes v0/conjoncture_collatz.md | 385 ++++++++++++++++++ 5 files changed, 385 insertions(+), 155 deletions(-) delete mode 100644 v0/Spécifications Mathématiques Collatz-Trapdoor.md create mode 100644 v0/Spécifications Mathématiques Collatz-Trapdoor.pdf create mode 100644 v0/Spécifications Mathématiques _ Protocole Collatz-Trapdoor v2.0.docx create mode 100644 v0/Spécifications Mathématiques _ Protocole Collatz-Trapdoor v2.0.pdf diff --git a/v0/Spécifications Mathématiques Collatz-Trapdoor.md b/v0/Spécifications Mathématiques Collatz-Trapdoor.md deleted file mode 100644 index 6a4a658..0000000 --- a/v0/Spécifications Mathématiques Collatz-Trapdoor.md +++ /dev/null @@ -1,155 +0,0 @@ -# **Spécifications Mathématiques : Protocole Collatz-Trapdoor** - -Ce document détaille les fondements arithmétiques du système cryptographique basé sur la dynamique de Collatz compressée. - -## **1\. Définition de l'Opérateur de Base** - -Soit ![][image1] l'ensemble des entiers impairs positifs. L'application de Collatz compressée ![][image2] est définie par : - -![][image3]Où ![][image4] est la **valuation 2-adique** de ![][image5] (l'exposant de la plus grande puissance de 2 divisant ![][image5]). - -## **2\. Construction de la Trajectoire (Clé Privée → Publique)** - -Soit ![][image6] la clé privée. On génère la clé publique en itérant ![][image7] fois l'application ![][image8]. - -### **Formule Explicite de la Trajectoire** - -Après ![][image7] étapes, le point d'arrivée ![][image9] peut être exprimé par la relation linéaire : - -![][image10]Où les paramètres structurels sont définis par : - -1. **Somme des Valuations :** ![][image11], avec ![][image12]. -2. **Terme Additif (Constante de Translation) :** $$ C\_k \= \\sum\_{j=0}^{k-1} 3^{k-1-j} \\cdot 2^{A\_j}![][image13] - -### **Données de la Clé Publique** - -La clé publique est le triplet ![][image14] où : - -* ![][image15] est le point d'arrivée. -* ![][image7] est le nombre d'itérations. -* ![][image16] est le **modulo de précision**. - -## **3\. Exemple Concret d'Application** - -### **A. Génération et Dérivation** - -Alice choisit ![][image17] et ![][image18]. Comme calculé précédemment, sa clé publique est ![][image19]. - -### **B. Chiffrement** - -Pour chiffrer un message ![][image20], Bob utilise un "sel" aléatoire ![][image21] et calcule : - -![][image22]Où ![][image23] est un nombre qui suit la même trajectoire que ![][image24] sur ![][image7] pas. - -### **C. Déchiffrement** - -Alice utilise ![][image24] pour soustraire la structure de Collatz de ![][image25] et retrouver ![][image20] par division modulaire. - -## **4\. Protocole de Signature avec Nonce (Non-Répudiation)** - -L'utilisation d'un **Nonce** (![][image26]) garantit que chaque signature est unique. La signature ![][image27] lie le secret ![][image24] au condensé du message ![][image28] et au nombre à usage unique ![][image26]. - -## **5\. Résistance Post-Quantique (Analyse Détaillée)** - -La résistance du protocole face à un ordinateur quantique repose sur deux piliers : - -### **A. Échec de l'Algorithme de Shor (Non-Périodicité)** - -L'algorithme de Shor casse le RSA car il peut trouver la "période" (le cycle) d'une fonction d'exponentiation modulaire. - -* **Dans Collatz :** La suite des valuations ![][image29] est apériodique et chaotique. Il n'y a pas de structure répétitive prévisible sur laquelle un ordinateur quantique peut s'appuyer pour réduire la complexité. - -### **B. Problème des Préimages dans un Graphe (Complexité de Grover)** - -L'algorithme de Grover permet de chercher un élément dans une base de données non triée avec une accélération quadratique (![][image30]). - -* **Le Labyrinthe Inverse :** Inverser ![][image31] revient à remonter un arbre binaire dont le nombre de nœuds est proportionnel à ![][image32]. -* **Résistance :** Même avec l'accélération de Grover, le nombre d'opérations reste de l'ordre de ![][image33]. Si ![][image34], l'effort requis (![][image35]) reste totalement hors de portée des capacités de calcul de l'univers, qu'elles soient quantiques ou classiques. - -### **C. Réduction au problème "Learning With Errors" (LWE)** - -Le terme additif ![][image36] agit comme une erreur (un bruit) injectée à chaque pas. Retrouver ![][image24] ressemble au problème de l'apprentissage avec erreurs, qui est l'une des bases les plus solides de la cryptographie post-quantique actuelle. - -## **6\. Performance et Sécurité** - -* **Compute :** Très efficace grâce aux opérations binaires (bit-shifts). -* **Exclusions :** Bannissement des nombres de Mersenne (![][image37]) et des trajectoires trop courtes. - -## **7\. Recommandations Finales** - -* **Taille de ![][image24] :** 2048 bits. -* **Horizon ![][image7] :** 256 itérations minimum (512 pour une sécurité PQ maximale). - -[image1]: - -[image2]: - -[image3]: - -[image4]: - -[image5]: - -[image6]: - -[image7]: - -[image8]: - -[image9]: - -[image10]: - -[image11]: - -[image12]: - -[image13]: - -[image14]: - -[image15]: - -[image16]: - -[image17]: - -[image18]: - -[image19]: - -[image20]: - -[image21]: - -[image22]: - -[image23]: - -[image24]: - -[image25]: - -[image26]: - -[image27]: - -[image28]: - -[image29]: - -[image30]: - -[image31]: - -[image32]: - -[image33]: - -[image34]: - -[image35]: - -[image36]: - -[image37]: \ No newline at end of file diff --git a/v0/Spécifications Mathématiques Collatz-Trapdoor.pdf b/v0/Spécifications Mathématiques Collatz-Trapdoor.pdf new file mode 100644 index 0000000000000000000000000000000000000000..62b84dac41199ab779886d7228acb91bbf6490bb GIT binary patch literal 144909 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zg6HAAGqMe^I1o(0b_7IILuCaD*`2`QSmy}>wqFDgFl4&|3vtzm{s}DZA9N3we1LdC z^rOJiMtx?{K9r|8q-S^vHfE@-fb=gw1WW^&1j%+D z>@`Fy*lM78gJ_W$k3({o2b&O;l?h13@-hzD47?1y59HWnq7RaoOPLhQ92t#dB3K(p zJ_8Kd&%BI5dvRXI(unr3p+>Za?G{=epjR;-gZY_}BKv@6C?uaj!eSm`q-gKV!yDg- z_6&FnNcyofvi0DNaI}vCwTf(8p22%q9^5{-T^`v7uvL5MpFm zU_Z15VFQO~z;%Wa7CZgPR)#jPd=WQSWE&tn(b&LH#4sMu8!Tx=Q%=C^9F!;;8z?LFi{09!wjRszL|p`sP$R{(zrIhO*MLnQrx253JGJ9bi^QxcG;#C!mZA;$}b znRo!B&=-I~4+$8>3gp_SC?E^L&j9Pn7tWCK3Y^pua-R$^DLGz%ffJeNixdn+f>scW zSrEwa(!ehmrofXM-0YlIuW(b=*Vi+1+O(qcj6iSUnl&5XU<=5i1IK@}T^-iwbbf{t ah4mZk+%|NcCxDj}JebP5x`w7k%KrkP6}70, +\qquad +N_0=\left\lfloor\frac{C_k}{\Delta_k}\right\rfloor+1, +\qquad +n\ge N_0\Rightarrow U^{(k)}(n)0. +] +Donc (n_40, +] +donc descente stricte. + +Forme affine et audit +Ici (k=4), (A_4=8), (2^{A_4}=256), (3^4=81), (C_4=85). +[ +U^{(4)}(n)=\frac{81n+85}{256}. +] +Inégalité : + +* (\dfrac{81n+85}{256}0, +] +donc descente stricte. + +Forme affine et audit +Ici (k=5), (A_5=8), (2^{A_5}=256), (3^5=243), (C_5=211), (\Delta=2^8-3^5=256-243=13). +[ +U^{(5)}(n)=\frac{243n+211}{256}. +] +Inégalité : + +* (\dfrac{243n+211}{256}0. +] + +Audit +Ici (k=5), (A_5=8), (C_5=227), (\Delta=13). +Seuil : + +* (N_0=\left\lfloor \dfrac{227}{13}\right\rfloor+1) +* (227=17\cdot 13+6), donc (\left\lfloor \dfrac{227}{13}\right\rfloor=17) +* (N_0=18) + +Clause (D) : +[ +\forall n\ \text{impair},\ n\equiv 175\pmod{512},\ n\ge 18\Rightarrow U^{(5)}(n)

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