**Motivations:** - Keep the current Collatz working materials (conjecture notes, partial certificate, trapdoor spec) versioned alongside the rest of v0. - Integrate new narrative bridges into the children-book draft. - Align the prototype write-up with sensor/feature-vector input framing. **Root causes:** - N/A (content/documentation update) **Correctifs:** - Clarify the prototype input-port description and add a sensor integration note. **Evolutions:** - Add a depth-16 partial Collatz certificate (Markdown + JSON) and its proof-style write-up. - Add Collatz conjecture working notes. - Add Collatz-Trapdoor mathematical specification draft. - Add two additional "bis" chapters to `v0/livre_enfant.md`. - Add a working notes dump. **Pages affectées:** - v0/Spécifications Mathématiques Collatz-Trapdoor.md - v0/collatz_certificat_partiel_depth16.json - v0/collatz_certificat_partiel_depth16.md - v0/conjoncture_collatz.md - v0/livre_enfant.md - v0/notes.md - v0/prototype_reel.md Co-authored-by: Cursor <cursoragent@cursor.com>
155 lines
70 KiB
Markdown
155 lines
70 KiB
Markdown
# **Spécifications Mathématiques : Protocole Collatz-Trapdoor**
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Ce document détaille les fondements arithmétiques du système cryptographique basé sur la dynamique de Collatz compressée.
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## **1\. Définition de l'Opérateur de Base**
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Soit ![][image1] l'ensemble des entiers impairs positifs. L'application de Collatz compressée ![][image2] est définie par :
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![][image3]Où ![][image4] est la **valuation 2-adique** de ![][image5] (l'exposant de la plus grande puissance de 2 divisant ![][image5]).
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## **2\. Construction de la Trajectoire (Clé Privée → Publique)**
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Soit ![][image6] la clé privée. On génère la clé publique en itérant ![][image7] fois l'application ![][image8].
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### **Formule Explicite de la Trajectoire**
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Après ![][image7] étapes, le point d'arrivée ![][image9] peut être exprimé par la relation linéaire :
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![][image10]Où les paramètres structurels sont définis par :
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1. **Somme des Valuations :** ![][image11], avec ![][image12].
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2. **Terme Additif (Constante de Translation) :** $$ C\_k \= \\sum\_{j=0}^{k-1} 3^{k-1-j} \\cdot 2^{A\_j}![][image13]
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### **Données de la Clé Publique**
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La clé publique est le triplet ![][image14] où :
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* ![][image15] est le point d'arrivée.
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* ![][image7] est le nombre d'itérations.
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* ![][image16] est le **modulo de précision**.
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## **3\. Exemple Concret d'Application**
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### **A. Génération et Dérivation**
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Alice choisit ![][image17] et ![][image18]. Comme calculé précédemment, sa clé publique est ![][image19].
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### **B. Chiffrement**
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Pour chiffrer un message ![][image20], Bob utilise un "sel" aléatoire ![][image21] et calcule :
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![][image22]Où ![][image23] est un nombre qui suit la même trajectoire que ![][image24] sur ![][image7] pas.
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### **C. Déchiffrement**
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Alice utilise ![][image24] pour soustraire la structure de Collatz de ![][image25] et retrouver ![][image20] par division modulaire.
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## **4\. Protocole de Signature avec Nonce (Non-Répudiation)**
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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].
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## **5\. Résistance Post-Quantique (Analyse Détaillée)**
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La résistance du protocole face à un ordinateur quantique repose sur deux piliers :
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### **A. Échec de l'Algorithme de Shor (Non-Périodicité)**
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L'algorithme de Shor casse le RSA car il peut trouver la "période" (le cycle) d'une fonction d'exponentiation modulaire.
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* **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.
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### **B. Problème des Préimages dans un Graphe (Complexité de Grover)**
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L'algorithme de Grover permet de chercher un élément dans une base de données non triée avec une accélération quadratique (![][image30]).
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* **Le Labyrinthe Inverse :** Inverser ![][image31] revient à remonter un arbre binaire dont le nombre de nœuds est proportionnel à ![][image32].
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* **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.
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### **C. Réduction au problème "Learning With Errors" (LWE)**
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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.
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## **6\. Performance et Sécurité**
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* **Compute :** Très efficace grâce aux opérations binaires (bit-shifts).
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* **Exclusions :** Bannissement des nombres de Mersenne (![][image37]) et des trajectoires trop courtes.
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## **7\. Recommandations Finales**
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* **Taille de ![][image24] :** 2048 bits.
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* **Horizon ![][image7] :** 256 itérations minimum (512 pour une sécurité PQ maximale).
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[image1]: <data:image/png;base64,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>
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[image2]: <data:image/png;base64,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>
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[image3]: <data:image/png;base64,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>
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[image12]: <data:image/png;base64,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>
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[image20]: <data:image/png;base64,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>
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[image21]: <data:image/png;base64,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>
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