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Exploring Cryptography Projects with Rust and Discrete Mathematics

Overview

Summary: This note documents my journey in exploring cryptographic concepts using Rust while applying and deepening my understanding of Discrete Mathematics. The primary motive behind this exercise is to actively learn by implementing mathematical principles in practical cryptographic projects. Each project is designed to reinforce concepts like number theory, modular arithmetic, and graph theory, ensuring that the learning is both practical and comprehensive.

Key Takeaways

  • Learning by doing: The best way to internalize Discrete Mathematics concepts is through real-world applications, and cryptography provides the perfect context.
  • Rust as a learning tool: Rust’s strict type system and focus on safety make it an ideal language for understanding the nuances of mathematical computations.
  • Connecting theory to practice: By implementing these projects, I aim to bridge the gap between theoretical knowledge and practical application.

Detailed Notes

1. Caesar Cipher & Frequency Analysis

  • Concepts: Modular Arithmetic, Probability, Combinatorics, Permutations
  • Summary: Implement a simple encryption system that shifts letters by a fixed number of positions. Frequency analysis is then used to decrypt messages by exploiting letter distribution patterns.
  • GitHub: Caesar Cipher Project

2. RSA Encryption Implementation

  • Concepts: Number Theory, Modular Arithmetic, Euler’s Totient Function
  • Summary: Implement the RSA encryption algorithm, which uses large primes and modular arithmetic to secure data through public and private keys.
  • GitHub: RSA Encryption Project

3. Diffie-Hellman Key Exchange

  • Concepts: Discrete Logarithms, Modular Arithmetic, Prime Numbers
  • Summary: Develop a secure key exchange mechanism where two parties can generate a shared secret using modular exponentiation, even over an insecure channel.
  • GitHub: Diffie-Hellman Project

4. Elliptic Curve Cryptography (ECC)

  • Concepts: Finite Fields, Group Theory, Algebraic Curves
  • Summary: Implement ECC to explore how elliptic curves provide cryptographic strength with smaller key sizes compared to traditional RSA.
  • GitHub: Elliptic Curve Cryptography Project

5. Password Hashing with Salt

  • Concepts: Probability, Combinatorics, Hash Functions, Set Theory
  • Summary: Secure password storage using hashing techniques, with added protection through a unique random salt to defend against dictionary and rainbow table attacks.
  • GitHub: Password Hashing Project

6. Digital Signatures (Simplified RSA)

  • Concepts: Modular Arithmetic, Number Theory, Hash Functions
  • Summary: Create a mechanism for authenticating messages by signing them with a private key and verifying the authenticity using the corresponding public key.
  • GitHub: Digital Signatures Project

7. Zero-Knowledge Proof Simulation

  • Concepts: Graph Theory, Probability, Combinatorics
  • Summary: Simulate a zero-knowledge proof where a prover can convince a verifier of their knowledge of a secret without revealing the secret itself.
  • GitHub: Zero-Knowledge Proof Project

8. Graph-Based Cryptographic Protocol

  • Concepts: Graph Theory, Combinatorial Optimization, Complexity Theory
  • Summary: Explore how graph structures, such as Hamiltonian cycles or isomorphism problems, can be used in cryptographic protocols.
  • GitHub: Graph-Based Cryptographic Protocol Project

By documenting this journey, I aim to maintain a structured record of my learning while inviting others to follow along and contribute to discussions. These notes will serve as a long-term reference for revisiting key concepts and reflecting on how they evolve with further exploration.