Objective: Explore the underlying mathematical principles that form the basis of cryptographic algorithms.
Introduction to Mathematical Foundations
- Purpose: Understanding the mathematics behind cryptography provides insight into how and why cryptographic methods are secure.
- Key Areas: Focus on number theory and algebra, which are crucial in understanding the algorithms used in cryptography.
Key Concepts and Theories
- Modular Arithmetic:
- Fundamental to many cryptographic algorithms.
- Involves arithmetic with a wrap-around feature at a certain value, known as the modulus.
- Prime Numbers and Factoring:
- The difficulty of factoring large prime numbers forms the basis of algorithms like RSA.
- Prime numbers are used because their properties make the factoring problem hard.
- Elliptic Curves:
- Used in algorithms like ECC (Elliptic Curve Cryptography).
- Involves points on a curve and the mathematics of their relationships.
- Discrete Logarithms:
- The discrete logarithm problem is a hard problem used in cryptographic algorithms, including Diffie-Hellman and some forms of ECC.
Hands-on Exercise: Exploring Modular Arithmetic
- Activity: Solve basic problems using modular arithmetic.
- Objective: Gain practical experience with the concept of modulus as used in cryptography.
Practical Application
- Algorithm Design: Understanding these concepts is essential for developing or analyzing cryptographic algorithms.
- Security Analysis: Helps in assessing the strength and vulnerabilities of cryptographic systems.
Further Reading and Resources
- “Introduction to Cryptography with Coding Theory” by Wade Trappe and Lawrence C. Washington.
- Online courses and resources on number theory and algebra in cryptography.