Mathematics for Cryptography
Introduction
Mathematics forms the backbone of modern cryptography. This guide covers the essential mathematical concepts you need to understand cryptographic systems and algorithms.
Number Theory Fundamentals
Prime Numbers
Prime numbers are the building blocks of modern cryptography. A prime number is a natural number greater than 1 that is only divisible by 1 and itself.
Key concepts:
- Prime factorization
- The Fundamental Theorem of Arithmetic
- Testing for primality
- Large prime numbers in cryptography
Modular Arithmetic
Modular arithmetic is essential for many cryptographic operations, especially in public-key cryptography.
Basic Operations
- Addition modulo n: (a + b) mod n
- Multiplication modulo n: (a × b) mod n
- Exponentiation modulo n: (a^b) mod n
Properties
- Commutative property: (a + b) mod n = (b + a) mod n
- Associative property: ((a + b) + c) mod n = (a + (b + c)) mod n
- Distributive property: (a × (b + c)) mod n = ((a × b) + (a × c)) mod n
Greatest Common Divisor (GCD)
The GCD is crucial for key generation and determining multiplicative inverses.
- Euclidean algorithm
- Extended Euclidean algorithm
- Bézout's identity
Algebraic Structures
Groups
Groups are fundamental to public-key cryptography.
Properties:
- Closure
- Associativity
- Identity element
- Inverse elements
Important examples:
- Multiplicative group of integers modulo n
- Elliptic curve groups
Fields
Fields provide the mathematical foundation for many cryptographic operations.
Types:
- Finite fields (Galois fields)
- Binary fields GF(2^n)
- Prime fields GF(p)
Number Theory for Public-Key Cryptography
Euler's Totient Function
φ(n) counts the numbers less than n that are coprime to n.
Properties:
- For prime p: φ(p) = p - 1
- For prime p, q: φ(pq) = (p-1)(q-1)
Fermat's Little Theorem
For prime p and integer a not divisible by p: a^(p-1) ≡ 1 (mod p)
Chinese Remainder Theorem (CRT)
Used for efficient implementations of RSA and other cryptographic algorithms.
Probability and Information Theory
Entropy
Measuring randomness and information content:
- Shannon entropy
- Min-entropy
- Perfect secrecy
Random Number Generation
- True random numbers vs. pseudo-random numbers
- Statistical tests for randomness
- Cryptographically secure pseudo-random number generators (CSPRNG)
Advanced Topics
Elliptic Curve Mathematics
- Point addition and multiplication
- Discrete logarithm problem
- Bilinear pairings
Lattice-Based Cryptography
- Lattice definitions and properties
- Hard lattice problems
- Learning With Errors (LWE)
Computational Complexity
Big O Notation
Understanding algorithmic efficiency:
- Time complexity
- Space complexity
- Common complexity classes
Cryptographic Hardness
Mathematical problems underlying cryptographic security:
- Integer factorization
- Discrete logarithm
- Elliptic curve discrete logarithm
Practice Problems and Examples
- Calculate GCD using Euclidean algorithm
- Find multiplicative inverses modulo n
- Implement modular exponentiation
- Verify Euler's totient function properties
- Solve simple CRT problems
Resources and Further Reading
- Number Theory and Cryptography textbooks
- Online practice platforms
- Research papers and publications
- Interactive tools and calculators
Glossary
- Modulo: The remainder operation
- Coprime: Numbers whose greatest common divisor is 1
- Group: A set with an operation that satisfies closure, associativity, identity, and inverse properties
- Field: A set with two operations (addition and multiplication) satisfying specific axioms
- Lattice: A discrete subgroup of R^n