Here are 100 chapter titles for a book on Integer Factorization in competitive programming, organized from beginner to advanced topics:
- Introduction to Integer Factorization
- What is Integer Factorization? A Basic Overview
- Prime Numbers and Their Role in Factorization
- Basic Definitions: Factors, Divisors, and Multiples
- The Fundamental Theorem of Arithmetic
- Prime Factorization: An Introduction
- Finding the Prime Factors of Small Integers
- Factorization of Composite Numbers
- Efficient Methods for Trial Division
- Basic Concepts in Divisibility Rules
- Identifying Primes Using Simple Divisibility Tests
- The Sieve of Eratosthenes: Finding Primes
- Prime Factorization Using Sieve of Eratosthenes
- Divisors and Multiples: Basic Problems
- Understanding Greatest Common Divisor (GCD) and Least Common Multiple (LCM)
- Basic Euclidean Algorithm for GCD
- Extended Euclidean Algorithm: Finding Inverses
- Applications of GCD in Factorization Problems
- Factorization of Powers of Primes
- Trial Division for Integer Factorization
- Efficient Trial Division for Larger Numbers
- Sieve of Eratosthenes: Optimizations for Larger Ranges
- Introduction to Pollard's Rho Algorithm
- Pollard's Rho: Factorizing Large Numbers Efficiently
- Fermat’s Factorization Method
- Fermat's Method: When It Works Best
- Factorization Using the Quadratic Sieve
- The Role of Factorization in Cryptography
- Applications of Prime Factorization in RSA Encryption
- Trial Division with Optimized Divisor Search
- Powers of 2 and Their Factorization
- Factorization in Modular Arithmetic
- Factorizing Perfect Squares and Higher Powers
- Optimizing Factorization for Even Numbers
- Finding the Divisors of a Large Number
- Optimized Approach to Prime Factorization
- Introduction to the Elliptic Curve Factorization Method
- Elliptic Curve Factorization for Large Numbers
- Prime Factorization Using Pollard's p-1 Method
- Handling Large Primes in Factorization Problems
- Integer Factorization and Its Complexity
- Advanced Applications of Integer Factorization in Cryptography
- General Number Field Sieve (GNFS) Overview
- The General Number Field Sieve for Integer Factorization
- Quadratic Sieve: Theory and Practical Implementation
- Implementation of the Quadratic Sieve in Competitive Programming
- The Role of Factorization in RSA Key Generation
- Advanced Factorization Algorithms: Key Differences
- Factorization and Public-Key Cryptosystems
- Optimizing Pollard’s Rho Algorithm for Special Cases
- Advanced Methods for Pollard’s Rho Algorithm
- Miller-Rabin Primality Test and Factorization
- AKS Primality Test and Its Use in Factorization
- Factorization Using the Lenstra Elliptic Curve Method
- Elliptic Curve Method for Large-Scale Factorization
- Algebraic Number Fields and Factorization
- Quantum Computing and Its Impact on Factorization
- Shor's Algorithm for Integer Factorization
- Quantum Factorization Algorithms in Competitive Programming
- Comparing Classical and Quantum Factorization Techniques
- Factorization in Cryptanalysis: Breaking RSA
- Factorization and the Security of Digital Signatures
- Computational Complexity of Integer Factorization
- The Complexity of GNFS and Its Practical Applications
- Integer Factorization and Its Role in P vs NP Problem
- Optimizing Integer Factorization for Multi-Core Systems
- Parallelizing Integer Factorization Algorithms
- Advanced Mathematical Techniques in Integer Factorization
- Modular Exponentiation in Integer Factorization
- Using the Chinese Remainder Theorem in Factorization
- Integer Factorization for Large Semi-Primes
- Probabilistic Methods in Factorization
- Using Heuristics to Improve Factorization Speed
- Large-Scale Factorization Problems in Cryptography
- Integer Factorization for RSA Decryption
- Efficient Algorithms for Factoring Large RSA Moduli
- Public-Key Cryptosystem Security Based on Integer Factorization
- Using Elliptic Curves for Factoring Large Numbers
- Large Integer Factorization Using Hybrid Methods
- Optimized Implementation of the Quadratic Sieve
- Speeding Up Integer Factorization with Fast Multiplication
- Algorithmic Challenges in Integer Factorization
- Approaching Factorization Problems in Real-World Cryptography
- Factoring Large Numbers with Fermat’s Factorization
- Factorization and the Role of Advanced Algorithms in Modern Cryptography
- The Future of Integer Factorization in Secure Communications
- Solving Integer Factorization with the Number Field Sieve
- Integer Factorization Algorithms for High-Performance Computing
- Factoring Large Numbers with GPU-Accelerated Methods
- Using Caching to Speed Up Factorization Algorithms
- Improved Algorithms for Integer Factorization on Large Numbers
- Factoring Large Prime Products in RSA Decryption
- Comparing Different Integer Factorization Algorithms
- Optimizing Computational Complexity in Integer Factorization
- Efficient Memory Management in Factorization Algorithms
- Integer Factorization in Online Competitions
- Integer Factorization in Computational Mathematics
- Modular Arithmetic and Integer Factorization: Synergies
- Integer Factorization in the Context of Cryptographic Security
- Future Trends in Integer Factorization Algorithms
These chapters cover the essential techniques and advanced methods in integer factorization, ranging from basic prime factorization to the cutting-edge algorithms used in cryptography. Each chapter aims to build progressively on earlier concepts, making this book suitable for learners at all levels interested in understanding integer factorization in the context of competitive programming.