Integer Factorization Algorithms: From Fermat to Quadratic Sieve

Integer factorization forms the mathematical backbone of modern computational number theory and public-key cryptography. Understanding how large composite numbers are decomposed into prime factors provides crucial insight into algorithmic complexity, modular arithmetic, and cryptographic security. This structured, text-based course provides a clear pathway from classical factorization concepts to modern sieve-based techniques. You will build intuitive knowledge of number-theoretic principles, learn how smooth numbers enable efficient relation generation, and analyze how linear algebra over finite fields allows you to extract factors from large integers. What you'll learn: - Understand foundational number theory terminology, divisibility rules, and prime properties - Apply Fermat's factorization method and recognize its computational strengths and limits - Explore Kraitchik's method and the utility of quadratic residues - Identify smooth numbers and adapt sieving strategies to collect factor relations - Construct the Quadratic Sieve workflow for factoring large composite integers - Solve binary linear systems to combine relations into non-trivial factor pairs - Analyze algorithm complexity, asymptotic bounds, and their significance in contemporary cryptography The course begins with foundational concepts, key terms, and algebraic principles before progressing step-by-step through practical algorithmic designs and complexity evaluations. This course is created for beginners in discrete mathematics, computer science students, and programmers looking to explore computational number theory without needing prior advanced cryptography coursework. Begin reading today to build a strong foundation in integer factorization and modern sieving algorithms.