Factorization and Primality Testing

1 Unique Factorization and the Euclidean Algorithm.- 1.1 A theorem of Euclid and some of its consequences.- 1.2 The Fundamental Theorem of Arithmetic.- 1.3 The Euclidean Algorithm.- 1.4 The Euclidean Algorithm in practice.- 1.5 Continued fractions, a first glance.- 1.6 Exercises.- 2 Primes and Perfect Numbers.- 2.1 The Number of Primes.- 2.2 The Sieve of Eratosthenes.- 2.3 Trial Division.- 2.4 Perfect Numbers.- 2.5 Mersenne Primes.- 2.6 Exercises.- 3 Fermat, Euler, and Pseudoprimes.- 3.1 Fermat's Observation.- 3.2 Pseudoprimes.- 3.3 Fast Exponentiation.- 3.4 A Theorem of Euler.- 3.5 Proof of Fermat's Observation.- 3.6 Implications for Perfect Numbers.- 3.7 Exercises.- 4 The RSA Public Key Crypto-System.- 4.1 The Basic Idea.- 4.2 An Example.- 4.3 The Chinese Remainder Theorem.- 4.4 What if the Moduli are not Relatively Prime?.- 4.5 Properties of Euler's ø Function.- Exercises.- 5 Factorization Techniques from Fermat to Today.- 5.1 Fermat's Algorithm.- 5.2 Kraitchik's Improvement.-5.3 Pollard Rho.- 5.4 Pollard p - 1.- 5.5 Some Musings.- 5.6 Exercises.- 6 Strong Pseudoprimes and Quadratic Residues.- 6.1 The Strong Pseudoprime Test.- 6.2 Refining Fermat's Observation.- 6.3 No "Strong" Carmichael Numbers.- 6.4 Exercises.- 7 Quadratic Reciprocity.- 7.1 The Legendre Symbol.- 7.2 The Legendre symbol for small bases.- 7.3 Quadratic Reciprocity.- 7.4 The Jacobi Symbol.- 7.5 Computing the Legendre Symbol.- 7.6 Exercises.- 8 The Quadratic Sieve.- 8.1 Dixon's Algorithm.- 8.2 Pomerance's Improvement.- 8.3 Solving Quadratic Congruences.- 8.4 Sieving.- 8.5 Gaussian Elimination.- 8.6 Large Primes and Multiple Polynomials.- 8.7 Exercises.- 9 Primitive Roots and a Test for Primality.- 9.1 Orders and Primitive Roots.- 9.2 Properties of Primitive Roots.- 9.3 Primitive Roots for Prime Moduli.- 9.4 A Test for Primality.- 9.5 More on Primality Testing.- 9.6 The Rest of Gauss' Theorem.- 9.7 Exercises.- 10 Continued Fractions.- 10.1 Approximating the Square Root of 2.- 10.2 The