Undergraduate Texts in Mathematics - Bressoud D.M. / Брессу Д.М. - Factorization and Primality Testing / Факторизация и тесты на простоту [1989, PDF/DjVu, ENG]

Preface
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 φ Function
4.6 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 Bhiscara-Brouncker Algorithm
10.3 The Bhascara-Brouncker Algorithm Explained
10.4 Solutions Really Exist
10.5 EXERCISES
11 Continued Fractions Continued, Applications
11.1 CFRAC
11.2 Some Observations on the Bhiscara-Brouncker Algorithm
11.3 Proofs of the Observations
11.4 Primality Testing with Continued Fractions
11.5 The Lucas-Lehmer Algorithm Explained
11.6 EXERCISES
12 Lucas Sequences
12.1 Basic Definitions
12.2 Divisibility Properties
12.3 Lucas’ Primality Test
12.4 Computing the V’s
12.5 EXERCISES
13 Groups and Elliptic Curves
13.1 Groups
13.2 A General Approach to Primality Tests
13.3 A General Approach to Factorization
13.4 Elliptic Curves
13.5 Elliptic Curves Modulo p
13.6 EXERCISES
14 Applications of Elliptic Curves
14.1 Computation on Elliptic Curves
14.2 Factorization with Elliptic Curves
14.3 Primality Testing
14.4 Quadratic Forms
14.5 The Power Residue Symbol
14.6 EXERCISES
The Primes Below 5000
Index

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