beta version of Victor Shoups book, "A Computational Introduction to Number Theory and Algebra"
From: Mads Rasmussen (mads_at_SPAMFILTER.opencs.com.br)
Date: 08/20/03
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Date: Wed, 20 Aug 2003 09:06:10 -0300
I think this was announced at crypto 2003 but anyway...
Victor Shoup is writing a new book, he has made a beta version available for
download at his site: www.shoup.net
here's the contents as you can see, very intereting:
1 Basic Properties of the Integers 1
1.1 Divisibility and Primality
1.2 Ideals and Greatest Common Divisors
1.3 More on Unique Factorization and Greatest Common Divisors
2 Congruences 7
2.1 Definitions and Basic Properties
2.2 Solving Linear Congruences
2.3 Residue Classes
2.4 Euler's phi-Function
2.5 Other Arithmetic Functions
3 Computing with Large Integers
3.1 Asymptotic Notation
3.2 Machine Models and Complexity Theory
3.3 Basic Integer Arithmetic
3.4 Computing in Zn
3.5 Notes
4 Euclid's Algorithm
4.1 The Basic Euclidean Algorithm
4.2 The Extended Euclidean Algorithm
4.3 Computing Modular Inverses and Chinese Remaindering
4.4 Speeding up Algorithms via Modular Computation
4.5 Rational Reconstruction and Applications
4.6 Notes
5 The Distribution of Primes
5.1 Chebyshev's Theorem on the Density of Primes
5.2 Bertrand's Postulate
5.3 The Sum p<=x 1/p
5.4 The Sieve of Eratosthenes
5.5 The Prime Number Theorem . . . and Beyond
5.6 Notes
6 Discrete Probability Distributions
6.1 Finite Probability Distributions: Basic Definitions
6.2 Conditional Probability and Independence
6.3 Random Variables
6.4 Expectation and Variance
6.5 Some Useful Bounds
6.6 The Birthday Paradox
6.7 Statistical Distance
6.8 Measures of Randomness and the Leftover Hash Lemma
6.9 Discrete Probability Distributions
6.10 Notes
7 Probabilistic Algorithms
7.1 Basic Definitions
7.2 Approximation of Functions
7.3 Flipping a Coin until a Head Appears
7.4 Generating a Random Number from a Given Interval
7.5 Generating a Random Prime
7.6 Generating a Random Non-Increasing Sequence
7.7 Generating a Random Factored Number
7.8 Notes
8 Abelian Groups
8.1 Definitions, Basic Properties, and Some Examples
8.2 Subgroups
8.3 Cosets and Quotient Groups
8.4 Group Homomorphisms and Isomorphisms
8.5 Cyclic Groups
8.6 The Structure of Finite Abelian Groups
9 Rings
9.1 Definitions, Basic Properties, and Examples
9.2 Polynomial rings
9.3 Ideals and Quotient Rings
9.4 Ring Homomorphisms and Isomorphisms
10 Probabilistic Primality Testing
10.1 Trial Division
10.2 The Structure of Z^*_n
10.3 The Miller-Rabin Test
10.4 Generating Random Primes using the Miller-Rabin Test
10.5 Perfect Power Testing and Prime Power Factoring
10.6 Factoring and Computing Euler's phi-Function are Equivalent
10.7 The RSA Cryptosystem
10.8 Notes
11 Computing Generators and Discrete Logarithms in Z^*_p
11.1 Finding a Generator for Z^*_p
11.2 Computing Discrete Logarithms Z^*_p
11.3 The Diffie-Hellman Key Establishment Protocol
11.4 Notes
12 Quadratic Residues and Quadratic Reciprocity
12.1 Quadratic Residues
12.2 The Legendre Symbol
12.3 The Jacobi Symbol
12.4 Notes
13 Computational Problems Related to Quadratic Residues
13.1 Computing the Jacobi Symbol
13.2 Testing Quadratic Residuosity
13.3 Computing Modular Square Roots
14 Vector Spaces and Algebras
14.1 Definitions, Properties, and Some Examples
14.2 Subspaces and Quotient Spaces
14.3 Vector Space Homomorphisms and Isomorphisms
14.4 Linear Independence, Bases, and Dimension
14.5 Algebras
15 Matrices over Fields
15.1 Basic Definitions and Properties
15.2 Matrices and Linear Maps
15.3 The Inverse of a Matrix
15.4 Gaussian Elimination
15.5 Applications of Gaussian Elimination
15.6 Notes
16 Subexponential-time Algorithms for Discrete Logarithms and Factoring
16.1 Smooth Numbers
16.2 An Algorithm for Discrete Logarithms
16.3 An Algorithm for Factoring Integers
16.4 Practical Improvements
16.5 Notes
17 More Rings
17.1 The Field of Fractions of an Integral Domain
17.2 Unique Factorization of Polynomials
17.3 Polynomial Congruences
17.4 Polynomial Quotient Algebras
17.5 General Properties of Extension Fields
17.6 Formal Derivatives
17.7 Formal Power Series and Laurent Series
17.8 Unique Factorization Domains
17.9 Constructing the Real Numbers
18 Polynomial Arithmetic and Applications
18.1 Basic Arithmetic
18.2 Euclid's Algorithm
18.3 Computing Modular Inverses and Chinese Remaindering
18.4 Rational Function Reconstruction and Applications
18.5 Notes
19 Finite Fields
19.1 The Characteristic and Cardinality of a Finite Field
19.2 Some Useful Divisibility Criteria
19.3 The Existence of Finite Fields
19.4 The Subfield Structure and Uniqueness of Finite Fields
19.5 Conjugates, Norms and Traces
20 Algorithms for Finite Fields
20.1 Testing and Constructing Irreducible Polynomials
20.2 Factoring Polynomials over Finite Fields: the Cantor-Zassenhaus
Algorithm
20.3 Factoring Polynomials over Finite Fields: Berlekamp's Algorithm
20.4 Notes
21 Deterministic Primality Testing
21.1 The Basic Idea
21.2 The Algorithm and its Analysis
21.3 Notes
A Notation and Useful Facts
Bibliography
- Next message: Herra Tossavainen: "Re: Please test this encryption"
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- Next in thread: Mark Wooding: "Re: beta version of Victor Shoups book, "A Computational Introduction to Number Theory and Algebra""
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