简介
this book is intended as a text for a course on cryptography with emphasis on algebraic methods. it is written so as to be accessible to graduate or advanced undergraduate students, as well as to scientists in other fields. the first three chapters form a self-contained introduction to basic concepts and techniques. here my approach is intuitive and informal. for example, the treatment of computational complexity in chapter 2, while lacking formalistic rigor, emphasizes the aspects of the subject that are most important in cryptography.
chapters 4-6 and the appendix contain material that for the most part has not previously appeared in textbook form. a novel feature is the inclusion of three types of cryptography - "hidden monomial" systems, combinatorial-algebraic sys-tems, and hyperelliptic systems - that are at an early stage of development. it is too soon to know which, if any, of these cryptosystems will ultimately be of practical use. but in the rapidly growing field of cryptography it is worthwhile to continually explore new one-way constructions coming from different areas of mathematics. perhaps some of the readers will contribute to the research that still needs to be done.
目录
Chapter 1. Cryptography
§1. Early History
§2. The Idea of Public Key Cryptography
§3. The RSA Cryptosystem
§4. Diffie-Hellman and the Digital Signature Algorithm
§5. Secret Sharing, Coin Flipping, and Time Spent on Homework
§6. Passwords, Signatures, and Ciphers
§7. Practical Cryptosystems and Useful Impractical Ones
Exercises
Chapter 2. Complexity of Computations
§ 1. The Big-O Notation
Exercises
§2. Length of Numbers
Exercises
§3. Time Estimates
Exercises
§4. P, NP, and NP-Completeness
Exercises
§5. Promise Problems
§6. Randomized Algorithms and Complexity Classes
Exercises
§7. Some Other Complexity Classes
Exercises
Chapter 3. Algebra
§1. Fields
Exercises
§2. Finite Fields
Exercises
§3. The Euclidean Algorithm for Polynomials
Exercises
§4. Polynomial Rings
Exercises
§5. Grobner Bases
Exercises
Chapter 4. Hidden Monomial Cryptosystems
§1. The Imai-Matsumoto System
Exercises
§2. Patarin's Little Dragon
Exercises
§3. Systems That Might Be More Secure
Exercises
Chapter 5. Combinatorial-Algebraic Cryptosystems
§1. History
§2. Irrelevance of Brassard's Theorem
Exercises
§3. Concrete Combinatorial-Algebraic Systems
Exercises
§4. The Basic Computational Algebra Problem
Exercises
§5. Cryptographic Version of Ideal Membership
§6. Linear Algebra Attacks
§7. Designing a Secure System
Chapter 6. Elliptic and Hyperelliptic Cryptosystems
§1. Elliptic Curves
Exercises
§2. Elliptic Curve Cryptosystems
Exercises
§3. Elliptic Curve Analogues of Classical Number Theory Problems
Exercises
§4. Cultural Background: Conjectures on Elliptic Curves
and Surprising Relations with Other Problems
§5. Hyperelliptic Curves
Exercises
§6. Hyperelliptic Cryptosystems
Exercises
Appendix. An Elementary Introduction to Hyperelliptic Curves
by Alfred J. Menezes, Yi-Hong Wu, and Robert J. Zuccherato
§1. Basic Definitions and Properties
§2. Polynomial and Rational Functions
§3. Zeros and Poles
§4. Divisors
§5. Representing Semi-Reduced Divisors
§6. Reduced Divisors
§7. Adding Reduced Divisors
Exercises
Answers to Exercises
Bibliography
Subject Index
§1. Early History
§2. The Idea of Public Key Cryptography
§3. The RSA Cryptosystem
§4. Diffie-Hellman and the Digital Signature Algorithm
§5. Secret Sharing, Coin Flipping, and Time Spent on Homework
§6. Passwords, Signatures, and Ciphers
§7. Practical Cryptosystems and Useful Impractical Ones
Exercises
Chapter 2. Complexity of Computations
§ 1. The Big-O Notation
Exercises
§2. Length of Numbers
Exercises
§3. Time Estimates
Exercises
§4. P, NP, and NP-Completeness
Exercises
§5. Promise Problems
§6. Randomized Algorithms and Complexity Classes
Exercises
§7. Some Other Complexity Classes
Exercises
Chapter 3. Algebra
§1. Fields
Exercises
§2. Finite Fields
Exercises
§3. The Euclidean Algorithm for Polynomials
Exercises
§4. Polynomial Rings
Exercises
§5. Grobner Bases
Exercises
Chapter 4. Hidden Monomial Cryptosystems
§1. The Imai-Matsumoto System
Exercises
§2. Patarin's Little Dragon
Exercises
§3. Systems That Might Be More Secure
Exercises
Chapter 5. Combinatorial-Algebraic Cryptosystems
§1. History
§2. Irrelevance of Brassard's Theorem
Exercises
§3. Concrete Combinatorial-Algebraic Systems
Exercises
§4. The Basic Computational Algebra Problem
Exercises
§5. Cryptographic Version of Ideal Membership
§6. Linear Algebra Attacks
§7. Designing a Secure System
Chapter 6. Elliptic and Hyperelliptic Cryptosystems
§1. Elliptic Curves
Exercises
§2. Elliptic Curve Cryptosystems
Exercises
§3. Elliptic Curve Analogues of Classical Number Theory Problems
Exercises
§4. Cultural Background: Conjectures on Elliptic Curves
and Surprising Relations with Other Problems
§5. Hyperelliptic Curves
Exercises
§6. Hyperelliptic Cryptosystems
Exercises
Appendix. An Elementary Introduction to Hyperelliptic Curves
by Alfred J. Menezes, Yi-Hong Wu, and Robert J. Zuccherato
§1. Basic Definitions and Properties
§2. Polynomial and Rational Functions
§3. Zeros and Poles
§4. Divisors
§5. Representing Semi-Reduced Divisors
§6. Reduced Divisors
§7. Adding Reduced Divisors
Exercises
Answers to Exercises
Bibliography
Subject Index
Algebraic aspects of cryptography
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