Modern Cryptography, Probabilistic Proofs and Pseudorandomness

ISBN-13: 9783642084324 / Angielski / Miękka / 2010 / 183 str.

Oded Goldreich

Opis

You can start by putting the DO NOT DISTURB sign. Cay, in Desert Hearts (1985). The interplay between randomness and computation is one of the most fas- cinating scientific phenomena uncovered in the last couple of decades. This interplay is at the heart of modern cryptography and plays a fundamental role in complexity theory at large. Specifically, the interplay of randomness and computation is pivotal to several intriguing notions of probabilistic proof systems and is the focal of the computational approach to randomness. This book provides an introduction to these three, somewhat interwoven domains (i.e., cryptography, proofs and randomness). Modern Cryptography. Whereas classical cryptography was confined to the art of designing and breaking encryption schemes (or "secrecy codes"), Modern Cryptography is concerned with the rigorous analysis of any system which should withstand malicious attempts to abuse it. We emphasize two aspects of the transition from classical to modern cryptography: ( 1) the wide- ning of scope from one specific task to an utmost wide general class of tasks; and (2) the move from an engineering-art which strives on ad-hoc tricks to a scientific discipline based on rigorous approaches and techniques.

Recenzje

From the reviews:

"This book is intended for a reader with some knowledge of the theory of computing. It is divided into three chapters, each a survey of one of the topics mentioned in the title. The chapters may be read independently, and are somewhat different in nature. ...The book ends with four appendices. The first summarises the probability theory, complexity theory and cryptography that is used in the body of the book. The second gives examples of randomisation in algorithmic problems, complexity theory and distributed computing. The third contains two proofs, not to be found in the literature, of well-known results. The last appendix lists other surveys by the author." (Simon R. Blackburn, Mathematical Reviews)

"Modern cryptography, probabilistic proofs and pseudorandomness are three areas in theoretical computer science that demonstrate the interplay between randomness and computations. ... This book is informative and rich in content. ... the most appealing feature of this book is that it leans toward the intuition and historical motivations around these topics. ... it is an excellent resource for students and researchers ... . this book will probably give you a good collection of background motivations and nice discussions as well." (Andrew C. Lee, SIGACT News, Vol. 34 (4), 2003)

Spis treści

Preface Chapter 1: The Foundations of Modern Cryptography 1.1 Introduction Part I: Basic Tools 1.2 Central Paradigms 1.2.1 Computational Difficulty 1.2.2 Computational Indistinguishability 1.2.3 The Simulation Paradigm 1.3 Pseudorandomness 1.3.1 The Basics 1.3.2 Pseudorandom Functions 1.4 Zero-Knowledge 1.4.1 The Basics 1.4.2 Some Variants Part II: Basic Utilities 1.5 Encryption 1.5.1 Definitions 1.5.2 Constructions 1.5.3 Beyond eavesdropping security 1.6 Signatures 1.6.1 Definitions 1.6.2 Constructions 1.6.3 Two variants 1.7 Cryptographic Protocols 1.7.1 Definitions 1.7.2 Constructions Part III: Concluding Comments 1.8 Some Notes 1.8.1 General notes 1.8.2 Specific notes 1.9 Historical Perspective 1.10 Two Suggestions for Future Research 1.11 Some Suggestions for Further Reading Chapter 2: Probabilistic Proof Systems 2.1 Introduction 2.2 Interactive Proof Systems 2.2.1 Definition 2.2.2 The Role of Randomness 2.2.3 The Power of Interactive Proofs 2.2.4 The Interactive Proof System Hierarchy 2.2.5 How Powerful Should the Prover be? 2.3 Zero-Knowledge Proof Systems 2.3.1 A Sample Definition 2.3.2 The Power of Zero-Knowledge 2.3.3 The Role of Randomness 2.4 Probabilistically Checkable Proof Systems 2.4.1 Definition 2.4.2 The Power of Probabilistically Checkable Proofs 2.4.3 PCP and Approximation 2.4.4 More on PCP itself 2.4.5 The Role of Randomness 2.5 Other Probabilistic Proof Systems 2.5.1 Restricting the Provers Strategy 2.5.2 Non-Interactive Probabilistic Proofs 2.5.3 Proofs of Knowledge 2.5.4 Refereed Games 2.6 Concluding Remarks 2.6.1 Comparison among the various systems 2.6.2 The Story 2.6.3 Open Problems Chapter 3: Pseudorandom Generators 3.1 Introduction 3.2 The General Paradigm 3.3 The Archetypical Case 3.3.1 A Short Discussion 3.3.2 Some Basic Observations 3.3.3 Constructions 3.3.4 Pseudorandom Functions 3.4 Derandomization of time-complexity classes 3.5 Space Pseudorandom Generators 3.6 Special Purpose Generators 3.6.1 Pairwise-Independence Generators 3.6.2Small-Bias Generators 3.6.3 Random Walks on Expanders 3.6.4 Samplers 3.6.5 Dispersers, Extractors and Weak Random Sources 3.7 Concluding Remarks 3.7.1 Discussion 3.7.2 Historical Perspective 3.7.3 Open Problems Appendix A: Background on Randomness and Computation A.1 Probability Theory -- Three Inequalities A.2 Computational Models and Complexity classes A.2.1 P, NP, and more A.2.2 Probabilistic Polynomial-Time A.2.3 Non-Uniform Polynomial-Time A.2.4 Oracle Machines A.2.5 Space Bounded Machines A.2.6 Average-Case Complexity A.3 Complexity classes -- Glossary A.4 Some Basic Cryptographic Settings A.4.1 Encryption Schemes A.4.2 Digital Signatures and Message Authentication A.4.3 The RSA and Rabin Functions Appendix B: Randomized Computations B.1 Randomized Algorithms B.1.1 Approx. Counting of DNF satisfying assignments B.1.2 Finding a perfect matching B.1.3 Testing whether polynomials are identical B.1.4 Randomized Rounding applied to MaxSAT B.1.5 Primality Testing B.1.6 Testing Graph Connectivity via a random walk B.1.7 Finding minimum cuts in graphs B.2 Randomness in Complexity Theory B.2.1 Reducing (Approximate) Counting to Deciding B.2.2 Two-sided error versus one-sided error B.2.3 The permanent: Worst-Case vs Average Case B.3 Randomness in Distributed Computing B.3.1 Testing String Equality B.3.2 Routing in networks B.3.3 Byzantine Agreement B.4 Bibliographic Notes Appendix C: Notes on two proofs C.1 Parallel repetition of interactive proofs C.2 A generic Hard-Core Predicate C.2.1 A motivating discussion C.2.2 Back to the formal argument C.2.3 Improved Implementation of Algorithm $A Appendix D: Related Surveys by the Author Bibliography (over 300 entries) '