Modern Cryptography, Probabilistic Proofs and Pseudorandomness

Cryptography is one of the most active areas in current mathematics research and applications. This book focuses on cryptography along with two related areas: the study of probabilistic proof systems, and the theory of computational pseudorandomness. Following a common theme that explores the interplay between randomness and computation, the important notions in each field are covered, as well as novel ideas and insights.

Contenu

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) '