| Foreword | p. ix |
| Efficient Distributed Computation Modulo a Shared Secret | p. 1 |
| Introduction | p. 1 |
| Previous Work | p. 2 |
| Organization of this Lecture | p. 3 |
| Preliminaries | p. 4 |
| The Network Model | p. 4 |
| Definitions and Notations | p. 4 |
| Building Blocks | p. 5 |
| Additive Sharing over Zq | p. 6 |
| Polynomial Sharing over Zq | p. 6 |
| Additive Sharing over Z | p. 7 |
| Polynomial Sharing over Z | p. 8 |
| Basic Protocols | p. 9 |
| Distributed Computation Modulo q | p. 9 |
| Joint Random Sharing over Zq | p. 10 |
| Joint Random Sharing of 0 in Zq | p. 10 |
| Computing Shares of the Inverse of a Shared Secret | p. 11 |
| Joint Random Invertible Element Sharing | p. 11 |
| A Different Approach | p. 11 |
| Converting among Different Secret Sharing Methods | p. 12 |
| Converting between Additive and Polynomial Shares | p. 12 |
| Converting between Integer Shares and Zq Shares | p. 13 |
| Computing Shares of the Binary Representation of a Secret | p. 16 |
| Approximate Truncation | p. 16 |
| Distributed Modular Reduction | p. 17 |
| Newton Iteration Method | p. 17 |
| First Step: Computing Shares of an Approximation of 1/p | p. 18 |
| Second Step: the Modular Reduction Protocol | p. 20 |
| Exponentiation with a Shared Exponent | p. 23 |
| Set Membership | p. 24 |
| Generating Shared Random Primes | p. 26 |
| The Basic Miller-Rabin Algorithm | p. 26 |
| Generation of a Shared Candidate Prime | p. 27 |
| Distributed Miller-Rabin Primality Test | p. 27 |
| Generation of Shared Random Safe Primes | p. 28 |
| Efficient Generation of Shared RSA Keys | p. 30 |
| Computing Inverses over a Shared Modulus | p. 30 |
| The Basic Idea | p. 30 |
| The Full Protocol | p. 31 |
| A Fundamental Lemma | p. 34 |
| References | p. 36 |
| Multiparty Computation, an Introduction | p. 41 |
| Introduction | p. 41 |
| What is Multiparty Computation? | p. 41 |
| The MPC and VSS Problems | p. 41 |
| Adversaries and their Powers | p. 42 |
| Models of Communication | p. 43 |
| Definition of Security | p. 44 |
| Results on MPC | p. 49 |
| Results for Threshold Adversaries | p. 49 |
| Results for General Adversaries | p. 50 |
| MPC Protocols | p. 51 |
| The Passive Case | p. 53 |
| The Active Case | p. 60 |
| Realization of FCom: Information Theoretic Scenario | p. 65 |
| Formal Proof for the FCom Realization | p. 75 |
| The Cryptographic Scenario | p. 76 |
| Using Encryption to Implement the Channels | p. 76 |
| Cryptographic Implementations of Higher-Level Functionalities | p. 77 |
| Protocols Secure for General Adversary Structures | p. 78 |
| Formal Details of the General Security Model for Protocols | p. 78 |
| The Real-Life Execution | p. 79 |
| The Ideal Process | p. 80 |
| The Hybrid Models | p. 82 |
| Composing Protocols | p. 83 |
| Composing Interfaces | p. 84 |
| References | p. 85 |
| Foundations of Modern Cryptography | p. 89 |
| Introduction | p. 89 |
| One-Way Functions | p. 89 |
| Definitions | p. 90 |
| Candidates from Number Theory | p. 92 |
| Weak vs. Strong One-Way Functions | p. 94 |
| Pseudo-Random Generators | p. 98 |
| Definitions | p. 99 |
| Constructions | p. 100 |
| A Cryptographic Application | p. 103 |
| Pseudo-Random Functions | p. 103 |
| Definitions | p. 104 |
| Constructions | p. 105 |
| Examples and Applications | p. 108 |
| Zero-Knowledge Protocols | p. 109 |
| Basic Definitions | p. 110 |
| Zero-Knowledge Proof Systems of Membership | p. 111 |
| Witness-Indistinguishable Proof Systems of Knowledge | p. 116 |
| Zero-Knowledge Proof Systems of Decision Power | p. 119 |
| Zero-Knowledge Transfers of Decision | p. 124 |
| References | p. 129 |
| Provable Security for Public Key Schemes | p. 133 |
| Introduction | p. 133 |
| Provable Security | p. 134 |
| Exact Security and Practical Security | p. 134 |
| Outline of the Notes | p. 135 |
| Related Work | p. 135 |
| Security Proofs and Security Arguments | p. 135 |
| Computational Assumptions | p. 135 |
| ""Reductionist"" Security Proofs | p. 136 |
| Practical Security | p. 136 |
| The Random-Oracle Model | p. 137 |
| The General Framework | p. 138 |
| A First Formalism | p. 138 |
| Digital Signature Schemes | p. 139 |
| Public-Key Encryption | p. 140 |
| The Computational Assumptions | p. 143 |
| Integer Factoring and the RSA Problem | p. 143 |
| The Discrete Logarithm and the Diffie-Hellman Problems | p. 145 |
| Digital Signature Schemes | p. 146 |
| Provable Security | p. 147 |
| DL-Based Signatures | p. 148 |
| RSA-Based Signatures | p. 154 |
| Public-Key Encryption | p. 163 |
| History | p. 163 |
| A First Generic Construction | p. 164 |
| OAEP: the Optimal Asymmetric Encryption Padding | p. 167 |
| REACT: a Rapid Enhanced-security Asymmetric Cryptosystem Transform | p. 179 |
| Conclusion | p. 184 |
| References | p. 185 |
| Efficient and Secure Public Key Cryptosystems | p. 191 |
| Efficient Integer Arithmetic | p. 191 |
| Modular Exponentiation | p. 191 |
| Window Methods | p. 192 |
| Montgomery Multiplication | p. 194 |
| Fast Variants of RSA Cryptosystem | p. 196 |
| PKCS #1 Version 2.1 | p. 196 |
| Multi-Exponent RSA | p. 197 |
| Size of Secret Primes | p. 199 |
| Comparison | p. 200 |
| Implementation Attack on RSA-CRT | p. 201 |
| EPOC Cryptosystem | p. 204 |
| EPOC 2 Cryptosystem | p. 204 |
| Reject Timing Attack on EPOC-2 | p. 207 |
| Relation to Other Cryptosystems | p. 212 |
| Other Encryption Primitives | p. 213 |
| Elliptic Curve Cryptosystem | p. 214 |
| Scalar Multiplication | p. 215 |
| Efficient Coordinate System | p. 217 |
| Side Channel Attacks on ECC | p. 218 |
| SPA on ECC | p. 218 |
| DPA and Countermeasures | p. 219 |
| Goubin's Power-Analysis Attack | p. 220 |
| Zero-Value Point Attack on ECC | p. 220 |
| Non-Zero Digit Methods | p. 226 |
| Montgomery Ladder Method | p. 226 |
| Non-Zero Window Method | p. 231 |
| References | p. 231 |
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