Homomorphic Encryptions

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Table of contents

1 Introduction 
1.1 Classical Cryptography
1.1.1 Confidentiality
1.1.2 Authentication
1.1.3 Provable Security
1.2 Advanced Security Goals
1.2.1 Decentralization
1.2.2 Anonymity
1.2.3 Traceability
1.3 Homomorphism
1.3.1 Homomorphic Encryptions
1.3.2 Homomorphic Signatures
1.3.3 Security with Homomorphism
1.3.4 Homomorphic Zero-Knowledge Proofs
1.4 Contributions
1.5 Organization of the Manuscript
2 Preliminaries 
2.1 Notations and Usefull Notions
2.2 Provable Security
2.3 Computational Assumptions
2.4 Cryptographic Primitives
2.4.1 (Homomorphic) Encryption
2.4.2 (Homomorphic) Signature
2.4.3 (Homomorphic) Proof
3 Decentralized Evaluation of Quadratic Polynomials on Encrypted Data 
3.1 Freeman’s Approach
3.1.1 Notations
3.1.2 Freeman’s Scheme with Projections
3.1.3 Homomorphic Properties
3.1.4 Security Properties
3.1.5 Re-Encryption
3.1.6 Verifiability
3.1.7 Distributed Decryption
3.2 Optimized Version
3.2.1 Instantiation
3.2.2 Security Properties
3.2.3 Decentralized Homomorphic Encryption
3.2.4 Efficiency
3.3 Applications
3.3.1 Encryption for Boolean Formulae
3.3.2 Group Testing on Encrypted Data
3.3.3 Consistency Model on Encrypted Data
4 Linearly-Homomorphic Signatures 
4.1 Definition, Properties and Security
4.2 Our One-Time LH-Sign Scheme
4.3 FSH LH-Sign Scheme
4.4 Square Diffie-Hellman
4.5 SqDH LH-Sign Scheme
4.5.1 A First Generic Conversion
4.5.2 A Second Generic Conversion
5 MixNet 
5.1 Our Scheme: General Description
5.2 Our Scheme: Full Description
5.3 Scalability
5.3.1 Constant-Size Proof
5.3.2 Efficiency
5.4 Security Analysis
5.4.1 Proof of Soundness
5.4.2 Proof of Privacy: Unlinkability
5.4.3 Proof of Correctness
5.5 Applications
5.5.1 Electronic Voting
5.5.2 Message Routing
6 Anonymous Credentials 
6.1 Overview of our New Primitives
6.1.1 Tag-based Signatures
6.1.2 Signatures with Randomizable Tags
6.1.3 Aggregate Signatures
6.2 Aggregate Signatures with Randomizable Tags
6.2.1 Anonymous Ephemeral Identities
6.2.2 Aggregate Signatures with Randomizable Tags
6.2.3 One-Time ART-Sign Scheme with Square Diffie-Hellman Tags (SqDH)
6.2.4 Bounded ART-Sign Scheme with Square Diffie-Hellman Tags (SqDH)
6.3 Multi-Authority Anonymous Crendentials
6.3.1 Definition
6.3.2 Security Model
6.3.3 Anonymous Credential from EphemerId and ART-Sign Scheme
6.4 SqDH-based Anonymous Credentials
6.4.1 The Basic SqDH-based Anonymous Credential Scheme
6.4.2 A Compact SqDH-based Anonymous Credential Scheme
6.5 Traceable Anonymous Credentials
6.5.1 Traceable Anonymous Credentials
6.5.2 Traceable SqDH-based Anonymous Credentials
6.5.3 Groth-Sahai Proof for Square Diffie-Hellman Tracing
6.6 Related Work
7 Conclusion 
A Joint Generation of Square Diffie-Hellman Tuples
B Another Bounded SqDH-Based ART-Sign