Multi-input functional encryption

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Table of contents

1 Introduction 
1.1 Tight Security
1.1.1 State of the Art in Tight Security
1.1.2 Contribution 1: Tightly CCA-Secure Encryption without Pairing
1.2 Functional Encryption
1.2.1 State of the Art in Functional Encryption
1.2.2 Contribution 2: Functional Encryption with New Features, and Richer Functionalities
1.2.3 Other contributions
2 Preliminaries 
2.1 Notations and Basics
2.1.1 Collision resistant hashing
2.1.2 Symmetric-Key Encryption
2.1.3 Authenticated Encryption
2.1.4 Public-Key Encryption
2.1.5 Key-Encapsulation Mechanism
2.2 Cryptographic Assumptions
2.2.1 Prime-Order Groups
2.2.2 Pairing Groups
2.2.3 Matrix Diffie-Hellman
2.2.4 Decisional Composite Residuosity
2.2.5 Learning With Errors
2.3 Definitions for Single-Input Functional Encryption
2.3.1 Security notions
2.4 Definitions for Multi-Input Functional Encryption
2.4.1 Security notions
2.4.2 Removing the extra condition generically
2.5 Definitions for Multi-Client Functional Encryption
2.6 Concrete Instances of Functional Encryption for Inner Products
2.6.1 Inner-Product FE from MDDH
2.6.2 Inner-Product FE from LWE
2.6.3 Inner-Product FE from DCR
3 Tightly CCA-Secure Encryption without Pairings 
3.1 Multi-ciphertext PCA-secure KEM
3.1.1 Our construction
3.1.2 Security proof
3.2 Multi-ciphertext CCA-secure Public Key Encryption scheme
3.2.1 Our construction
3.3 Security proof of PKE
4 Multi-Input Inner-Product Functional Encryption from Pairings 
4.1 Selectively-Secure, Private-Key MIFE for Inner Products
4.1.1 Selectively-secure, multi-input scheme from single-input scheme
4.1.2 Putting everything together
4.2 Achieving Adaptive Security
5 Multi-Input Inner-Product Functional Encryption without Pairings 
5.1 From Single to Multi-Input FE for Inner Product
5.1.1 Information-Theoretic MIFE with One-Time Security
5.1.2 Our Transformation for Inner Product over ZL
5.1.3 Our Transformation for Inner Product over Z
5.2 Concrete instances of FE for Inner Product
5.2.1 Inner Product FE from MDDH
5.2.2 Inner Product FE from LWE
5.2.3 Inner Product FE from DCR
6 Multi-Client Inner Product Functional Encryption 
6.1 MCFE with one-AD-IND-weak security
6.2 From one to many ciphertext for MCFE
6.3 Secret Sharing Encapsulation
6.3.1 Definitions
6.3.2 Construction of the Secret Sharing Encapsulation
6.4 Strengthening the Security of MCFE Using SSE
6.4.1 Generic construction of xx-AD-IND security for MCFE
6.5 Decentralizing MCFE
6.5.1 Distributed Sum
6.5.2 Our DSum Protocol
6.5.3 Security Analysis
6.5.4 Application to DMCFE for Inner Products
7 Functional Encryption for Quadratic Functions 
7.1 Private-key FE with one-SEL-IND security
7.2 Public-key FE
8 Conclusion 
8.1 Summary of the Contributions
8.2 Open Problems