Code Based Secret Sharing Schemes: Applied Combinatorial Coding Theory

by Alexis Bonnecaze, Patrick Solé, Romar Dela Cruz, Selda Calkavur

Length: 198 pages
Edition: 1
Language: English
Publisher: World Scientific Pub Co Inc
Publication Date: 2022-05-17
ISBN-10: 981124832X
ISBN-13: 9789811248320
Sales Rank: #0 (See Top 100 Books)

Secret sharing schemes form one of the most important topic in Cryptography. These protocols are used in many areas, applied mathematics, computer science, electrical engineering. A secret is divided into several pieces called shares. Each share is given to a user of the system. Each user has no information about the secret, but the secret can be retrieved by certain authorized coalition of users. This book is devoted to such schemes inspired by Coding Theory. The classical schemes of Shamir, Blakley, Massey are recalled. Survey is made of research in Combinatorial Coding Theory they trigerred, mostly self-dual codes, and minimal codes. Applications to engineering like image processing, and key management of MANETs are highlighted.

Cover Page
Title Page
Copyright Page
Foreword
Preface
Contents
1. Foundations
    1.1 Access Structures
    1.2 Secret-Sharing Schemes and Examples
    1.3 Alternative Definitions
    1.4 Security Models
    1.5 Shamir Scheme and Applications
    1.6 Basics of Coding Theory
    1.7 Code-Based Constructions of Secret-Sharing Schemes
        1.7.1 Construction 1
        1.7.2 Construction 2
    1.8 Multisecret-Sharing Schemes
    References
2. Massey Scheme
    2.1 On the Number of Minimal Codewords
        2.1.1 Introduction
        2.1.2 Maximum number of minimal codewords
        2.1.3 Minimum number of minimal codewords
    2.2 Secret Sharing Schemes Based on Self-dual Codes
        2.2.1 Massey scheme and self-dual codes
        2.2.2 An extension of Massey scheme
    References
3. Blakley Secret-Sharing Scheme
    3.1 Linear Codes
        3.1.1 LCD codes
    3.2 Ramp Secret-Sharing Schemes
    3.3 Multisecret-Sharing Schemes Based on Linear Codes
        3.3.1 Scheme description
        3.3.2 Secret distribution
        3.3.3 Secret recovery
    3.4 Statistics on Coalitions
        3.4.1 Security analysis
        3.4.2 Information theoretic efficiency
        3.4.3 Comparison with other schemes
        3.4.4 Conclusion and open problems
    3.5 A New Approach to Construct a Secret-Sharing Scheme Based on Blakley’s Method
        3.5.1 Proposed scheme
        3.5.2 Security analysis
        3.5.3 Conclusion
    3.6 Some Multisecret-Sharing Schemes over Finite Fields
        3.6.1 Notation
        3.6.2 Scheme description
        3.6.3 Statistics on coalitions
        3.6.4 Security analysis
        3.6.5 Information theoretic efficiency
        3.6.6 Comparison with other schemes
        3.6.7 Conclusions
    References
4. Alternative Schemes
    4.1 Codes and Coset Decoding
        4.1.1 Coset decoding
        4.1.2 Coset leader
    4.2 Multisecret-Sharing Schemes and Error Correcting Codes
        4.2.1 Scheme description
        4.2.2 Statistics on coalitions
        4.2.3 Democracy in secret-sharing
        4.2.4 Comparison with other schemes
        4.2.5 Conclusion
    4.3 A New Secret-Sharing Scheme Based on Polynomials over Finite Fields
        4.3.1 Polynomials over finite fields
        4.3.2 The scheme
        4.3.3 Properties and security
        4.3.4 Conclusion
    4.4 Roots of Irreducible Polynomials
        4.4.1 Traces and norms
        4.4.2 Secret-sharing schemes
        4.4.3 The schemes
        4.4.4 First scheme
        4.4.5 Second scheme
        4.4.6 Conclusion
    4.5 Secret-Sharing Schemes and Syndrome Decoding
        4.5.1 Syndrome decoding
        4.5.2 Why use syndrome decoding?
        4.5.3 Conclusion
    4.6 Secret Sharing, Zero Sum Sets, and Hamming Codes
        4.6.1 Algebraic preliminaries
        4.6.2 Integer residue rings
        4.6.3 Zero-sum sets
        4.6.4 Secret-sharing schemes
        4.6.5 The scheme
        4.6.6 Coding interpretation
        4.6.7 Random choice attack
        4.6.8 Information rate
        4.6.9 Comparison with other schemes
        4.6.10 Combination with Shamir’s scheme
        4.6.11 Conclusions
    4.7 The Least Squares Solutions in Code-Based Multisecret-Sharing Scheme
        4.7.1 Introduction
        4.7.2 Preliminaries
        4.7.3 Multisecret-sharing schemes and least-squares solutions in linear codes
        4.7.4 Comparison with other schemes
        4.7.5 Conclusion and open problems
    References
5. Applications
    5.1 On Key Distribution in MANETs
        5.1.1 Identity-based cryptography in MANETs
        5.1.2 Secret-sharing schemes without trusted party
        5.1.3 Hierarchical threshold secret sharing
        5.1.4 Conclusion
    5.2 Absolute Time for Round-Based Timestamping Schemes
        5.2.1 Introduction
        5.2.2 Preliminary
        5.2.3 Timestamping scheme and its security requirements
        5.2.4 Construction
        5.2.5 Security analysis
        5.2.6 Eliminating trust in the TSA
        5.2.7 Conclusion
    5.3 An Image Secret-Sharing Method Based on Shamir Secret Sharing
        5.3.1 Review of Shamir’s secret-sharing scheme
        5.3.2 Proposed method
        5.3.3 Application of some secret sharing schemes
        5.3.4 Proposed scheme
        5.3.5 Secret retrieval procedure
        5.3.6 Advantages
        5.3.7 Security analysis
        5.3.8 Conclusion
    References
Index