Tate Pairing Implementation for Hyperelliptic y 2 = x p – x + d 1st edition by Iwan Duursma, Hyang Sook Lee ISBN 3540205920 9783540205920

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ISBN 10:       3540205920
ISBN 13: 978-3540205920
Author:         Iwan Duursma, Hyang-Sook Lee
 

Tate Pairing Implementation for Hyperelliptic y 2 = x p – x + d  1st Table of contents:

1. Introduction

   • 1.1 Overview of Hyperelliptic Curves
   • 1.2 Introduction to the Tate Pairing
   • 1.3 Relevance of the Curve $y2=xp-x+d$
   • 1.4 Contributions of the Paper
   • 1.5 Organization of the Paper

2. Preliminaries

   • 2.1 Mathematical Background of Hyperelliptic Curves
     • 2.1.1 Definition and Properties of Hyperelliptic Curves
     • 2.1.2 The Curve $y2=xp-x+d$
   • 2.2 Elliptic and Hyperelliptic Curve Cryptography
   • 2.3 The Tate Pairing: Definition and Properties
     • 2.3.1 Tate Pairing on Elliptic Curves
     • 2.3.2 Generalization to Hyperelliptic Curves
   • 2.4 Computational Complexity Considerations

3. Mathematical Formulation of the Curve $y2=xp-x+d$

   • 3.1 Curve Definition and Structure
   • 3.2 Singularities and Smoothness of the Curve
   • 3.3 Points on the Curve and Group Law
   • 3.4 Special Properties of the Curve for Pairing-Based Cryptography

4. Tate Pairing on Hyperelliptic Curves

   • 4.1 The Tate Pairing: Generalization to Hyperelliptic Curves
   • 4.2 Pairing Computations in Characteristic ppp
   • 4.3 Efficient Computation of the Tate Pairing on $y2=xp-x+d$
     • 4.3.1 Algorithmic Approach to Tate Pairing
     • 4.3.2 Optimization Techniques for Efficient Pairing Computation
   • 4.4 Pairing-Based Cryptography on Hyperelliptic Curves

5. Implementation of the Tate Pairing

   • 5.1 Choice of Parameters for Implementation
   • 5.2 Efficient Algorithms for Tate Pairing Computation
     • 5.2.1 Miller’s Algorithm for Pairing Computation
     • 5.2.2 Final Exponentiation and Optimization
   • 5.3 Software Implementation Details
     • 5.3.1 Choice of Cryptographic Libraries and Tools
     • 5.3.2 Code Optimization for Performance
   • 5.4 Hardware Implementation Considerations
     • 5.4.1 Optimizing for FPGA and ASIC Architectures
     • 5.4.2 Comparison with Elliptic Curve Pairings

6. Security Considerations

   • 6.1 Security of the Tate Pairing on Hyperelliptic Curves
   • 6.2 Attacks on Pairing-Based Cryptosystems
     • 6.2.1 Move to Cryptographic Attacks: Miller’s Algorithm Vulnerabilities
     • 6.2.2 Side-Channel Attacks and Countermeasures
   • 6.3 Potential Application Areas and Security Implications

7. Performance Analysis

   • 7.1 Computational Efficiency: Time and Space Complexity
   • 7.2 Experimental Results on Pairing Computation
     • 7.2.1 Benchmarks on Various Hardware Platforms
     • 7.2.2 Comparison with Other Curve Families
   • 7.3 Scalability of the Implementation

8. Applications of Tate Pairing on Hyperelliptic Curves

   • 8.1 Pairing-Based Cryptosystems
   • 8.2 Identity-Based Encryption Using the Tate Pairing
   • 8.3 Applications in Zero-Knowledge Proofs and Digital Signatures
   • 8.4 Post-Quantum Cryptography Considerations

9. Extensions and Future Work

   • 9.1 Generalizing the Approach to Other Hyperelliptic Curves
   • 9.2 Improvements in Pairing Efficiency and Security
   • 9.3 Future Applications in Cryptography and Beyond
   • 9.4 Open Problems and Challenges in Pairing Implementation

10. Conclusion

• 10.1 Summary of Contributions
• 10.2 Final Thoughts on Hyperelliptic Curve Pairings
• 10.3 Future Directions for Research and Implementation

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