The AGM-X 0(N) Heegner Point Lifting Algorithm and Elliptic Curve Point Counting 1st edition by David Kohel ISBN 3540205920 9783540205920

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Product details:

ISBN 10:       3540205920
ISBN 13: 978-3540205920
Author:         David R. Kohel
 

The AGM-X 0(N) Heegner Point Lifting Algorithm and Elliptic Curve Point Counting  1st Table of contents:

1. Introduction

   • 1.1 Overview of Elliptic Curves
   • 1.2 Heegner Points and Their Relevance in Cryptography
   • 1.3 The AGM-X₀(N) Heegner Point Lifting Algorithm
   • 1.4 Significance of Elliptic Curve Point Counting
   • 1.5 Contributions of the Paper
   • 1.6 Organization of the Paper

2. Preliminaries

   • 2.1 Elliptic Curves: Definition and Properties
   • 2.2 Point Counting on Elliptic Curves
   • 2.3 Heegner Points and their Role in Number Theory
   • 2.4 The Arithmetic-Geometric Mean (AGM) Method
   • 2.5 The AGM-X₀(N) Lifting Algorithm Overview

3. The AGM-X₀(N) Heegner Point Lifting Algorithm

   • 3.1 Mathematical Formulation of Heegner Points
   • 3.2 Heegner Points and the Lifting Process
   • 3.3 Implementation of the AGM-X₀(N) Algorithm
     • 3.3.1 Computational Approach
     • 3.3.2 Optimizations and Complexity Analysis
   • 3.4 Theoretical Foundations of the AGM-X₀(N) Lifting
   • 3.5 The Interaction between Heegner Points and Elliptic Curves

4. Elliptic Curve Point Counting

   • 4.1 Challenges in Elliptic Curve Point Counting
   • 4.2 Algorithms for Point Counting
     • 4.2.1 Schoof’s Algorithm and Variants
     • 4.2.2 The AGM Approach to Point Counting
     • 4.2.3 The Role of Heegner Points in Point Counting
   • 4.3 Efficiency Considerations in Point Counting
     • 4.3.1 Time and Space Complexity of Point Counting Algorithms
     • 4.3.2 Optimizing Point Counting with Heegner Point Lifting
   • 4.4 Comparison with Other Point Counting Methods

5. Applications in Cryptography

   • 5.1 Elliptic Curve Cryptography (ECC) and its Importance
   • 5.2 Point Counting in ECC: Theoretical and Practical Implications
   • 5.3 Cryptographic Applications of Heegner Points
   • 5.4 Implementation of AGM-X₀(N) for ECC
     • 5.4.1 Key Generation and Public Key Infrastructure
     • 5.4.2 Signature Generation and Verification
   • 5.5 Enhancing Security with Heegner Point Lifting

6. Performance Evaluation

   • 6.1 Experimental Setup and Benchmarking
   • 6.2 Evaluation of AGM-X₀(N) Algorithm Performance
     • 6.2.1 Time Complexity Analysis
     • 6.2.2 Comparison with Existing Algorithms
   • 6.3 Memory and Computational Requirements
   • 6.4 Scalability of Point Counting with AGM-X₀(N)

7. Security Considerations

   • 7.1 Security of the AGM-X₀(N) Heegner Point Lifting Algorithm
   • 7.2 Attacks on Point Counting Algorithms and Mitigation Strategies
   • 7.3 Role of Heegner Points in Enhancing ECC Security
   • 7.4 Cryptographic Protocols and Security in Practical Systems

8. Extensions and Future Work

   • 8.1 Generalizations of the AGM-X₀(N) Algorithm
   • 8.2 Heegner Points on Other Types of Curves
   • 8.3 Integration with Other Cryptographic Primitives
   • 8.4 Future Directions for Research in Point Counting and Heegner Lifting

9. Conclusion

   • 9.1 Summary of Results
   • 9.2 Contributions to Elliptic Curve Cryptography and Number Theory
   • 9.3 Future Research Outlook

10. References

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