An Exploration of Mathematical Algorithms Behind Digital Security in Financial Transactions

1. Introduction 1.1 Background and Motivation 1.2 Purpose and Significance 1.3 Research Questions 2. Fundamentals of Digital Security 2.1 Overview of Digital Security 2.2 Importance in Financial Transactions 2.3 Common Vulnerabilities 3. Mathematical Foundations 3.1 Number Theory Basics 3.2 Cryptographic Functions 3.3 Algebraic Structures 4. Symmetric Key Algorithms 4.1 DES and AES Overview 4.2 Mathematical Principles 4.3 Application in Finance 5. Asymmetric Key Algorithms 5.1 RSA Algorithm Mechanics 5.2 Elliptic Curve Cryptography 5.3 Use Cases in Transactions 6. Hash Functions and Message Authentication 6.1 Secure Hash Algorithms 6.2 Digital Signatures 6.3 Integrity in Data Exchange 7. Quantum Cryptography Impact 7.1 Quantum Computing Basics 7.2 Potential Threats 7.3 Advancements in Quantum Security 8. Case Studies in Digital Security 8.1 Historical Security Breaches 8.2 Analysis of Financial Institutions 8.3 Lessons Learned 9. Conclusion and Future Research 9.1 Summary of Findings 9.2 Implications for Financial Security 9.3 Directions for Future Study