Cryptography number theory and ai security: Foundations and applications(Math and Artificial Intelligence)

In today's interconnected digital society, the protection of information is not just a technical requirement but a cornerstone of trust, governance, and progress. Every email we send, every financial transaction we make, every online search query we type, and every AI system that analyzes data in real time-all depend on cryptographic security. Cryptography, once regarded as a highly specialized tool of spies, mathematicians, and military strategists, has now become a pervasive and indispensable technology that underpins the very infrastructure of our global digital economy. At the heart of cryptography lies number theory-a branch of pure mathematics that investigates the properties of integers, prime numbers, modular arithmetic, and abstract algebraic structures. What was once considered "the queen of mathematics" and "an art for art's sake" has become one of the most powerful weapons against cybercrime. Techniques such as modular exponentiation, prime testing, elliptic curve arithmetic, and the discrete logarithm problem are not only abstract mathematical curiosities but also the lifeblood of secure communication, online banking, e-commerce, blockchain, and digital identity. However, the story does not stop at cryptography and number theory. The dawn of the Artificial Intelligence (AI) era has added a new layer of complexity and opportunity to the field of security. AI has empowered cybersecurity systems with the ability to detect anomalies, predict attacks, defend against malware, and even break weak cryptographic algorithms. But at the same time, AI introduces new threats-adversarial AI attacks, data poisoning, model inversion, and large-scale misinformation campaigns-all of which demand new security paradigms that combine mathematics, algorithms, and intelligent defense mechanisms. This book, "Cryptography, Number Theory, and AI Security: Foundations and Applications", is designed as a bridge between classical mathematical security principles and modern AI-driven defense strategies. It offers a comprehensive exploration of the fundamentals of cryptography, the mathematical structures underlying secure systems, and the emerging role of AI in both defending and attacking these systems.Why This Book?The motivation behind this book is rooted in the urgent necessity to educate the next generation of cybersecurity professionals, mathematicians, and AI researchers about the convergence of three critical domains: Cryptography - The science of securing communication. Number Theory - The mathematical foundation of modern encryption. AI Security - The defense and offense mechanism in the age of intelligent cyberattacks. Most books in the current literature focus on one of these domains in isolation. Cryptography textbooks tend to emphasize classical and modern ciphers but often neglect the growing influence of AI. Number theory texts explore mathematical elegance but seldom address real-world cryptographic implementations. AI security resources focus heavily on intrusion detection and anomaly detection but rarely link back to the mathematical principles of encryption and secure computation. This book is different. It offers a holistic approach: It teaches number theory not as an abstract subject but as a tool for cryptography. It explains cryptography not just as a system of locks and keys but as a living technology facing AI-driven adversaries. It highlights AI security not just as a buzzword but as a mathematically grounded, cryptographically informed discipline. By unifying these perspectives, the book provides students, researchers, and professionals with a robust conceptual and practical toolkit that can be applied in diverse fields-ranging from financial security systems..