Elliptic Curves: The Core Tool of Cryptography

Elliptic curves are special mathematical curves that play a key role in cryptography. The article explains how elliptic curves form Abelian groups with commutative and associative properties, ranging from informal definitions (y² = x³ + ax + b) to formal definitions (smooth projective algebraic curves). Group operations are implemented through geometric methods (such as reflecting the intersection point of a line through two points) or algebraic formulas, making Elliptic Curve Cryptography (ECC) practically feasible. ECC is widely used in blockchain, digital signatures, and other fields, providing efficient and secure encryption solutions. Understanding elliptic curves helps grasp the cutting-edge developments in modern cryptography, especially crucial for secure communication in AI systems.