up:: [[Cryptographic Algorithms]] # Elliptic Curve Cryptography (ECC) Elliptic Curve [[Cryptography]] (ECC) is a type of [[Asymmetric Encryption|public key cryptography]] that uses the mathematics behind elliptic curves over finite fields. ECC offers a similar level of security as traditional methods like [[RSA]] but with much shorter key lengths, making it more efficient. ## Key Features - **Efficiency**: Requires smaller key sizes compared to non-ECC [[cryptography]] (like [[RSA]]) for equivalent security, resulting in faster computations, less energy consumption, and lower storage requirements. - **Elliptic Curves**: Defined algebraically as cubic equations in two variables. The structure of these curves provides the foundation for ECC. - **Point Multiplication**: Central operation in ECC, where a point on the curve is multiplied by a number to produce another point on the curve. ## Mathematical Basis - **Curve Equation**: Most ECC applications use curves defined by the equation y^2 = x^3 + ax + b. - **Finite Fields**: ECC operates over finite fields, which limit the set of possible values. This ensures calculations wrap around upon reaching a certain value. - **[[Diffie-Hellman]] Key Exchange**: ECC can be used in the [[Elliptic Curve Diffie-Hellman]] ([[Elliptic Curve Diffie-Hellman|ECDH]]) key agreement protocol, enabling two parties to establish a shared secret. - **[[Digital Signature|Digital Signatures]]**: [[Elliptic Curve Digital Signature Algorithm]] ([[Elliptic Curve Digital Signature Algorithm|ECDSA]]) is a widely adopted standard for creating [[Digital Signature|digital signatures]] using ECC. ## Implications - **Quantum Vulnerability**: Like other [[public key]] cryptosystems, ECC is vulnerable to [[Quantum Computing|quantum computers]] due to [[Shor's algorithm]]. - **Cryptographic Agility**: As with other cryptosystems, it's vital to stay updated with recommended curve parameters and key lengths as computational capabilities advance. ## Current Status - **Widespread Adoption**: Due to its efficiency advantages, ECC is used in many security standards and protocols, including SSL/TLS certificates, [[Bitcoin]]'s signature [[algorithm]], and secure messaging apps. - **Research & Development**: Newer curves and associated protocols are being researched and standardized to ensure ongoing security and resistance against specific attack vectors. ## Related Concepts - **[[Asymmetric Encryption]]**: Cryptographic methods that use two distinct keys – one private and one public. - **[[Quantum Computing]]**: The study and development of computers based on [[quantum mechanics]]. - **[[RSA]]**: A widely used public-key cryptosystem not based on elliptic curves.