## Overview ECDSA is a variant of the [[Digital Signature Algorithm]] ([[Digital Signature Algorithm|DSA]]) that utilizes [[Elliptic Curve Cryptography]] ([[Elliptic Curve Cryptography|ECC]]) to provide stronger security with shorter key lengths. This [[algorithm]] is crucial for systems where computational power and storage are limited due to its efficiency. ## Key Concepts - **Elliptic Curve**: A smooth curve representing solutions to a specific algebraic equation. In [[Elliptic Curve Cryptography|ECC]], points on the curve are used for cryptographic processes. - **[[Private Key]]**: A secret random number used to generate the [[digital signature]]. - **[[Public Key]]**: A point on the elliptic curve, derived from the private key, used for signature verification. - **Signature**: A pair of numbers, commonly denoted (r, s), generated from the private key and the hash of the message. ## How ECDSA Works 1. **Key Generation**: A private key is chosen as a random number, and the corresponding public key is computed as a point on the elliptic curve. 2. **Signing**: Using the private key, the message's hash, and a random number (nonce), the (r, s) signature pair is generated. 3. **Verification**: A recipient can verify the signature using the signer's public key, the message's hash, and the (r, s) signature pair. ## Benefits - **Efficiency**: Offers the same level of security as traditional methods but with much shorter key lengths. - **Scalability**: Suitable for systems with limited resources like smart cards and IoT devices. - **Security**: Resistant to several types of cryptographic attacks when implemented correctly. ## Common Uses - **[[Blockchain Technology|Blockchain]]**: Used in [[Cryptocurrency|cryptocurrencies]] like [[Bitcoin]] for transaction signatures. - **Secure Communications**: For digitally signing messages and documents to prove [[authenticity]] and integrity. - **Certificates**: Used in SSL/TLS for website security. ## Vulnerabilities - **Nonce Reuse**: Reusing a nonce while signing different messages can reveal the private key. - **Weak Curves**: Some elliptic curves are more susceptible to attacks. Using well-established curves is crucial. - **Implementation Flaws**: Incorrect implementations can introduce vulnerabilities. ## Related Concepts - **[[Elliptic Curve Cryptography]] ([[Elliptic Curve Cryptography|ECC]])**: The broader field of [[cryptography]] that ECDSA is a part of. - **[[Digital Signature Algorithm|DSA]]**: The original [[digital signature]] [[algorithm]] on which ECDSA is based. - **ECDH ([[Elliptic Curve Diffie-Hellman]])**: An elliptic curve-based key exchange protocol.