ECDSA

ECDSA stands for Elliptic Curve Digital Signature Algorithm. It is a variant of the Digital Signature Algorithm (DSA) that uses elliptic curves over finite fields to provide digital signatures with comparable security using much smaller key sizes. ECDSA is widely used to assure authenticity and integrity of messages in protocols and software, including TLS, SSH, PGP, and many blockchain systems.

Key pairs in ECDSA consist of a private key d and a public key Q = dG, where G is a fixed base point on a chosen elliptic curve. The public key is derived from the private key, and signatures are produced with the private key while verification uses the public key. The security of ECDSA relies on the difficulty of the elliptic curve discrete logarithm problem.

Signature generation uses a per-signature random value k. Given a message, its hash z, and the private key d, one computes r = (kG).x mod n and s = k^{-1}(z + rd) mod n, where n is the order of G. The signature is the pair (r, s). Verification uses the public key Q and the signature: compute w = s^{-1} mod n, u1 = z w mod n, u2 = r w mod n, then X = u1G + u2Q; the signature is valid if r ≡ X.x mod n.

Security considerations include the need for proper nonce generation; reusing k or weak randomness can expose the private key. Deterministic nonces per RFC 6979 reduce this risk. ECDSA is standardized in several specifications (e.g., FIPS 186-4, ANSI X9.62) and supports multiple curves such as secp256r1 (P-256), secp384r1 (P-384), and secp256k1, the latter being prominent in cryptocurrency. Encoding of signatures typically uses two integers (r, s) and can be represented in DER or other formats.