ECDSA (Elliptic Curve Digital Signature Algorithm) is the elliptic curve analogue of DSA, standardised in FIPS 186 and the IETF, that produces digital signatures using a private key and verifies them with the corresponding public key. It is the most widely deployed signature algorithm in X.509 certificates (P-256 with SHA-256 is the default for certificate authorities issuing TLS certificates), in code signing (Authenticode, macOS, Linux package signing), in TLS 1.3 certificate authentication, in SSH host keys and user keys (though Ed25519 is increasingly preferred), and in blockchain and cryptocurrency systems. An ECDSA signature over a message m with private key d on curve with base point G produces a pair (r, s), where r is the x-coordinate of an ephemeral public key k × G and s encodes the relationship between the message hash, r, the private key d, and the nonce k. Verification requires only the public key Q = d × G and is fast; signing requires the private key and a nonce.
Diffie-Hellman (DH) is a key exchange protocol published by Whitfield Diffie and Martin Hellman in 1976 — the first public description of asymmetric cryptography and one of the most consequential cryptographic publications in history. Its fundamental contribution is solving the key establishment problem: two parties who have never communicated before, communicating over a channel that an adversary can fully observe, can nonetheless agree on a shared secret that the adversary cannot determine. The security of finite-field DH rests on the discrete logarithm problem: given g^a mod p and g^b mod p (the public values exchanged), computing g^ab mod p (the shared secret) requires solving for either a or b, which is computationally infeasible for sufficiently large groups. The 1976 original uses multiplicative groups of integers modulo a prime p; the security level is determined by the size of p (currently 2048-bit minimum, 3072-bit recommended) and the group’s structure. Finite-field DH is still deployed in TLS 1.2 DHE cipher suites and legacy IPsec configurations, but has been supplanted in new deployments by Elliptic Curve Diffie-Hellman (ECDH) and specifically by X25519, which provide equivalent security at dramatically smaller key sizes.
