ECDH

Elliptic Curve Diffie-Hellman (ECDH) is a public-key key exchange protocol that enables two parties to establish a shared secret over an insecure channel using the properties of elliptic curves. The derived secret can then be used to derive symmetric encryption keys for subsequent communication. When ephemeral keys are used, the protocol provides forward secrecy.

ECDH relies on the difficulty of the elliptic curve discrete logarithm problem. Each participant generates a private key as a randomly chosen integer and a corresponding public key obtained by multiplying the private key with a fixed base point on an agreed elliptic curve. The two public keys are exchanged.

To obtain the shared secret, each party multiplies the received public key by its own private key. The result is the same for both sides because scalar multiplication on the curve is commutative in this sense. The shared secret is then processed by a key derivation function to produce symmetric keys and, optionally, an agreed protocol transcript.

ECDH is widely used in secure communications protocols, notably as the key agreement component of TLS with ephemeral keys (ECDHE). It can be combined with digital signatures (for example, servers sign the ephemeral public key to authenticate the exchange). Curve choices such as Curve25519 and secp256r1 are common due to security and efficiency.

Security considerations include selecting well-supported curves, ensuring proper random key generation, and protecting private keys from leakage. Implementations must be resistant to side-channel attacks and misconfigurations. ECDH offers smaller key sizes for equivalent security levels compared with traditional DH, enabling faster computations and smaller bandwidth.

See also: Elliptic curve cryptography, Elliptic Curve Digital Signature Algorithm, Diffie-Hellman, ECDHE.