Elliptic curve Diffie–Hellman

Elliptic curve Diffie–Hellman (ECDH) is an anonymous key agreement protocol that allows two parties, each having an elliptic curve public–private key pair, to establish a shared secret over an insecure channel. This shared secret may be directly used as a key, or to derive another key which can then be used to encrypt subsequent communications using a symmetric key cipher. It is a variant of the Diffie–Hellman protocol using elliptic curve cryptography.

The following example will illustrate how a key establishment is made. Suppose Alice wants to establish a shared key with Bob, but the only channel available for them may be eavesdropped by a third party. Initially, the domain parameters (that is, ${\displaystyle (p,a,b,G,n,h)}$ in the prime case or ${\displaystyle (m,f(x),a,b,G,n,h)}$ in the binary case) must be agreed upon. Also, each party must have a key pair suitable for elliptic curve cryptography, consisting of a private key ${\displaystyle d}$ (a randomly selected integer in the interval ${\displaystyle [1,n-1]}$) and a public key ${\displaystyle Q}$ (where ${\displaystyle Q=dG}$, that is, the result of adding ${\displaystyle G}$ together ${\displaystyle d}$ times). Let Alice's key pair be ${\displaystyle (d_{A},Q_{A})}$ and Bob's key pair be ${\displaystyle (d_{B},Q_{B})}$. Each party must know the other party's public key prior to execution of the protocol.

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