Definition:Diffie-Hellman-Merkle Key Exchange

Definition

The Diffie-Hellman-Merkle key exchange is a safe method of exchanging a key.

Suppose Alice and Bob with to communicate securely.

They decide to use the order $p - 1$ cyclic group $G$ of multiplication modulo $p$ consisting of the non-zero elements of a finite field of order $p$

They choose a generator $g$ of $G$.

Alice chooses an integer $a$ and calculates $A = g^a$ modulo $p$.

Alice sends $A$ to Bob.

Bob chooses an integer $b$ and calculates $B = g^b$ modulo $p$.

Bob sends $B$ to Bob.

Both can calculate the key $g^{a b}$ modulo $p$:

$K = B^a$, calculated by Alice

and:

$K = A^b$, calculated by bob.

An eavesdropper may know $p$, $g$, $A$ and $B$ but cannot calculate $K$ without solving the discrete logarithm problem.

If $p$ is large, the discrete logarithm problem is very difficult to solve.

So $K$ can be used as the basis of a private key for Alice and Bob.