The Diffie–Hellman Key Exchange (DHKE) is a method that allows two sides to agree on a shared secret over a public channel. This method is based on the discrete logarithm problem which is believed to be hard to solve.

Textbook definition

Parameters

Name	Description
\(p\)	The prime modulus
\(g\)	A generator of the multiplicative group of integers modulo \(p\)
\(a\)	Alice’s private number: \(1 < a < p - 1\)
\(b\)	Bob’s private number: \(1 < b < p - 1\)
\(A\)	Alice’s public number: \(A = g^a \mod p\)
\(B\)	Bob’s public number: \(B = g^b \mod p\)
\(s\)	The shared secret: \(s = B^a = A^b = g^{ab} \mod p\)

Key exchange

Suppose a situation where Alice and Bob want to create a shared secret key.

1. They choose both a big prime number \(p\) and a generator \(g\) on which they agree on.
2. They create private keys \(a\) and \(b\) respectively, with \(a, b \in GF(p)\)
3. They both compute their corresponding public keys \(A\) and \(B\) and send them to each other over the public channel.

\[ \begin{aligned} A &= g^a \mod p \\ B &= g^b \mod p \\ \end{aligned} \]

4. They can now both compute the shared secret key \(s\) as

\[ s = B^a = A^b = g^{ab} \mod p \]

They can now use the shared secret \(s\) to derive a symmetric key for AES (for example), and use it to encrypt their future messages.

Tools

• Discrete logarithm calculator
• Wikipedia - Diffie–Hellman Key Exchange