Diffie–Hellman key exchange

Example

The Diffie–Hellman key exchange is an example of public-key cryptography. Two parties, Alice and Bob, use it to agree on a secret key over a public channel. They agree in public on a prime $p$ and an integer $g$, often a primitive root modulo $p$. Alice chooses a secret integer $a$ and sends $A = g^a \bmod p$. Bob chooses a secret integer $b$ and sends $B = g^b \bmod p$. Alice computes $B^a \bmod p$, and Bob computes $A^b \bmod p$. Both are equal to $g^{ab} \bmod p$, the shared secret. A small case: $p = 11$, $g = 2$, $a = 3$ and $b = 4$. Then $A = 8$, $B = 16 \bmod 11 = 5$, and $5^3 \equiv 8^4 \equiv 4 \pmod{11}$.