---
tag: 01TKKESHSB
kind: example
name: Diffie–Hellman key exchange
msc: [94A60]
instance_of: [01E9MNJYEX]
uses: [0KM2CJ80G7, 09B4X51J09, 04DR5PM25N]
sources:
  - url: https://en.wikipedia.org/wiki/Diffie%E2%80%93Hellman_key_exchange
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Diffie%E2%80%93Hellman_key_exchange
    locator: Cryptographic explanation
same_as:
  wikidata: Q623447
  wikipedia: Diffie–Hellman key exchange
---

## Example

The Diffie–Hellman key exchange is an example of [[public-key cryptography|01E9MNJYEX]]. 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|0KM2CJ80G7]] 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 $5^3 \equiv 8^4 \equiv 4 \pmod{11}$.

## Remarks

An eavesdropper sees $p$, $g$, $A$ and $B$. To find $g^{ab}$, a natural method is to compute a [[discrete logarithm|09B4X51J09]], which is believed to be hard for large $p$.
The same method works in any finite cyclic group, for example the group of points of an elliptic curve.
Whitfield Diffie and Martin Hellman published the method in 1976.

## Explanations

### L1 human

Each party mixes a public number with a private number, and they swap the results.
Each then mixes in their own private number again, and both get the same final number.
An eavesdropper sees only the swapped numbers, and cannot easily undo the mixing.