# Diffie-Hellman key exchange

> Diffie-Hellman lets two parties agree on a shared secret over a channel anyone can read, by mixing private exponents into public numbers an eavesdropper cannot unmix.

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## Mix a secret in public

*Interactive, play it in a browser: https://shapelessai.com/vizipedia/diffie-hellman-key-exchange#mix-a-secret-in-public*

## How the exchange works

Alice and Bob agree in public on a prime modulus p and a base g. Each picks a secret exponent and sends g raised to it, mod p; each then raises what the other sent to their own secret, and both land on the same number, written in RFC 2631 as ZZ = (yb ^ xa) mod p = (ya ^ xb) mod p [1]. Wikipedia's small example uses p = 23 and g = 5 [2] and ends with a shared secret of 18 [3]. The goal is a secret that stays unavailable to eavesdroppers [4], which then keys a fast symmetric cipher [5]. Paint is the classic picture: anyone listening knows only the common color and the two mixes [6].

*Version 1, claude-opus-5-5 for @vizipedia.*

1. [RFC 2631: Diffie-Hellman Key Agreement Method (IETF, 1999)](https://www.rfc-editor.org/rfc/rfc2631.txt) "ZZ = (yb ^ xa) mod p = (ya ^ xb) mod p" (quote found)
2. [Diffie-Hellman key exchange (Wikipedia)](https://en.wikipedia.org/wiki/Diffie%E2%80%93Hellman_key_exchange) "Alice and Bob publicly agree to use a modulus p = 23 and base g = 5" (quote found)
3. [Diffie-Hellman key exchange (Wikipedia)](https://en.wikipedia.org/wiki/Diffie%E2%80%93Hellman_key_exchange) "Alice and Bob now share a secret (the number 18)" (quote found)
4. [RFC 2631: Diffie-Hellman Key Agreement Method (IETF, 1999)](https://www.rfc-editor.org/rfc/rfc2631.txt) "agree upon a shared secret in such a way that the secret will be unavailable to eavesdroppers" (quote found)
5. [Diffie-Hellman key exchange (Wikipedia)](https://en.wikipedia.org/wiki/Diffie%E2%80%93Hellman_key_exchange) "This key can then be used to encrypt subsequent communications using a symmetric-key cipher." (quote found)
6. [Diffie-Hellman key exchange (Wikipedia)](https://en.wikipedia.org/wiki/Diffie%E2%80%93Hellman_key_exchange) "If a third party listened to the exchange, they would only know the common color" (quote found)

## Why Eve is stuck

Eve sees p, g and both public values. Getting a secret exponent back from them is the discrete logarithm problem [1], which is currently considered difficult when the group is large enough [2]; Wikipedia recommends primes of at least 2048 bits [3]. A fast discrete log algorithm would break this and many other public key systems [4], and Shor's algorithm on a quantum computer is one [5]. The bare exchange also proves nothing about who is on the other end: it is non-authenticated, and serves as the basis for authenticated protocols [6].

*Version 1, claude-opus-5-5 for @vizipedia.*

1. [Diffie-Hellman key exchange (Wikipedia)](https://en.wikipedia.org/wiki/Diffie%E2%80%93Hellman_key_exchange) "Such a problem is called the discrete logarithm problem." (quote found)
2. [Diffie-Hellman key exchange (Wikipedia)](https://en.wikipedia.org/wiki/Diffie%E2%80%93Hellman_key_exchange) "This is currently considered difficult for groups whose order is large enough." (quote found)
3. [Diffie-Hellman key exchange (Wikipedia)](https://en.wikipedia.org/wiki/Diffie%E2%80%93Hellman_key_exchange) "it is recommended to use prime numbers of at least 2048 bits in length" (quote found)
4. [Diffie-Hellman key exchange (Wikipedia)](https://en.wikipedia.org/wiki/Diffie%E2%80%93Hellman_key_exchange) "An efficient algorithm to solve the discrete logarithm problem would make it easy to compute a or b" (quote found)
5. [Diffie-Hellman key exchange (Wikipedia)](https://en.wikipedia.org/wiki/Diffie%E2%80%93Hellman_key_exchange) "Quantum computers can break public-key cryptographic schemes, such as RSA, finite-field DH and elliptic-curve DH key-exchange protocols, using Shor's algorithm" (quote found)
6. [Diffie-Hellman key exchange (Wikipedia)](https://en.wikipedia.org/wiki/Diffie%E2%80%93Hellman_key_exchange) "exchange itself is a non-authenticated key-agreement protocol, it provides the basis for a variety of authenticated protocols" (quote found)

## Where it runs today

TLS 1.3 lists (EC)DHE, Diffie-Hellman over finite fields or elliptic curves, as a key exchange mode [1]; implementations must support P-256 and should support X25519 [2]. Using fresh keys per session gives forward secrecy: the private keys are discarded once agreement is complete [3]. Signal's X3DH runs several elliptic curve Diffie-Hellman exchanges on X25519 or X448 [4]. The method was published by Whitfield Diffie and Martin Hellman in 1976 [5]; Hellman counts Ralph Merkle as a co-inventor of public key cryptography [6].

*Version 1, claude-opus-5-5 for @vizipedia.*

1. [RFC 8446: The Transport Layer Security (TLS) Protocol Version 1.3 (IETF, 2018)](https://www.rfc-editor.org/rfc/rfc8446.txt) "(EC)DHE (Diffie-Hellman over either finite fields or elliptic curves)" (quote found)
2. [RFC 8446: The Transport Layer Security (TLS) Protocol Version 1.3 (IETF, 2018)](https://www.rfc-editor.org/rfc/rfc8446.txt) "MUST support key exchange with secp256r1 (NIST P-256) and SHOULD support key exchange with X25519" (quote found)
3. [Diffie-Hellman key exchange (Wikipedia)](https://en.wikipedia.org/wiki/Diffie%E2%80%93Hellman_key_exchange) "the private keys are discarded once key agreement is complete" (quote found)
4. [The X3DH Key Agreement Protocol (Signal)](https://signal.org/docs/specifications/x3dh/) "The Elliptic Curve Diffie-Hellman function will be either the X25519 or X448 function" (quote found)
5. [Diffie-Hellman key exchange (Wikipedia)](https://en.wikipedia.org/wiki/Diffie%E2%80%93Hellman_key_exchange) "It is named after Whitfield Diffie and Martin Hellman who published it in 1976." (quote found)
6. [Martin E. Hellman's home page (Stanford)](https://www-ee.stanford.edu/~hellman/) "best known for his invention, with Diffie and Merkle, of public key cryptography" (quote found)

## The worked example: p = 23, g = 5

*Figure: https://shapelessai.com/vizipedia/diffie-hellman-key-exchange#the-worked-example-p-23-g-5*