# Diffie-Hellman

May 20, 2023

The Diffie-Hellman key exchange, often abbreviated as DH, is a cryptographic protocol that enables two parties, commonly referred to as Alice and Bob, to securely establish a shared secret key over a public communications channel. Named after its inventors, Whitfield Diffie and Martin Hellman, the protocol was first published in 1976 and has since become a fundamental component of numerous cryptographic systems, including the widely used Transport Layer Security (TLS) protocol.

## Overview

The main objective of the Diffie-Hellman key exchange is to allow two parties to derive a shared secret key that can be used for secure communication over an insecure channel. The strength of the protocol lies in its ability to prevent eavesdroppers, who might be intercepting the public communications, from obtaining the shared secret key. This is achieved through the use of mathematical properties that make it computationally infeasible for an attacker to compute the shared secret key from the public information exchanged between Alice and Bob.

The Diffie-Hellman protocol relies on modular arithmetic and the properties of finite cyclic groups, specifically the difficulty of the discrete logarithm problem. The protocol can be implemented using different mathematical structures, such as multiplicative groups of finite fields or elliptic curve groups. In this article, we will describe the original Diffie-Hellman protocol based on the multiplicative group of a finite field.

## The Protocol

The Diffie-Hellman key exchange protocol can be summarized in the following steps:

1. Initialization: Alice and Bob agree on two large prime numbers, `p` and `g`, where `g` is a primitive root modulo `p`. These numbers can be publicly known and shared between different instances of the protocol.
2. Private key generation: Alice and Bob each independently choose a random large number as their private key. Alice chooses `a`, and Bob chooses `b`. These private keys must remain secret and never be shared or transmitted over the public channel.
3. Public key generation: Alice computes her public key, `A`, as `A = g^a mod p`, and Bob computes his public key, `B`, as `B = g^b mod p`. Both Alice and Bob then exchange their public keys over the public channel.
4. Shared secret computation: Alice computes the shared secret key, `s`, as `s = B^a mod p`, and Bob computes the same shared secret key as `s = A^b mod p`. Due to the properties of modular arithmetic, both computations yield the same result, which is `s = g^(ab) mod p`. This shared secret key can now be used for secure communication between Alice and Bob.

## Security

The security of the Diffie-Hellman key exchange relies on the difficulty of solving the discrete logarithm problem. In the context of the protocol, this problem can be stated as follows: Given the public parameters `p`, `g`, `A`, and `B`, find the shared secret key `s`. This is equivalent to finding either Alice’s or Bob’s private key, `a` or `b`, respectively.

While there is no known efficient algorithm to solve the discrete logarithm problem in general, the security of the protocol depends on the choice of the parameters `p` and `g`. For example, the protocol is vulnerable to attacks if the prime number `p` is not large enough or if `g` is not a primitive root modulo `p`. To ensure the security of the protocol, it is recommended to use prime numbers with a length of at least 2048 bits and to follow established security guidelines when choosing the parameters.

It is important to note that the Diffie-Hellman key exchange only provides security against passive eavesdroppers who intercept the public communications between Alice and Bob. The protocol does not provide authentication, meaning that an active attacker could potentially impersonate one or both of the parties and perform a man-in-the-middle attack. To mitigate this risk, the Diffie-Hellman protocol can be combined with digital signatures or other authentication mechanisms to ensure the authenticity of the communicating parties.

## Variants and Applications

Since its inception, the Diffie-Hellman key exchange has been adapted and extended in various ways to improve its efficiency and security or to apply it to different cryptographic settings. Some notable variants and related protocols include:

• Elliptic Curve Diffie-Hellman (ECDH): This variant of the protocol is based on the properties of elliptic curve groups instead of the multiplicative group of a finite field. ECDH offers comparable security to the original Diffie-Hellman protocol with significantly smaller key sizes, resulting in faster computations and lower bandwidth requirements.
• Authenticated Diffie-Hellman: As mentioned earlier, the basic Diffie-Hellman protocol does not provide authentication. Authenticated variants of the protocol, such as the Station-to-Station (STS) protocol or the SIGMA protocol, combine the key exchange with digital signatures or other authentication mechanisms to prevent man-in-the-middle attacks.
• Diffie-Hellman in TLS: The Diffie-Hellman key exchange is used in various versions of the Transport Layer Security (TLS) protocol, which secures communication over the internet. TLS can use either the original Diffie-Hellman protocol or the Elliptic Curve Diffie-Hellman protocol for key exchange, combined with various authentication and encryption mechanisms.