Diffie–Hellman Key Exchange

Diffie–Hellman Key Exchange (DHKE)

**Diffie–Hellman Key Exchange** (DHKE) is a cryptographic method to **securely exchange cryptographic keys** (key agreement protocol) over a public (insecure) channel in a way that overheard communication does not reveal the keys. The exchanged keys are used later for encrypted communication (e.g. using a symmetric cipher like AES).

The Diffie–Hellman (DH) method is **anonymous key agreement scheme**: it allows two parties that have no prior knowledge of each other to jointly establish a **shared secret key over an insecure channel**.

Note that the DHKE method is **resistant to** **sniffing attacks** (data interception), but it is vulnerable to **man-in-the-middle attacks** (attacker secretly relays and possibly **alters the communication** between two parties).

Key Exchange by Mixing Colors

The Diffie–Hellman Key Exchange protocol is very similar to the concept of "**key exchanging by mixing colors**", which has a good visual representation, which simplifies its understanding. This is why we shall first explain how to exchange a secret color by **color mixing**.

The design of color mixing key exchange scheme assumes that if we have two liquids of different colors, we can **easily mix the colors** and obtain a new color, but the reverse operation is almost impossible: **no way to separate the mixed colors** back to their original color components.

This is the color exchange **scenario**, step by step:

**Alice**and**Bob**, agree on an arbitrary**starting (shared) color**that does not need to be kept secret (e.g.*yellow*).**Alice**and**Bob**separately select a**secret color**that they keep to themselves (e.g.*red*and*sea green*).- Finally
**Alice**and**Bob****mix**their secret color together with their mutually shared color. The obtained mixed colors area ready for public exchange (in our case*orange*and*light sky blue*).

The next steps in the color exchanging scenario are as follows:

**Alice**and**Bob**publicly**exchange**their two**mixed colors**.- We assume that there is no efficient way to extract (separate) the secret color from the mixed color, so third parties who know the mixed colors cannot reveal the secret colors.

- Finally,
**Alice**and**Bob**mix together the color they received from the partner with their own secret color.- The result is the
**final color mixture**(*yellow-brown*) which is identical to the partner's color mixture. - It is the
**securely exchanged shared key**.

If a third parties have intercepted the color exchanging process, it would be computationally difficult for them to determine the secret colors.

The **Diffie-Hellman Key Exchange** protocol is based on similar concept, but uses **discrete logarithms** and **modular exponentiations** instead of color mixing.

The Diffie-Hellman Key Exchange (DHKE) Protocol

Now, let's explain how the **DHKE** protocol works.

The Math behind DHKE

(**ga**)**b** mod **p** = (**gb**)**a** mod **p**

where **g**, **a**, **b** and **p** are positive integers.

If we have **A** = **ga** mod **p** and **B** = **gb** mod **p**, we can calculate **gab** mod **p**, without revealing **a** or **b** (which are called **secret exponents**).

In computing theory, these is no efficient algorithm which can find a secret exponent. If we have **m**, **g** and **p** from the below equation:

there is no efficient (fast) algorithm to find the secret exponent **s**. This is known as the Discrete Logartihm Problem (DLP).

Discrete Logarithm Problem (DLP)

The **Discrete Logarithm Problem (DLP)** in computer science is defined as follows:

- By given element
and value*b*=*a*find the exponent*bx*(if it exists)*x*

The exponent *x* is called **discrete logarithm**, i.e. **x** = *log***b**(**a**). The elements *a* and *b* can be simple integers modulo *p* (from the group ℤ/pℤ) or elements of finite cyclic multiplicative group **G** (modulo *p*), where *p* is typically a prime number.

In cryptography, many algorithms rely on the **computational difficulty of the DLP problem** over carefully chosen group, for which **no efficient algorithm exists**.

The DHKE Protocol

Now, after we are familiar with the above mathematical properties of the modular exponentiations, we are ready to explain **the DHKE protocol**. This is how it works:

Let's explain each step of this key-exchange process:

- Alice and Bob agree to use two public integers:
**modulus p**and**base g**(where**p**is prime, and**g**is a primitive root modulo**p**).- For example, let
**p**= 23 and**g**= 5. - The integers
**g**and**p**are public, typically hard-coded constants in the source code.

- Alice chooses a
**secret integer a**(e.g.**a**= 4), then calculates and sends to Bob the number**A = ga mod p**.- The number
**A**is public. It is sent over the public channel and its interception cannot reveal the secret exponent**a**. - In our case we have:
**A**= 54 mod 23 = 4.

- Bob chooses a
**secret integer b**(e.g.**b**= 3), then calculates and sends to Alice the number**B = gb mod p**.- In our case we have:
**B**= 53 mod 23 = 10

- Alice computes s = Ba mod p
- In our example:
**s**= 104 mod 23 =**18**

- Bob computes s = Ab mod p
- In our example:
**s**= 43 mod 23 =**18**

- Alice and Bob now share a
**secret number s****s**= Ab mod p = Ba mod p = (ga)b mod p = (gb)a mod p = gab mod p =**18**- The shared secret key
**s**cannot be computed from the publicly available numbers**A**and**B**, because the secret exponents**a**and**b**cannot be efficiently calculated.

In the most common implementation of DHKE (following the RFC 3526) the base is **g** = **2** and the modulus **p** is a large **prime number** (1536 ... 8192 bits).

Security of the DHKE Protocol

The DHKE protocol is based on the practical difficulty of the Diffie–Hellman problem, which is a variant of the well known in the computer science DLP (discrete logarithm problem), for which no efficient algorithm still exists.

DHKE exchanges a **non-secret sequence of integer numbers** over insecure, public (sniffable) channel (such as signal going through a cable or propagated by waves in the air), but does not reveal the secretly-exchanged shared private key.

Again, be warned that DHKE protocol in its classical form is **vulnerable to** **man-in-the-middle attacks**, where a hacker can intercept and modify the messages exchanged between the parties.

Finally, note that the integers **g**, **p**, **a** and **p** are typically very big numbers (1024, 2048 or 4096 bits or even bigger) and this makes the **brute-force attacks** non-sense.

DHKE - Live Example

As live example, you can play with this online DHKE tool: http://www.irongeek.com/diffie-hellman.php

ECDH - Elliptic Curves-based Diffie-Hellman Key Exchange Protocol

The **Elliptic-Curve Diffie–Hellman (ECDH)** is an anonymous key agreement protocol that allows two parties, each having an **elliptic-curve public–private key pair**, to establish a shared secret over an insecure channel.

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Contents

Diffie–Hellman Key Exchange (DHKE)

Key Exchange by Mixing Colors

The Diffie-Hellman Key Exchange (DHKE) Protocol

The Math behind DHKE

Discrete Logarithm Problem (DLP)

The DHKE Protocol

Security of the DHKE Protocol

DHKE - Live Example

ECDH - Elliptic Curves-based Diffie-Hellman Key Exchange Protocol