This article is a part of a series on Cryptography. Use the navigation boxes to view the rest of the articles.

How can two people in a crowded room derive a secret that only the pair know, without revealing the secret to anyone else that might be listening?

That is exactly the scenario the Diffie-Hellman Key Exchange exists to solve.

The **Diffie-Hellman Key Exchange is a means for two parties to jointly establish a shared secret over an unsecure channel**, without having any prior knowledge of each other.

They never actually exchange the secret, just some values that both combine which let them attain the same resulting value.

Conceptually, the best way to visualize the Diffie-Hellman Key Exchange is with the ubiquitous paint color mixing demonstration. It is worth quickly reviewing it if you are unfamiliar with it.

However, in this article we want to go a step further and actually show you the math in the Diffie-Hellman Key Exchange.

### DH Math

Before you get into the math of Diffie-Hellman, you will want to have a basic understanding of what a **Prime** number is, and what the **Modulus** operation is (aka, remainder division). Both of these terms have been defined in another article.

Below is an infographic outlining all the steps of the Diffie-Hellman exchange between Alice and Bob.

Notice how both Alice and Bob were able to attain the same Shared Secret of 3. Anyone listening in on their DH Key exchange would only know the Public Values, and the starting P and G values. There is no consistent way to combine those numbers (13, 6, 2, 9) to attain 3.

### DH Numbers

In our example, we used a Prime number of 13. Since this Prime number is also is used as the Modulus for each calculation, the entire key space for the resulting Shared Secret can only ever be 0-12. The bigger this number, the more difficult a time an attacker will have in brute forcing your shared secret.

Obviously, we were using very small numbers above to help keep the math relatively simple. True DH exchanges are doing math on numbers which are vastly larger. There are three typical sizes to the numbers in Diffie-Hellman:

DH Group 1 | 768 bits |

DH Group 2 | 1024 bits |

DH Group 5 | 1536 bits |

The bit-size is a reference to the Prime number. This directly equates to the entire key space of the resulting Shared Secret. To give you an idea of just how large this key space is:

In order to fully write out a 768 bit number, you would need 232 decimal digits.

In order to fully write out a 1024 bit number, you would need 309 decimal digits.

In order to fully write out a 1536 bit number, you would need 463 decimal digits.

### Using the Shared Secret

Once the Shared Secret has been attained, it typically becomes used in the calculation to establish a joint Symmetric Encryption key and/or a joint HMAC Key – also known as Session Keys.

But it is important to point out that the Shared Secret itself should not directly be used as the Secret Key. If it were, all you can be assured of is that throughout the secure conversation you are still speaking to the same party that was on the other side of the Diffie-Hellman exchange.

However, you still have no confirmation or assurance as to who the other party is. Just that no one else can all of a sudden pretend to be them in the middle of your secure conversation.

The generation of the actual Session Keys should include the DH Shared Secret, along with some other value that would only be known to the intended other party, like something from the Authentication scheme you chose.

Great series on Cryptography! I found you guys via the ARP video on youTube which was very good too. I hope you guys will add other topics soon. Looking forward to it!

Glad you enjoyed the videos and the article series, Roger. The Network Address Translation articles are releasing this week! Sign up to get an e-mail when it comes out.

Ed- In the Alice / Bob example, I randomly selected a different private key to do the math as an exercise. I ran into a strange combination with these numbers-

I freakishly random selected Alice private key as 8 and Bob’s private key as 3. When I calculate the public keys, it turns out that Alice gets a public key of 3 and Bob has a public key of 8.

Alice Public key = 6^8 mod 13 = 3

Bob Public key = 6^3 mod 12 = 8

Scratching my head what’s so strange about these numbers, to reveal the private keys. Hope I did that math right.

Any thoughts are appreciated. Again, great series. Thanks much.

– m

Hi M,

Yea, I’m getting the same 3/8 8/3 oddity there.

All I can say is you run into these strange occurrences when you’re using small numbers. When you’re doing MOD P with a P of 13, it means there are only 13 possible values that can be the result. So you’re bound to run into a bunch of overlaps.

With “big number” DH, there are also certain combinations that are known to be not secure, and most implementations of DH explicitly/manually account for avoiding those combinations. That is part of why it’s dangerous to try to “roll your own Crypto” — I could create a simple bash script or excel spreadsheet to do the math above, but I’ll likely miss out on industry best practices in regards to DH.

Glad you enjoyed the series =)