Theoretical Version Introduction
- The core idea of the Diffie-Hellman key exchange is to enable two parties to jointly generate a secret key without needing to send this secret key directly to each other.
Key Exchange Steps
- Choosing a Public Base: Initially, both parties agree on a public base number (g) and a large prime number (p), both of which are known to potential eavesdroppers.
- Generating Private Keys: Each party randomly selects a private key (denoted as a and b respectively) which remains secret and is not shared with anyone.
- Calculating and Sharing Public Keys: Each party uses their private key and the agreed-upon public base to calculate a public key. The client calculates A = g^a mod p, and the server calculates B = g^b mod p. Then, the client and server exchange these public keys with each other.
- Generating a Shared Secret Key: Each party can use the other’s public key and their own private key to calculate the same shared secret key. The client calculates S = B^a mod p, and the server calculates S = A^b mod p. Due to the properties of mathematics, the S values calculated by both sides are the same, and this S value is the shared secret key.
Simplified Version Introduction
For those who are still confused after the above explanation, do not be discouraged. It simply means that your life is filled with a variety of experiences, and you’ve spent less time on mathematics. In fact, if you don’t need to use this concept, it’s perfectly fine not to understand it for a lifetime. However, if you’re curious, here’s another way to explain it:
Key Exchange Steps
- Choosing Private Colors: Alice and Bobeach secretly choose a color (this corresponds to their private keys). Alice chooses red, and Bob chooses blue. They do not reveal their chosen colors to anyone.
- Preparing and Sending Mixed Colors: Alice mixes her secret color (red) (紅色) 和 with a public color (yellow) to create Alice’s mixed color (orange), which she then sends to Bob through an insecure mailing system. Similarly, Bob mixes his secret color (blue) with the public color (yellow) to create Bob’s mixed color (green), which he mails to Alice.
- Mixing Colors Again: When Alice receives Bob’s mixed color, she mixes it with her own secret color (red). Likewise, when Bob receives Alice’s mixed color, he mixes it with his own secret color (blue).
Result
- Alice’s final color
- = Bob’s mixed color (green) + her own secret color (red)
- = (Bob’s secret color (blue) + public color (yellow) ) + her own secret color (red)
- Bob’s final color
- = Alice’s mixed color (orange) + his own secret color (blue)
- = (Alice’s secret color (red) + public color (yellow) ) + his own secret color (blue)
- Finally, both Alice and Bob end up with the same final color, which is red ＋yellow ＋blue ＝ dark brown
- This final color corresponds to the shared secret key. Even if someone intercepts all their mail, without knowing their secret colors, they cannot replicate this final color.
Communicating Practically
- From then on, Alice would use a special material orange paper to mail letters to Bob，which Bob has to sprinkle blue mysterious ink on to reveal the dark brown text
- And Bob would use a special material green paper to mail letters to Alice, which Alice has to sprinkle red mysterious ink on to reveal the dark brown text
Recommended Reading
- For readers wanting to understand more details, I recommend this article:
- https://www.practicalnetworking.net/series/cryptography/diffie-hellman/
- Also, don’t miss the explanation in this tutorial video after the 1:15 mark, it’s quite straightforward and easy to understand.