gen-define-diffie-hellman-key-exchange-and-give-one-short-worked-c524

Define Diffie-Hellman key exchange and give one short worked example.

# Diffie-Hellman Key Exchange **Definition:** Diffie-Hellman is a cryptographic protocol that allows two parties to establish a shared secret key over an insecure channel without having previously shared any secrets. The security relies on the difficulty of computing discrete logarithms. ## How it Works 1. **Public parameters:** Both parties agree on a large prime number *p* and a generator *g* 2. **Private keys:** Each party generates a private random number 3. **Public keys:** Each party computes and exchanges a public value 4. **Shared secret:** Each party combines their private key with the other's public key to arrive at the same shared secret ## Worked Example **Setup:** - Public prime: *p* = 23 - Public generator: *g* = 5 **Alice's steps:** - Chooses private key: *a* = 6 - Computes public key: *A* = g^a mod p = 5^6 mod 23 = **8** - Sends 8 to Bob **Bob's steps:** - Chooses private key: *b* = 15 - Computes public key: *B* = g^b mod p = 5^15 mod 23 = **19** - Sends 19 to Alice **Shared secret calculation:** - Alice computes: *B^a* mod *p* = 19^6 mod 23 = **2** - Bob computes: *A^b* mod *p* = 8^15 mod 23 = **2** **Result:** Both parties now share the secret key **2** without ever transmitting it!