Diffie-Hellman Key Exchange Protocol

Goal

To have an efficient security communication, we need to have a shared secret. with a unsecure channel

How does it work

We use the following operation: g ^ s % p

• is hard to inverse (discrete logarithm problem)
• can be composed
  • (g ^ s1 % p)^ s2 % p == (g ^ s2 % p)^ s1 % p

Steps

• Both users agree on the parameters g and p. They can be publicly shared
• Then each user fetch a secret number (s1, s2) (private key)
• With that, each user generates the public key g ^ s % p
• and you get the public key of the other user and apply the pk ^ s %p
• and since both users used the same parameters but in different order…

Example

example of:

• p = 23
• g = 5

p = 23  # small prime
g = 5   # generator
 
# Alice's keys
alice_private = 6
alice_public = pow(g, alice_private, p)  # g^a mod p
 
# Bob's keys  
bob_private = 15
bob_public = pow(g, bob_private, p)      # g^b mod p
 
# Shared secret calculation
alice_shared = pow(bob_public, alice_private, p)    # (g^b)^a mod p
bob_shared = pow(alice_public, bob_private, p)      # (g^a)^b mod p
 
print(f"Alice public: {alice_public}")
print(f"Bob public: {bob_public}")
print(f"Shared secret: {alice_shared} == {bob_shared}")