The Diffie-Hellman key exchange algorithm establishes a shared secret key between two parties over an unsecured communication channel.
In this article, we will solve a couple of Diffie-Hellman key exchange algorithm examples. It will give you a better idea about this algorithm.
Example 1
John and Ricky use the Diffie-Hellman protocol with a common prime number p = 11 and a primitive root g = 6. If John’s private key ‘a’ = 3 and Ricky’s private key ‘b’ = 2 then find out:
- John’s public key ‘A’
- Ricky’s public key ‘B’
- Shared secret key
Solution:
Step 1. Calculate John’s public key ‘A’ using:
A = g ^ a mod p
= 6 ^ 3 mod 11
= 216 mod 11
A = 7
You can use this online Modulo calculator to find out the mod.
Step 2. Calculate Ricky’s public key ‘B’ using:
B = g ^ b mod p
= 6 ^ 2 mod 11
= 36 mod 11
B = 3
Step 3. Calculate John’s shared secret key using Ricky’s public key B:
S = B ^ a mod p
= 3 ^ 3 mod 11
= 27 mod 11
= 5
Shared secret key of John = 5
Step 4. Calculate Ricky’s shared secret key using John’s public key A:
S = A ^ b mod p
= 7 ^ 2 mod 11
= 49 mod 11
= 5
Shared secret key of Ricky = 5
John and Ricky have the same shared secret key i.e. 5 without transferring their private key.
Example 2
User 1 and User 2 decided to use the Diffie-Hellman algorithm to establish a shared secret key. They selected their prime number p = 13 and its primitive root g = 2. User 1’s private key is 5 and User 2’s private key is 6. Find out:
- The public key of User 1 & User 2
- Shared secret key between User 1 & User 2
Solution:
Given:
- prime number (p) = 13
- primitive root (g) = 2
- User 1 private key (a) = 5
- User 2 private key (b) = 6
Step 1: Calculate the public key of User 1:
Let the public key of User 1 be ‘A’.
A = g ^ a mod p
= 2 ^ 5 mod 13
= 32 mod 13
A = 6
Step 2: Calculate the public key of User 2:
Let the public key of User 2 be ‘B’:
B = g ^ b mod p
= 2 ^ 6 mod 13
= 64 mod 13
B = 12
Step 3: Calculate User 1’s shared secret using User 2’s public key B:
S = B ^ a mod p
= 12 ^ 5 mod 13
= 248832 mod 13
= 12
Step 4: Calculate User 2’s shared secret key using User 1’s public key A:
S = A ^ b mod p
= 6 ^ 6 mod 13
= 46656 mod 13
= 12
Hence, both User 1 and User 2 have the same shared secret key i.e. 12.
Example 3
Alice and Bob want to establish a shared secret key using Diffie-Hellman key exchange protocol. Both decided on a prime number ‘p’ as 23 and its primitive root ‘g’ as 5. Alice’s private key is 9 and Bob’s private key is 11. Find out the public key of Alice & Bob and also the shared secret key between them.
Solution:
Given:
- Prime number (p) = 23
- Primitive root (g) = 5
- Alice’s private key (a) = 9
- Bob’s private key (b) = 11
Step 1: Calculate Alice’s public key using:
Let Alice’s public key be ‘A’.
A = g ^ a mod p
= 5 ^ 9 mod 23
= 1953125 mod 23
A = 11
Alice’s public key = 11
Step 2: Calculate Bob’s public key using:
Let Bob’s public key be ‘B’.
B = g ^ b mod p
= 5 ^ 11 mod 23
= 48828125 mod 23
B = 22
Bob’s public key = 22
Step 3: Calculate Alice’s shared secret key using Bob’s public key B:
S = B ^ a mod p
= 22 ^ 9 mod 23
= 1,207,269,217,792 mod 23
S = 22
Alice’s shared secret key = 22
Step 4: Calculate Bob’s shared secret key using Alice’s public key A:
S = A ^ b mod p
= 11 ^ 11 mod 23
= 285,311,670,611 mod 23
S = 22
Bob’s shared secret key = 22
Hence, Alice and Bob have the same shared secret key i.e. 22 using Diffie-Hellman key exchange algorithm.
If you found any error in the above examples, please let us know in the comment section below.