1. Diffie-Hellman Algorithm
Network Security

2. Diffie-Hellman Algorithm
Objectives of the Topic
After completing this topic, a student will be able to
describe the Diffie-Hellman algorithm.

3. Diffie-Hellman Algorithm
Figures and material in this topic have been adapted from
“Network Security Essentials : Applications and Standards”, 2014, by William Stallings.

4. Diffie-Hellman Algorithm
The first published public-key algorithm that defined public-key cryptography was published by Diffie and Hellman.
It is generally referred to as the Diffie –Hellman key exchange.

5. Diffie-Hellman Algorithm
A number of commercial products employ this key exchange technique.

6. Diffie-Hellman Algorithm
Purpose of the algorithm is to enable two users to exchange a secret key securely that then can be used for subsequent encryption of messages
The algorithm itself is limited to the exchange of the keys.

7. Diffie-Hellman Algorithm
Depends for its effectiveness on the difficulty of computing discrete logarithms.

8. Diffie-Hellman Algorithm
The Algorithm
Lets assume that there are two publicly known numbers: a prime number q and an integer α that is a primitive root of q.

9. Diffie-Hellman Algorithm
For a prime number p, if α is a primitive root of p, then α, α2,…, αp-1 are distinct (mod p).
E.g. for prime number 19, its primitive roots are 2, 3, 10, 13, 14, and 15.
Suppose the users A and B wish to create a shared key.

10. Diffie-Hellman Algorithm
User A selects a random integer XA < q and computes YA = αXA mod q.
Similarly, user B independently selects a random integer XB < q and computes YB = αXB mod q.

11. Diffie-Hellman Algorithm
Each side keeps the X value private and makes the Y value available publicly to the other side.
Thus, XA is A’s private key and YA is A’s corresponding public key,
The same applies for B.

12. Diffie-Hellman Algorithm
User A computes the key as K = (YB)XA mod q and
User B computes the key as K = (YA)XB mod q.
The result is that the two sides have exchanged a secret value.

13. Diffie-Hellman Algorithm

14. Diffie-Hellman Algorithm
Example:
Lets take q = 353 and a primitive root of 353, α= 3.
A and B select private keys XA = 97 and XB = 233, respectively.

15. Diffie-Hellman Algorithm
Each computes its public key:
A computes YA = 397 mod 353 = 40.
B computes YB = 3233 mod 353 = 248.
After they exchange public keys, each can compute the common secret key:

16. Diffie-Hellman Algorithm
A computes K = (YB)XA mod 353 = mod 353 = 160.
B computes K = (YA)XB mod 353 = mod 353 = 160.
an attacker would have available the following information:
q = 353; α = 3; YA = 40; YB = 248.

17. Diffie-Hellman Algorithm
In this simple example, it would be possible by brute force to determine the secret key 160.

18. Diffie-Hellman Algorithm
Security of the Diffie-Hellman algorithm:
While it is relatively easy to calculate exponentials modulo a prime, it is very difficult to calculate discrete logarithms.
For large primes, the latter task is considered infeasible.
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