1. Diffie-Hellman Key Exchange
CSCI 5857: Encoding and Encryption

2. Outline Key exchange without public/private keys
Public and private components of Diffie-Hellman
Secure information exchange in Diffie-Hellman
Underlying mathematics
Man-in-the-middle attack
Station-to-station key exchange

3. Diffie-Hellman Key Exchange
Common goal of public key encryption: Securely agree upon a symmetric key
Bob generates symmetric key KS
Encrypts with Alice’s public key KAPU and sends to Alice
Alice decrypts with her private key KAPR
Then use KS to exchange information (using AES, 3DES, etc.)
Problem: What if neither Alice or Bob have a public key?
Diffie-Hellman key exchange (1976 – precedes RSA)
Allows two people to securely generate a symmetric key without a preexisting public key
Based on modular logarithms

4. Secure Key Generation
Alice, Bob exchange data to securely generate a value
Data transmitted doesn’t allow others to find that value
That value used as symmetric key to send further information
Public info
Public info
Private info
Private info
generator
generator D P E P
Esymmetric (P, kS)

5. Public and Private Information
Public information (known to Alice, Bob, and everyone):
p: large prime number (at least 1024 bits)
g: Primitive root “generator” (g < p)
Private information
x: random number created (and only known) by Alice
y: random number created (and only known) by Bob
x and y used to generate shared key k
Knows p, g
Generates x
Knows p, g
Generates y

6. Key Generation Alice computes R1 = gx mod p Bob computes R2 = gy mod p
Alice sends R1 to Bob
Bob sends R2 to Alice

7. Security of Key Generation
Darth cannot derive x from R1 or y from R2
Would have to solve modular logarithm problem
x = logg (R1 mod p)
y = logg (R2 mod p)

8. Key Computation Alice computes k = R2 x mod p
Bob computes k = R1 y mod p
Alice, Bob now have shared key k
Nobody else can compute without knowing x or y
No secret information transmitted!

9. Diffie-Hellman Mathematics
Alice’s POV: k = R2 x mod p
= (gy mod p)x mod p
= gyx mod p
Bob’s POV: k = R1 y mod p
= (gx mod p)y mod p
= gxy mod p
gyx mod p = gxy mod p = k

10. Diffie-Hellman Example
Public key: g = 7, p = 23
Chooses x = 3
R1 = 73 mod 23 = 21
Chooses y = 6
R2 = 76 mod 23 = 4 4 21
K = 43 mod 23 = 18
K = 216 mod 23 = 18

11. Man-in-the-Middle Attack
Most serious weakness in Diffie-Hellman
Assumes Darth has ability to:
Intercept messages between Alice and Bob
Masquerade as Alice or Bob to send messages to the other
“I am Bob”
“I am Alice”

12. Man-in-the-Middle Attack
Darth generates own random value z
Computes own R3 = gz mod p from public values of p, g
Goal: Trick Alice and Bob into using keys he has created from z

13. Man-in-the-Middle Attack
Darth intercepts R1 sent by Alice and R2 sent by Bob
Computes kAlice = R1 z mod p
Computes kBob = R2 z mod p R1 R2
z R3
kAlice kBob x y

14. Man-in-the-Middle Attack
Darth sends R3 to Alice posing as Bob
Darth sends R3 to Bob posing as Alice
Alice computes kAlice = R3 x mod p
Bob computes kBob = R3 y mod p R3 R3
kBob
kAlice
kAlice kBob

15. Man-in-the-Middle Attack
Darth can read messages sent by Alice and Bob!
Example: Message sent from Alice to Bob
Alice encrypts with kAlice believing it is secure
Darth intercepts and decrypts with kAlice
Re-encrypts with kBob and sends to Bob (posing as Alice
C = E(P, kAlice)
C = E(P, kBob)
P = D(C, kAlice)

16. Station-to-Station Key Agreement
Participants in Diffie-Hellman must authenticate their identities
Only solution to Man-in-the-Middle attack
Authentication usually based on certificates
Signed by trusted authorities
Contain public keys for participants

17. Station-to-Station Key Agreement