Evaluation of Authentication Algorithms for Small Devices

2.2 SSH authentication using Diffie-Hellman Key
Exchange
Here we will have a closer look at the SSH-Protocol, and specially the Diffie-Hellman
Key Exchange[5]. The SSH-Protocol is a typical example for a Challenge-Response
Protocol, where the claimant is challenged by the verifier to respond with username
and password. In a first phase the protocol uses the Diffie-Hellman-Keyexchange
to establish a session key, which is used to encrypt further traffic. We assume this
protocol being secure, under the assumption that there exists no effective algorithm
to calculate a discrete logarithm, and not man in the middle attack only passive
listening.
The Algorithm works as follows:
• Alice (A) selects a big value p, a generator value g and a Secret x with 1 ≤
x ≤ (p − 1)
• A calculates u = g x mod p
• A sends g,p,u to Bob (B)
• B selects a secret y
• B calculates k = z y and v = g y mod p
• B sends v to A
• A calculates k = v x
In the end both have the same secret the can use as a key k beeing:
k = g xy mod p = (g x mod p) y = (g y mod p) x
The secret can be used for further secure communications e.g. to exchange authentically
certificates or cryptographic keys.
Part I
A
B
1 p ∈ prime −→ 2 p ∈ prime 1
1 g ∈ R {2 . . . p − 2} −→ g ∈ R {2 . . . p − 2} 1
1 x ∈ R {2 . . . p − 2} y ∈ R {2 . . . p − 2} 1
(t + h − 2) u = g x mod p −→
←− v = g y mod p (t + h − 2)
+(2n − n)(n + 3)
t + h − 2 k = (g y ) x mod n k = (g x ) y mod n t + h − 2
n 2 + 7n + 7 total multiplications n 2 + 7n + 7