In this problem, we explore the Diffie-Hellman (DH) public-key encryption
algorithm, which allows two entities to agree on a shared key. The
DH algorithm makes use of a large prime number p and another large number
g less than p. Both p and g are made public (so that an attacker would know
them). In DH, Alice and Bob each independently choose secret keys, SA and
SB, respectively. Alice then computes her public key, TA, by raising g to SA
and
then taking mod p. Bob similarly computes his own public key TB by raising g
to SB and then taking mod p. Alice and Bob then exchange their public keys
over the Internet. Alice then calculates the shared secret key S by raising TB
to
SA and then taking mod p. Similarly, Bob calculates the shared key S´ by
raising TA to SB and then taking mod p.
a. Prove that, in general, Alice and Bob obtain the same symmetric key, that
is, prove S = S´.
b. With p = 11 and g = 2, suppose Alice and Bob choose private keys SA = 5
and SB = 12, respectively. Calculate Alice’s and Bob’s public keys, TA and
TB . Show all work.
c. Following up on part (b), now calculate S as the shared symmetric key. Show
all work.
d. Provide a timing diagram that shows how Diffie-Hellman can be attacked by
a man-in-the-middle. The timing diagram should have three vertical lines, one
for Alice, one for Bob, and one for the attacker Trudy
