1) Alice and Bob use the Diffie–Hellman key exchange technique with a common prime q = 1 5 7 and a primitive root a = 5. a. If Alice has a private key XA = 15, find her public key YA. b. If Bob has a private key XB = 27, find his public key YB. c. What is the shared secret key between Alice and Bob?

2) Alice and Bob use the Diffie-Hellman key exchange technique with a common prime q = 2 3 and a primitive root a = 5 . a. If Bob has a public key YB = 1 0 , what is Bob’s private key YB? b. If Alice has a public key YA = 8 , what is the shared key K with Bob? c. Show that 5 is a primitive root of 23.

3) In the Diffie–Hellman protocol, each participant selects a secret number x and sends the other participant ax mod q for some public number a. What would happen if the participants sent each other xa for some public number a instead? Give at least one method Alice and Bob could use to agree on a key. Can Eve break your system without finding the secret numbers? Can Eve find the secret numbers?

4) This problem illustrates the point that the Diffie–Hellman protocol is not secure without the step where you take the modulus; i.e. the “Indiscrete Log Problem” is not a hard problem! You are Eve and have captured Alice and Bob and imprisoned them. You overhear the following dialog. Bob: Oh, let’s not bother with the prime in the Diffie–Hellman protocol, it will make things easier. Alice: Okay, but we still need a base a to raise things to. How about a = 3? Bob: All right, then my result is 27. Alice: And mine is 243. What is Bob’s private key XB and Alice’s private key XA? What is their secret combined key? (Don’t forget to show your work.)

5) Section 10.1 describes a man-in-the-middle attack on the Diffie–Hellman key exchange protocol in which the adversary generates two public–private key pairs for the attack. Could the same attack be accomplished with one pair? Explain.

6) Is (5, 12) a point on the elliptic curve y2 = x 3 + 4 x – 1 over real numbers?

7) This problem performs elliptic curve encryption/decryption using the scheme outlined in Section 10.4. The cryptosystem parameters are E11(1, 7) and G = (3, 2). B’s private key is nB = 7. a. Find B’s public key PB. b. A wishes to encrypt the message Pm = (10, 7) and chooses the random value k = 5. Determine the ciphertext Cm. c. Show the calculation by which B recovers Pm from Cm.

8) The following is a first attempt at an elliptic curve signature scheme. We have a global elliptic curve, prime p, and “generator” G. Alice picks a private signing key XA and forms the public verifying key YA = XAG. To sign a message M: ■ Alice picks a value k. ■ Alice sends Bob M, k, and the signature S = M – kXAG. ■ Bob verifies that M = S + kYA. a. Show that this scheme works. That is, show that the verification process produces an equality if the signature is valid. b. Show that the scheme is unacceptable by describing a simple technique for forging a user’s signature on an arbitrary message.

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