Java Security

Hello everyone. I am a first time poster here. I'm just a few weeks away from college graduation and I've got a killer exam in cryptography due soon. I'm sure there must be people out there who enjoy doing problems like this. I'll post them below. If anyone knows the answers or can help me out, I'd really appreicate it. Thanks!!

1. (20) Ф(n) is the number of positive integers less than n and relatively prime to n. Determine the following
a. Ф(18)
b. Ф(5)
It can be shown that if gcd(m,n) = 1 then Ф(mn) = Ф(m) Ф(n). Use this property and the above result to determine the following
c. Ф(41)
d. Ф(90)


2. (20) This is an example of RSA algorithm in page 270 in which p = 17, q = 11, and e = 7. It is shown that d = 23. If the plaintext is 48, please calculate the ciphertext using the public key KU = {7, 187}. Verify your result by using the private key KR = {23, 187} to decrypt the ciphetext.


3. (20) In an RSA system, the public key of a given user is e = 5, n = 35. What is the private key of this user?


4. (20) What are two different uses of public-key cryptography related to key distribution? List four general categories of schemes for the distribution of public keys.


5. (20) Briefly explain Diffie-Hellman key exchange. User A and B use the Diffie-Hellman key exchange technique a common prime q = 71 and a primitive root α7.
a. If user A has private key X= 5, what is A�s public key Y? AA
b. If user B has private key X= 12, what is A�s public key Y? BB
c. What is shared secret key?