1. Suppose that n = 10088821 is a product of two distinct primes, and Φ(n) = 10082272. Determine the prime factors of n.

2. It is easy to show that the converse of Fermat's Theorem does not hold; i.e., the congruence aⁿ ≡ a (mod n) for all a does not imply that n is prime. A composite number nthat satisfy this congruence is called a pseudoprime to base a.

Verify that 341 is a pseudoprime to base 2.
Verify that 91 is a pseudoprime to base 3.
Show that if p is prime and 2p - 1 iscomposite, then2p - 1is a pseudoprime to base 2.

3. Show that every prime is either in the form 4k + 1 or 4k + 3, where k is a positive integer.

4. Find the results of the following, using Euler's theorem:
12-1 mod 77
16-1 mod 323
20-1 mod 403
44-1 mod 667

5. What are the last three digits of 7⁸⁰³ ?

6. Perform the encryption and decryption using the RSA algorithm for the following:
p=11;q=13;e=11;M=7
p=11;q=13;e=11;M=7

7. Please answer the following questions.
List and briefly describe categories of security mechanisms.
List and briefly describe categories of security services.
What is the role of a compression function in a hash function?
What are the three broad categories of applications of public-key cryptosystems?