Exercise 2. This software computes the elements of the subgroup U(n)k = {xk | x U(n)} of U(n) and its order. Run the program for (n,k) = (27,3), (27,5), (27,7), and (27,11). Do you see a relationship connecting|U(n)| and |U(n)k|, phi(n), and k? Make a conjecture. Run the program for (n,k) = (25,3), (25,5), (25,7), and (25,11). Do you see a relationship connecting |U(n)| and |U(n)k|, phi(n), and k? Make a conjecture. Run the program for (n,k) = (32,2), (32,4), and (32,8). Is your conjecture valid for U(32,16)? If not, restrict your conjecture. Run the program for (n,k) = (77,2), (77,3), (77,5), (77,6), (77,10), and (77,15)? Do you see a relationship among U(77,6) and U(77,2), and U(77,3)? What about U(77,10), U(77,2), and U(77,5)? What about U(77,15), U(77,3), and U(77,5)? Make a conjecture. Use the theory developed in this chapter about expressing U(n) as external direct products of cyclic groups of the form Zn to analyze these groups to verify your conjectures.Exercise 3. This software implements the algorithm given in Chapter 8 to express U(n) as an external direct product of groups of the form Zk. Assume that n is given in prime-power factorization form. Run your program for 3 . 5 . 7, 16 . 9 . 5, 8 . 3 . 25, 9 . 5 . 11, and 2 . 27 . 125. [ NOTE: Please enter the prime-power factorization form with a `period(".")' in between the integers and without any space. Also, this program has been written to accept n as any integer, i.e., instead of entering n in the factored form as 3 . 5 . 7 you could enter 105 . ]Exercise 5. This program implements the RSA public key cryptography scheme. The user enters two primes p and q, an r that is relatively prime to m = lcm(p -1,q -1), and the message M to be sent. Then the program computes the s which is the inverse of r mod m, and the value of Mr mod pq. Then the user can input those numbers and have the computer raise the numbers to the s power to obtain the original input.

Week 4 notes (Test was on Wednesday so this only includes Friday’s lecture) Horsetails -are everywhere except Asia and Australia -1 genus has 30 species -Were diverse during the carboniferous period Seeds -contains embryo surrounded by nutrients with a covering to protect it -Plants were able to become dominant produces with the evolution of the seed -seeds...