Comp 3704
Introduction to Cryptography
Homework 6
Due Tuesday, February 18, 2003

1. Read the article handed out in class.
2. List the elements of Z₁₂^* and give all of their orders. Do the same for Z₁₁^* and Z₁₅^*. Which of these three multiplicative groups are cyclic? For each of these groups that are cyclic, list all of the primitive elements. Then pick one of the primitive elements, and show how to get all of the other primitive elements from it. The purpose of this problem is to get you to experiment with the structure of these groups, and to better understand the concepts "cyclic" and "primitive element".
3. Show that 2 is a primitive root mod 29.
4. Calculate 35²⁶⁶⁶ mod 72. Use Corollary 5.5 (p. 164) to simplify the calculation. Now explain why Corollary 5.5 cannot be used to calculate 10²⁶⁶⁶ mod 72.
5. Calculate 45²¹² mod 211. Take advantage of Fermat's little theorem to do this problem.
6. Do problem 5.10 from the text. This is a very important problem. It shows that RSA really is a cryptosystem. Optional extension to this problem: problem 5.11
7. Do problem 5.6 from the text. (optional: do 5.7 as well)
8. Use Euler's criterion to determine if 5 is a quadratic residue mod 67. Please note that 67 is indeed an odd prime.