Problems

1.5. As we learned in this chapter, modular arithmetic is the basis of many cryptosystems. As a consequence, we will address this topic with several problems in this and upcoming chapters. Lets start with an easy one: Compute the result without a calculator. 1. 15 29 mod 13 2. 2 29 mod 13 3. 2 3 mod 13 4. -11 3 mod 13 The results should be given in the range from 0,1,..., modulus-1. Briefly describe the relation between the different parts of the problem.

1.6. Compute without a calculator: 1. 1/5 mod 13 2. 1/5 mod 7 3. 3 2/5 mod 7

1.7. We consider the ring Z4. Construct a table which describes the addition of all elements in the ring with each other:

+	0 1 2 3
0 1 2 3	0 1 2 3 1 1 2 2

1. Construct the multiplication table for Z4. 2. Construct the addition and multiplication tables for Z5. 3. Construct the addition and multiplication tables for Z6. 4. There are elements in Z4 and Z6 without a multiplicative inverse. Which elements are these? Why does a multiplicative inverse exist for all nonzero elements in Z5?