How many elements does each of the multiplicative groups have?

Other by Geoffrey

8.1. Understanding the functionality of groups, cyclic groups and subgroups is im­portant for the use of public-key cryptosystems based on the discrete logarithm problem. That’s why we are going to practice some arithmetic in such structures in this set of problems.

Let’s start with an easy one. Determine the order of all elements of the multi­plicative groups of:

1.765

  1. Z;
  2. Zj3

Create a list with two columns for every group, where each row contains an element a and the order ord(a).

(Hint: In order to get familiar with cyclic groups and their properties, it is a good idea to compute all orders “by hand”, i.e., use only a pocket calculator. If you want to refresh your mental arithmetic skills, try not to use a calculator whenever possible, in particular for the first two groups.)

8.2. We consider the group Z53. What arc the possible element orders? How many elements exist for each order?

8.3. We now study the groups from Problem 8.2.

  1. How many elements does each of the multiplicative groups have?
  2. Do all orders from above divide the number of elements in the corresponding multiplicative group?
  3. Which of the elements from Problem 8.1 are primitive elements?

Verify for the groups that the number of primitive elements is given by 0 (1 Zr* l)