17.2.4. Easy Computation in Certain Cyclic Groups

Try these first

1. 1.

   What is the discrete logarithm of the identity element in a cyclic group?

   Hint

   Recall the properties of group elements.

2. 2.

   Explain why every integer except zero can be a generator in Z_p.

   Hint

   Think about prime properties.

3. 3.

   What is the discrete logarithm of an element in a cyclic group?

   • An integer x where g^x = y
   • An element in the group
   • The order of the group

   Hint

   Consider the definition provided earlier.

4. 4.

   True or False: The discrete logarithm can be computed easily in any cyclic group.

   • True
   • False

   Hint

   Recall the discussion about groups that complicate these calculations.

5. 5.

   Analyze the discrete log problem in both easy and hard cyclic groups. Provide examples for each and explain the underlying properties.

   Hint

   Focus on properties of prime moduli versus composite structures.

6. 6.

   Given a cyclic group defined by a prime number p, devise a method for securely exchanging keys using the discrete log approach.

   Hint

   Reflect on the practicality of sending public values openly.