17.2. More Applications of Groups

Try these first

1. 1.

   What is the definition of a discrete logarithm?

   Hint

   Think about how traditional logarithms work.

2. 2.

   In a cyclic group, what does a generator do?

   Hint

   Recall the example of powers in mathematics.

3. 3.

   What does the discrete logarithm represent in a cyclic group?

   • The base of the logarithm
   • The exponent that satisfies g^x = y
   • The value of y

   Hint

   Consider what exponent means in mathematics.

4. 4.

   True or False: The Diffie-Hellman protocol allows two parties to establish a shared key without sharing their private keys.

   • True
   • False

   Hint

   Think about the purpose of the protocol.

5. 5.

   Suppose you have a cyclic group of order 7, with generator g = 3. Calculate the discrete logarithm of y = 5.

   Hint

   List the powers of g until you reach y.

6. 6.

   Explain and prove why the discrete logarithm problem remains hard in multiplicative groups modulo a large prime.

   Hint

   Focus on the nature of cyclic groups and modular arithmetic!