17.2.3. Computational Difficulty of Discrete Logarithm

Try these first

1. 1.

   Define a discrete logarithm within a cyclic group.

   Hint

   Think about how logarithms work in general.

2. 2.

   What is the complexity of the brute-force algorithm for computing discrete logarithms?

   Hint

   Consider how many elements are in a cyclic group.

3. 3.

   What does the discrete logarithm measure?

   • The number of elements in a group
   • The exponent of a generator yielding a group element
   • The sum of a series
   • The average of a group

   Hint

   Relate it to the concept of logarithms you already know.

4. 4.

   True or False: The difficulty of solving DLP is crucial for the security of cryptographic systems.

   • True
   • False

   Hint

   Think about how secure communications depend on problem complexity.

5. 5.

   Consider a cyclic group of order 7 generated by 3. What is the discrete logarithm of 6 to the base 3?

   Hint

   Calculate powers of the generator one by one.

6. 6.

   Explain how the difficulty of the DLP can lead to vulnerabilities in cryptographic protocols. Provide an example.

   Hint

   Reflect on real-world implications of breaking encryption schemes.