17.2.2. Definition of Discrete Logarithm

Try these first

1. 1.

   What is the basic definition of a cyclic group?

   Hint

   Think about how one element can generate all others.

2. 2.

   What is the discrete logarithm of 1 to any base?

   Hint

   Remember the property of logarithms.

3. 3.

   What is the discrete logarithm in a cyclic group?

   • The base of the logarithm
   • The exponent that generates an element
   • The order of the group

   Hint

   Think of which part of the equation represents the exponent.

4. 4.

   The logarithm of 1 in a cyclic group is always _____.

   • True
   • False

   Hint

   Recall the property of logarithms.

5. 5.

   Given a cyclic group of order 11 and generator g = 2, calculate the discrete logarithm of y = 8.

   Hint

   Iterate powers of the generator until you match y.

6. 6.

   Explain why it is computationally harder to determine the discrete logarithm in groups defined by multiplication modulo a prime compared to addition modulo a prime.

   Hint

   Think about how the outputs of multiplication can change drastically with slight changes in the input.