Practice Difficult Computation In Certain Cyclic Groups (17.2.5)

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6 sample questions

Question 1.
What is the discrete logarithm of 1 to any base g in a cyclic group?
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Think about the exponent needed to get the identity element.
Question 2.
Is it possible for a cyclic group to have no generators?
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Recall the definition of cyclic groups.
3Question 3.
What is the discrete logarithm of 1 for any base g?
- True
- False
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Recall the fundamental property of logarithms.
4Question 4.
In the group Z_5, if g=2 and y=4, what is the discrete log?
- 1
- 2
- 3
- 4
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Calculate powers of g modulo 5.
Question 5.
In a large cyclic group Z_{23} where g=5, compute the discrete log of y=17 using brute force and provide a step-by-step explanation.
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Keep calculating powers until matching y=17.
Question 6.
Explain why an efficient algorithm for computing discrete logarithm in Z*_p (where p is prime) is currently conjectured to exist or not.
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Focus on patterns in behavior.

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