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Find all natural numbers n such that mod (2", 7) = 4. Use your result to complete the following sentence: "Under multiplication mod 7, the discrete logarithm(s) of 4 with base 2 are ...." Find all natural numbers n such that mod (2", 7) = 3. Use your result to complete the following sentence: "Under multiplication mod 7, the discrete logarithm(s) of 3 with base 2 are ...." Find all nonzero elements of Z7 \ {0} which have no discrete logarithms with base 2. Find all nonzero elements of Z7\{0} which have no discrete logarithms with base 3.

Find all natural numbers n such that mod (2", 7) = 4. Use your result to complete the following sentence: "Under multiplication mod 7, the discrete logarithm(s) of 4 with base 2 are ...." Find all natural numbers n such that mod (2", 7) = 3. Use your result to complete the following sentence: "Under multiplication mod 7, the discrete logarithm(s) of 3 with base 2 are ...." Find all nonzero elements of Z7 \ {0} which have no discrete logarithms with base 2. Find all nonzero elements of Z7\{0} which have no discrete logarithms with base 3.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please do part ABCD and please show step by step and explain

In previous math courses you learned that the inverse operation of expo- nentiation is taking the logarithm: for example, 23 = 8 while log₂ 8 = 3. It is possible to do the same with discrete exponentiation: an inverse operation to discrete exponentiation is referred to as 'finding a discrete logarithm or (DL)'. Note that since discrete exponentiation involves raising to a power which is a natural number, a DL will always be a natural number. For ex- ample, since mod (25, 7) = 4, we could say that under multiplication mod 7, 5 is a DL of 4 with base 2. Now why have we been saying, "a DL" rather than "the DL"? Because there happens to be more than one: Exercise 16.1.5. (a) Find all natural numbers n such that mod (2", 7) = 4. Use your result to complete the following sentence: "Under multiplication mod 7, the discrete logarithm(s) of 4 with base 2 are ...." (b) Find all natural numbers n such that mod (2, 7) = 3. Use your result to complete the following sentence: "Under multiplication mod 7, the discrete logarithm(s) of 3 with base 2 are ...." (c) Find all nonzero elements of Z7\{0} which have no discrete logarithms with base 2. (d) Find all nonzero elements of Z7\{0} which have no discrete logarithms with base 3.
Transcribed Image Text:In previous math courses you learned that the inverse operation of expo- nentiation is taking the logarithm: for example, 23 = 8 while log₂ 8 = 3. It is possible to do the same with discrete exponentiation: an inverse operation to discrete exponentiation is referred to as 'finding a discrete logarithm or (DL)'. Note that since discrete exponentiation involves raising to a power which is a natural number, a DL will always be a natural number. For ex- ample, since mod (25, 7) = 4, we could say that under multiplication mod 7, 5 is a DL of 4 with base 2. Now why have we been saying, "a DL" rather than "the DL"? Because there happens to be more than one: Exercise 16.1.5. (a) Find all natural numbers n such that mod (2", 7) = 4. Use your result to complete the following sentence: "Under multiplication mod 7, the discrete logarithm(s) of 4 with base 2 are ...." (b) Find all natural numbers n such that mod (2, 7) = 3. Use your result to complete the following sentence: "Under multiplication mod 7, the discrete logarithm(s) of 3 with base 2 are ...." (c) Find all nonzero elements of Z7\{0} which have no discrete logarithms with base 2. (d) Find all nonzero elements of Z7\{0} which have no discrete logarithms with base 3.
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