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Exercise 16.1.6. 550 CHAPTER 16 FURTHER TOPICS IN CRYPTOGRAPHY (a) We may define a function f : Z7\ {0} → Z7\ {0} by the equation: f(n)= mod(2,7). Use parts (a) and (b) of Exercise 16.1.5 to prove that f is neither one-to-one nor onto. (b) We may also define a function g: Z7\ {0} → Z7\ {0} by the equation: g(n)= mod (3, 7). Prove or disprove: g is one-to-one. (c) With the same g as in part (b), prove or disprove: g is onto.

Exercise 16.1.6. 550 CHAPTER 16 FURTHER TOPICS IN CRYPTOGRAPHY (a) We may define a function f : Z7\ {0} → Z7\ {0} by the equation: f(n)= mod(2,7). Use parts (a) and (b) of Exercise 16.1.5 to prove that f is neither one-to-one nor onto. (b) We may also define a function g: Z7\ {0} → Z7\ {0} by the equation: g(n)= mod (3, 7). Prove or disprove: g is one-to-one. (c) With the same g as in part (b), prove or disprove: g is onto.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Please do Exercise 16.1.6 part ABC and show step by step and explain

Exercise 16.1.6. 550 CHAPTER 16 FURTHER TOPICS IN CRYPTOGRAPHY (a) We may define a function f : Z7\ {0} → Z7\ {0} by the equation: f(n)= mod(2,7). Use parts (a) and (b) of Exercise 16.1.5 to prove that f is neither one-to-one nor onto. (b) We may also define a function g: Z7\ {0} → Z7\ {0} by the equation: g(n)= mod (3, 7). Prove or disprove: g is one-to-one. (c) With the same g as in part (b), prove or disprove: g is onto.
Transcribed Image Text:Exercise 16.1.6. 550 CHAPTER 16 FURTHER TOPICS IN CRYPTOGRAPHY (a) We may define a function f : Z7\ {0} → Z7\ {0} by the equation: f(n)= mod(2,7). Use parts (a) and (b) of Exercise 16.1.5 to prove that f is neither one-to-one nor onto. (b) We may also define a function g: Z7\ {0} → Z7\ {0} by the equation: g(n)= mod (3, 7). Prove or disprove: g is one-to-one. (c) With the same g as in part (b), prove or disprove: g is onto.
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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