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Problem 3 Weak collisions The cryptographic relevance of this problem will become evident when we cover hash functions in class. For each question below, provide a brief explanation and a compact formula for your answer. Let n be a positive integer. Consider an experiment involving a group of participants, where we assign each participant a number that is randomly chosen from the set {1,2,...,n} (so all these assignments are independent events). Note that we allow for the possibility of assigning the same number to two different participants. Now pick your favourite number N between 1 and n. When any one of the participants is assigned the number N, we refer to this as a weak collision (with N). In this problem, we determine how to ensure at least a 50% chance of a weak collision in our experiment. b. a. What is the probability that a given participant is assigned your favourite number N? What is the probability that a given participant is not assigned the number N? c. Suppose k people participate in the experiment (for some positive integer k). What is the probability that none of them is assigned the number N, i.e. that there is no weak collision? d. Suppose again that k people participate in the experiment (for some positive integer k). What is the probability that at least one of them is assigned the number N, i.e. that a weak collision occurs? e. Intuitively, the more people participate, the likelier we encounter a weak collision. We wish to find the minimum number K of participants required to ensure at least a 50% chance of a weak collision. Suppose n = 12. What is the threshold K in this case? Give a numerical value that is an integer. f. Generalizing part (e) from n = 12 to arbitrary n, prove that if the number of participants is at least log(2)n≈ 0.69n (where "log" refers to the natural logarithm), then there is a better than 50% chance of a weak collision. Use (without proof) the inequality 1-x 0. (3)

Problem 3 Weak collisions The cryptographic relevance of this problem will become evident when we cover hash functions in class. For each question below, provide a brief explanation and a compact formula for your answer. Let n be a positive integer. Consider an experiment involving a group of participants, where we assign each participant a number that is randomly chosen from the set {1,2,...,n} (so all these assignments are independent events). Note that we allow for the possibility of assigning the same number to two different participants. Now pick your favourite number N between 1 and n. When any one of the participants is assigned the number N, we refer to this as a weak collision (with N). In this problem, we determine how to ensure at least a 50% chance of a weak collision in our experiment. b. a. What is the probability that a given participant is assigned your favourite number N? What is the probability that a given participant is not assigned the number N? c. Suppose k people participate in the experiment (for some positive integer k). What is the probability that none of them is assigned the number N, i.e. that there is no weak collision? d. Suppose again that k people participate in the experiment (for some positive integer k). What is the probability that at least one of them is assigned the number N, i.e. that a weak collision occurs? e. Intuitively, the more people participate, the likelier we encounter a weak collision. We wish to find the minimum number K of participants required to ensure at least a 50% chance of a weak collision. Suppose n = 12. What is the threshold K in this case? Give a numerical value that is an integer. f. Generalizing part (e) from n = 12 to arbitrary n, prove that if the number of participants is at least log(2)n≈ 0.69n (where "log" refers to the natural logarithm), then there is a better than 50% chance of a weak collision. Use (without proof) the inequality 1-x 0. (3)

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Problem 3 Weak collisions The cryptographic relevance of this problem will become evident when we cover hash functions in class. For each question below, provide a brief explanation and a compact formula for your answer. Let n be a positive integer. Consider an experiment involving a group of participants, where we assign each participant a number that is randomly chosen from the set {1,2,...,n} (so all these assignments are independent events). Note that we allow for the possibility of assigning the same number to two different participants. Now pick your favourite number N between 1 and n. When any one of the participants is assigned the number N, we refer to this as a weak collision (with N). In this problem, we determine how to ensure at least a 50% chance of a weak collision in our experiment. b. a. What is the probability that a given participant is assigned your favourite number N? What is the probability that a given participant is not assigned the number N? c. Suppose k people participate in the experiment (for some positive integer k). What is the probability that none of them is assigned the number N, i.e. that there is no weak collision? d. Suppose again that k people participate in the experiment (for some positive integer k). What is the probability that at least one of them is assigned the number N, i.e. that a weak collision occurs? e. Intuitively, the more people participate, the likelier we encounter a weak collision. We wish to find the minimum number K of participants required to ensure at least a 50% chance of a weak collision. Suppose n = 12. What is the threshold K in this case? Give a numerical value that is an integer. f. Generalizing part (e) from n = 12 to arbitrary n, prove that if the number of participants is at least log(2)n≈ 0.69n (where "log" refers to the natural logarithm), then there is a better than 50% chance of a weak collision. Use (without proof) the inequality 1-x<e¯¤ for x > 0. (3)
Transcribed Image Text:Problem 3 Weak collisions The cryptographic relevance of this problem will become evident when we cover hash functions in class. For each question below, provide a brief explanation and a compact formula for your answer. Let n be a positive integer. Consider an experiment involving a group of participants, where we assign each participant a number that is randomly chosen from the set {1,2,...,n} (so all these assignments are independent events). Note that we allow for the possibility of assigning the same number to two different participants. Now pick your favourite number N between 1 and n. When any one of the participants is assigned the number N, we refer to this as a weak collision (with N). In this problem, we determine how to ensure at least a 50% chance of a weak collision in our experiment. b. a. What is the probability that a given participant is assigned your favourite number N? What is the probability that a given participant is not assigned the number N? c. Suppose k people participate in the experiment (for some positive integer k). What is the probability that none of them is assigned the number N, i.e. that there is no weak collision? d. Suppose again that k people participate in the experiment (for some positive integer k). What is the probability that at least one of them is assigned the number N, i.e. that a weak collision occurs? e. Intuitively, the more people participate, the likelier we encounter a weak collision. We wish to find the minimum number K of participants required to ensure at least a 50% chance of a weak collision. Suppose n = 12. What is the threshold K in this case? Give a numerical value that is an integer. f. Generalizing part (e) from n = 12 to arbitrary n, prove that if the number of participants is at least log(2)n≈ 0.69n (where "log" refers to the natural logarithm), then there is a better than 50% chance of a weak collision. Use (without proof) the inequality 1-x<e¯¤ for x > 0. (3)
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