# hw708 - very useful fact known as the union bound Pr E 1...

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CS 4803 CNS: Computer and Network Security April 3, 2008 Homework 7 Lecturer: Sasha Boldyreva Due: April 10, 2008 Assignment 7.01 Do the assigned reading. Assignment 7.02 Indicate how much time did you spend on this homework. In all problems do not just give an answer, but show your reasoning/work. Problem 7.1, 12 points. Problem 18 from Section 6.2 of Rosen’s textbook. Problem 7.2, 5 points. Problem 20 from Section 6.2 of Rosen’s textbook. Problem 7.3, 5 points. Problem 24 (d) and (e) from Section 3.6 of Rosen’s textbook. Show the intermediate results. Problem 7.4, 5 points. Run the Extended GCD algorithm by hand on input (1529,14039) from the previous problem and show the intermediate and ﬁnal results. Problem 7.5, 5 points. Prove that for any integers a, b, c, if c | ( ab ), and gcd(a, c) = 1, then c | b . Problem 7.6, 10 points. In the proof of the Birthday paradox we used the following
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Unformatted text preview: very useful fact known as the union bound: Pr( E 1 ∪ E 2 . . . E n ) ≤ Pr( E 1 ) + Pr( E 2 ) + . . . Pr( E n ) , where E 1 , E 2 , . . . E n are events associated with the same sample space. Prove it by induc-tion and use the base case n = 2. Problem 7.7, 5 points. Here is a sample proof that contains an error. Explain why the proof is not correct. Theorem: If a | b and b | c , then a | c . Proof: Since a | b , b = ak . Since b | c, c = bk . Therefore c = bk = ( ak ) k = ak 2 . Therefore a | c . Problem 7.8, 5 points. Prove that if n is an integer that is not a multiple of 3, then n 2 ≡ 1 mod 3. Problem 7.9, 5 points. Suppose that all you know about high tide is that its ex-pected height is 1 meter, and the height is always non-negative. What can we say about the probability that you see high tide more than 2 meters?...
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