hw2 - of the letters of the alphabet in this sample of...

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CS 555, Spring, 2010, Homework 2 Due: February 10, 2010, 9:30 AM. 1. Friedman’s original method of ±nding the length t of the key in a Vigen` ere cipher used this de±nition of the Index of Coincidence ( IC ). For 0 i < n , let F i be the frequency of occurrence of the i -th letter of the (ciphertext) alphabet (which has n letters) in a ciphertext of length N . De±ne IC as IC = n - 1 X i =0 F i ( F i - 1) 2 ! ± N ( N - 1) 2 = 1 N ( N - 1) n - 1 X i =0 F i ( F i - 1) . Then IC represents the probability that two letters (such as the third and tenth letters) chosen at random in the ciphertext are the same letter. One can estimate IC theoretically in terms of N and t . For English and a polyalphabetic cipher with key length t , the expected value of IC is 1 t N - t N - 1 (0 . 065) + t - 1 t N N - 1 (0 . 038) . (a) Assuming IC equals its expected value, ±nd t in terms of IC and N . (b) One hundred characters of ciphertext from a suspected Vigen` ere cipher were intercepted by one of your agents. Here is the frequency of occurrence
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Unformatted text preview: of the letters of the alphabet in this sample of ciphertext: A B C D E F G H I J K L M 2 10 2 5 3 1 8 2 2 5 1 3 1 N O P Q R S T U V W X Y Z 2 1 10 8 1 8 5 2 1 3 5 1 8 Compute the Index of Coincidence IC for this sample. (c) Assuming the IC computed in Part (b) is near its expected value, what is your best guess for t ? Remember that your answer to Part (b) is just an approximation. (d) Now suppose that the Kasiski method applied to the same ciphertext of length 100 characters suggests that t is a divisor of 18. Would this informa-tion change your answer to Part (c)? If so, what is your new guess for t ? Explain your answer. 2. Use Shannon’s theorem to prove that Vernam’s cipher is perfectly secret. Your proof should be shorter and simpler than the proof of this fact given in class and in the slides. 1...
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