# hw8 - Computer Science 340 Reasoning about Computation...

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Computer Science 340 Reasoning about Computation Homework 8 Due at the beginning of class on Wednesday, November 21, 2007 Problem 1 The ciphertext below QFL HCVPS PX V ANSWLCEZK NCJVS; PQ XQVCQX QFL BPSZQL RNZ JLQ ZT PS QFL BNCSPSJ VSW WNLX SNQ XQNT ZSQPK RNZ JLQ QN DKVXX was created by using a substitution cipher on the alphabet of 26 letters. Recover the plaintext and show all your steps. (Hint: J decodes to G.) Problem 2 1. Find the smallest integer n > 1 such that 2 n = 1 (mod 33). What is the value of 2 273648273465737284375838005837397466 (mod33)? Justify your answer. 2. Compute 3 n (mod 11) for n = 1 , 2 , . . . , 10. What is the value of 3 2736482734657372843758380058373974 (mod11)? Justify your answer. 3. Compute 5 2000378 (mod 127) as simply as you can. 4. Compute 2 5432675 (mod 13). Problem 3 Prove without factoring that 65 is not prime, using Fermat’s theorem: a p - 1 = 1 (mod p ) for all integers 0 < a < p .

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Problem 4 The entropy H ( X ) of a random variable X defined as p i log(1 /p i ), where Pr[ X = k ]= p k . For example, the entropy of the outcome H, T of a fair coin is (1 / 2) log(2)+(1 / 2) log(2) = 1.
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• Fall '07
• CharikarandChazelle
• Cryptography, Probability theory, JLQ QN DKVXX, JLQ ZT PS, RNZ JLQ

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