# extra - that monks in a temple tower were given 64 rings at...

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CS346 Cryptography, Fall 2009 EXTRA CREDIT LAST CHANCE TO SUBMIT: DECEMBER 1, IN CLASS 1. EXTRA CREDIT from HW1 (10 points) A recreational problem, called the Tower of Hanoi problem, was invented by the French mathematician Lucas in 1883. It is described as follows. There are three vertical posts and n rings, all of diﬀerent sizes, placed on one of the posts from largest on the bottom to smallest on the top. (No larger ring is placed upon a smaller one.) The object of the game is to move all rings from from the given post to another post, subject to the following rules. 1. Only one ring may be moved at a time. 2. A ring may never be placed over a smaller ring. A. (8 points) Determine the number of moves required to transfer n rings from one post to another. NOTE: For full credit, argue that this number of moves is suﬃcient, and also show (by a separate argument) that it is not possible to solve the problem with fewer moves. B. (2 points) Using the above, answer the following question. Ancient folklore tells us,
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Unformatted text preview: that monks in a temple tower were given 64 rings at the beginning of time. They were told to play the above game, and that the world would end when they were ﬁnished. Assuming that the monks worked in shifts twenty-four hours per day, moving one ring per second without any errors, how many years does the world last? 2. EXTRA CREDIT from HW2 (10 points) Prove that the number of iterations of the Euclidean algorithm to ﬁnd gcd( a,b ) is O (log b ). (Note that this holds for both b ≤ a and b ≥ a .) 3. EXTRA CREDIT from HW3 (10 points) Describe the most eﬃcient algorithm you can for computing x b mod n , if b is an arbi-trary integer. Your solution should be eﬃcient enough to use for RSA encoding. HINT: We have already seen in class how to do this if b is of the form b = 2 l for some integer l ....
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## This note was uploaded on 10/07/2010 for the course C S 52475 taught by Professor Gal during the Fall '10 term at University of Texas.

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