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sol2 - CS346 Cryptography Fall 2009 Homework 2 SOLUTIONS...

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CS346 Cryptography, Fall 2009 Homework 2, SOLUTIONS 1. (5 points) Give an algorithm that on input integers x and m computes x 37 mod m . As a primitive step you may assume an algorithm for modular multiplication: on input y and z it computes ( y · z ) mod m . Your algorithm should use as few modular multiplications as possible (7 is the best possible). You may write the algorithm informally. Solution: x 2 = x × x (mod m ) x 4 = x 2 × x 2 (mod m ) x 8 = x 4 × x 4 (mod m ) x 16 = x 8 × x 8 (mod m ) x 32 = x 16 × x 16 (mod m ) x 36 = x 32 × x 4 (mod m ) x 37 = x 36 × x (mod m ) Note that x i = x i (mod m ) 2. (6 points) Find the greatest common divisors of the following pairs, without factoring. Use the Euclidean algorithm instead. For each pair, give the sequence of r i -s produced by the algorithm. NOTE: you should be able to do this by hand, without a computer or calculator. A. gcd (103927 , 102313) Solution: r 0 = 103927, r 1 = 102313, r 2 = 1614, r 3 = 631, r 4 = 352, r 5 = 279, r 6 = 73, r 7 = 60, r 8 = 13, r 9 = 8, r 10 = 5, r 11 = 3, r 12 = 2, r 13 = 1, r 14 =1, r 15 = 0. So, gcd(103927 , 102313) = r 14 = 1.
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