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# hw2 - r i-s produced by the algorithm NOTE you should be...

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CS346 Cryptography, Fall 2010 Homework 2, Due September 30 1. (10 points) As a reminder, here is a description of the Euclidean algorithm: To compute the greatest common divisor of two positive integers r 0 and r 1 , where r 0 > r 1 , the algorithm produces the following numbers: r i +2 = r i mod r i +1 Note that r i +2 is the remainder of the division of r i by r i +1 . The process stops when r m +1 = 0, and returns r m as the answer. The correctness of the Euclidean algorithm depends on the fact gcd ( r 0 ,r 1 ) = gcd ( r 1 ,r 2 ) = ... = gcd ( r i ,r i +1 ) Prove that this is true. 2. (10 points) Find the greatest common divisors of the following pairs, without factoring. Use the Euclidean algorithm instead. For each pair, give the sequence of
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Unformatted text preview: 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) B. gcd (145393 , 82933) 3. (10 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. 4. (5 points) Problem 18 (Chapter 6.8) 5. (5 points) Problem 12 (Chapter 6.8)...
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