503_HW5 - : (Z 8 , + 8 ) .- For the group (Z 8 , + 8 ) ,...

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UML CS Analysis of Algorithms 91.503 (Section 201) Fall, 2001 Homework Set #5            Assigned: Wednesday, 11/7       Due: Tuesday, 11/13 (start  of lecture)    This assignment covers material on Number-Theoretic Algorithms from Chapter 33 of our text. Note: Partial credit for wrong answers is only given if work is shown. 1. (25 points) Prime Factorization & Euclid’s GCD Algorithm: - Find the prime factorization of 660 - Find the prime factorization of 2520 - From these prime factorizations, find the greatest common divisor gcd(660, 2520) - List the recursive calls that are executed by Euclid’s greatest common divisor algorithm (p. 810 of text) for a = 660 and b = 2520. What is the final value of gcd(660, 2520) ? Note that this should be the same as the value obtained above using prime factorization. Hint: we suggest writing a short program to execute the algorithm. 2. (25 points) Finite Groups: - List all the elements of the finite additive group modulo 8
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Unformatted text preview: : (Z 8 , + 8 ) .- For the group (Z 8 , + 8 ) , create a table similar to Figure 33.2 (a) on p. 815 of our text to show the result of each possible pairwise addition of group elements. - List all the elements of the set Z* 8 for the finite multiplicative group modulo 8 : (Z* 8 , . 8 ) . Recall that, in general, the elements should be a subset of: consisting of those elements of Z n that are relatively prime with respect to n . That is, they satisfy:- Calculate the number of elements of (Z* 8 , . 8 ) using Eulers phi function: Note that this should be the same as the size obtained above by building the set Z* 8 .- For the group (Z* 8 , . 8 ) , create a table similar to Figure 33.2 (b) on p. 815 of our text to show the result of each possible pairwise multiplication of group elements. [ ] } 1 ) , gcd( : { * = = n a Z a Z n n n } 1 , , 1 , {-= n Z n -= n p p n n | 1 1 ) (...
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