312f10mid2 - a 4 1 mod 10. (b) (10 points) What are the...

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My NAME is: Problem 1 2 3 4 5 Total Score MAT 312 Applied Algebra Midterm 2 November 2, 2010 No books or notes may be consulted during this test. You may use a programmable graphing calculator, but no “computer algebra” systems. Explain your answers carefully. Total score = 100. 1. (a) (15 points) Solve the congruence equation 40 x 3 mod 177 (b) (15 points) Find all solutions to the congruence equation 30 x 9 mod 177 2. Reminder: The Chinese Remainder algorithm for the solution of a system of congru- ences: x a 1 mod m 1 , x a 2 mod m 2 , ..., x a n mod m n where m 1 ,m 2 ,... ,m n are all relatively prime, is x = a 1 M 1 y 1 + a 2 M 2 y 2 + · · · + a n M n y n , where M i = m 1 · · · ˆ m i · · · m n (i.e. m i has been left out of the product), and y i is the multiplicative inverse of M i modulo m i . (a) (15 points) Solve: x 1 mod 6 x 2 mod 7 x 3 mod 5 . 1
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(b) (5 points) Adapt the algorithm to solve 2 x 2 mod 12 x 2 mod 7 x 3 mod 5 3. (a) (10 points) Explain why if a is not divisible by 2 or by 5, then
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Unformatted text preview: a 4 1 mod 10. (b) (10 points) What are the last three digits of 377 400 ? Explain your work carefully! 4. Given the permutation = p 1 2 3 4 5 6 7 8 9 2 5 8 4 6 1 3 9 7 P , (a) (10 points) calculate 3 and -1 (Use matrix or cycle notation, as you prefer). (b) (10 points) Write as a product of disjoint cycles. 5. (10 points) Here is the mathematical center of the RSA algorithm: You know that N = pq is the product of 2 large primes. A number x , which is smaller than p and smaller than q , has been encoded as y = x a mod N . You know ( N ) , and that ( a, ( N )) = 1. Explain carefully how you get x mod N back from y . Explain why the competition, even knowing N and a , cannot decode y . END OF EXAMINATION 2...
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312f10mid2 - a 4 1 mod 10. (b) (10 points) What are the...

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