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# 1st - CSci 5471 Modern Cryptography Spring 2010 Homework#1...

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CSci 5471: Modern Cryptography, Spring 2010 Homework #1 Yongdae Kim Due: Feb. 2 9:00 AM Type up your solution (either text or pdf) and send it to me and TA by email. ( kyd(atmark)cs.umn.edu, yongdaek(atmark)gmail.com,hkang(atmark)cs.umn.edu ) Show all steps. Please ask any questions, if you find any problems. Total 100 points + 20 points extra credit 1. Discrete Math and Number Theory (a) (5 pts) Prove or disprove: n 3 + ( n + 1) 3 + ( n + 2) 3 is always mutiple of 9 . (Hint) Proof by cases (b) (5 pts) Show that if 2 n - 1 is a prime, then n is a prime. (Hint) Proof by contradiction. (c) (5 pts) Show that for all integers n , 4 6 | n 2 + 2 . (d) (10 points) Let n be a positive integer. The Euler function φ ( n ) is defined as the number of non-negative integers k less than n which are relatively prime to n . Prove that i. (3 points) φ ( p ) = p - 1 for all prime p . ii. (7 points) φ ( p c ) = p c - 1 ( p - 1) for all prime p and all positive integer c . (e) (10 pts) Show that in any set of n + 1 positive integers not exceeding 2 n , there must be two integers that are relatively prime. (Hint) Use Pigeonhole Principles.
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