# 4th - CSci 5471: Modern Cryptography, Fall 2010 Homework #4...

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CSci 5471: Modern Cryptography, Fall 2010 Homework #4 Yongdae Kim Due: Mar. 23 9:00 AM Show all steps. Please ask any questions, if the problems are not clear. – Send me your solution by e-mail. Total 100 + 40 points. 1. Number Theory (45 points): Try to use theorems you learned. Use of calculator for this homework won’t help your preparation for upcoming quiz. (a) (5 points) Solve for x using Chinese remainder theorem. 5 · x 2 (mod 8) 7 · x 2 (mod 9) x 0 (mod 11) (b) (5 points) Let m,n be the integers such that gcd( m,n ) = 1 . When does gcd( m,n ) = gcd( m + n,m - n ) hold? Justify your claim. (c) (5 points) Find the remainder of 10 10 + 10 10 2 + ··· + 10 10 10 upon division by 7. (d) (6 points) Compute 2 28224 (mod 113 · 127) . (e) (7 points) Compute 2 2 64 (mod 180) . (f) (7 points) Find all generators in Z * 19 . (g) (10 points) Let p be an odd prime. Show that if a h = 1 mod p then a ph = 1 mod p 2 . 2. Performance Degradation when Key Size Increases (10 points) We consider number of bit operations instead of number of multiplications for this problem. Suppose modular multiplication (i.e. a · b (mod n ) with | a | = | b | = | n | ) requires 4 log 2 2 ( n ) -bit operations in the worst case. Furthermore, we assume that square-and-multiply algorithm is used for modular exponentiation. Compared with 512 bit modular exponentiation, how many times is 1024-bit modular exponentiation slower?

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3. Database Security (35 points)
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## 4th - CSci 5471: Modern Cryptography, Fall 2010 Homework #4...

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