Zhou_Zhang_HW3

Zhou_Zhang_HW3 - Zhou, HuanXian (0225418) Zhang, Huan...

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Zhou, HuanXian (0225418) Zhang, Huan (0336630) CS392 HW3 HW #3: Number Theory/Design Principles/Threat Modeling CS 392/6813: Computer Security Fall 2010 [100pts] DUE 10/22/2010 at Midnight Problem 1 [10*3=30pts] 1) We discussed how to use the Euclidian algorithm to compute the GCD of two numbers. Use this algorithm to compute the following values. Show all of the steps involved. a. The GCD of 56 and 125. Euclidian algorithm: If y = ax + b then gcd(x,y) = gcd(x,b) 125 = (a)*56 + b 125 = int(125/56)*56 + 125%56 125 = 2*56 + 13 gcd(56 ,13) 56 = (a)*13 + b 56 = 4*13 + 4 gcd(13,4) = 1 gcd(56, 125) = gcd(56,13) = gcd(13 ,4) = 1 b. The GCD of 52 and 876. 876 = (a)*52 + b 876 = int(876/52)*52 + 876%52 876 = 16*52 + 44 gcd(52 ,44) 52 = (a)*44 + b 52 = 1*44 + 8 gcd(44, 8) 44 = 5*8 + 4 gcd(8, 4) = 4 gcd(52, 876) = gcd(52 ,44) = gcd(44, 8) = gcd(8,4) = 4 2) We also discussed the use of the Extended Euclidian algorithm to calculate modular inverses. Use this algorithm to compute the following values. Show all of the steps involved.
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a. 957895 -1 (mod 1229399450) Since the gcd of 957895 and 1229399450 is not 1, they are not relatively prime to each other, thus they don’t have modular inverse. b. 13 -1 (mod 31) Refer to: http://math.ucdenver.edu/~wcherowi/courses/m5410/exeucalg.html 31 = 2*13+5 p0 = 0 13 = 2*5+3 p1 = 1 5 = 1*3+2 p2 = 0 – 1(2) mod 31 = -2 mod 31 = 29 3 = 1*2+1 p3 = 1 – 29(2) mod 31 = -57 mod 31 = 5 2 = 2*1+0 p4 = 29 – 5(1) mod 31 = 24 mod 31 = 24
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Zhou_Zhang_HW3 - Zhou, HuanXian (0225418) Zhang, Huan...

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