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crypto hw - Zhou HuanXian ID 0225418 CS393 Crypto HW...

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Zhou, HuanXian ID: 0225418 CS393 Crypto HW PROBLEM #1 Using the prime numbers p=13 q=3 Compute an RSA public and private key pair of (n,e) and (n,d) respectively. Using the last two digits of your Poly student ID, XY calculate XYmod38 as the message m. So for example if the last two digits of your Poly ID are 99 then 99mod38=23 and you would use 23 as the message m. Verify decryption using the private key (n,d) Show all steps in the computation. Be verbose. When calculating large exponents in a modulus be sure to use the binary expansion technique. Show all steps of this binary expansion. n = p∙q = 13*3 =39 Φ(n) = 12*2 =24 Choose e = 7, relatively prime to 24 d = e - 1 mod Φ = 7 - 1 mod 24 = ( * + )/ Φ y 1 e where y = 1,2,3,4 …. = (24*y + 1)/7 = 31 when y = 9 Public key: (39,7) Private key: (39,31) M = 18 mod 38 = 18 Y = E(m) = me mod n = 187 mod 39 = + + 184 2 1 mod 39 = ( 184 182 ) 181 mod 39 = 181 mod 39 18 = 182 mod 39 12 = 184 mod 39 27 = ( ) 27∙12∙18 mod 39 = ( ) 5832 mod 39 = 21 D(y) = yd mod n = 2131 mod 39 = + + + + 2116 8 4 2 1 mod 39 = ( 2116 218 214 212 ) 211 mod 39 = 211 mod 39 21 = 212 mod 39 12 = 214 mod 39 27 = 218 mod 39 27 = 2116 mod 39 27 = ( ) 21∙12∙27∙27∙27
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