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RSA – An Example Here is an example of the RSA scheme in action. I have chosen primes that are large enough to be interesting but small enough that you can do all of the arithmetic with a pen, paper and calculator. Key Generation 1. 2. 3. 4. 5. p = 31 q = 23 (chosen at random) n = 31*23 = 713 r = 30*22 = 660 e = 223 (chosen at random) d = 367 (computed using Euclids algorithm as below) 1 = 7 – 3(2) 1 = 7 – 3(9-7) = 4(7) – 3(9) 1 = 4(214-9(23))-3(9) = 4(214) – 95(9) 1 = 4(214) – 95(223 – 214) = 99(214) – 95(223) 1 = 99(660 – 2(223))-95(223) 1 = 99(660)-293(223) so the inverse of 223 mod 660 = -293 = 367 660 = 223 * 2 + 214 223 = 214 * 1 +9 214 = 9 * 23 +7 9=7*1+2 7 = 2 * 3 +1 6. Bob’s private key = 367 Bob’s public key = (223, 713) Encryption Alice wishes to send Bob the message m = 439. She computes c=439223 mod 713 = 284 (see fast exponent ion table below) Alice sends the ciphertext 284 to Bob. Decryption Bob receives the ciphertext 284 from Alice. He computes 284367 mod 713 = 439 (see fast exponentiation table below) He knows the message is 439 439223 mod 713 y u 1 439 439 211 652 315 36 118 683 377 98 242 98 98 335 335 284 n 223 111 55 27 13 6 3 1 0 284367 mod 713 y u 1 284 284 87 466 439 656 211 94 315 94 118 397 377 652 242 652 98 439 n 367 183 91 45 22 11 5 2 1 0