Euclid's Algorithm and RSA Encryption Scheme: Solving Problems and Decrypting Messages, Exercises of Advanced Algorithms

Download Euclid's Algorithm and RSA Encryption Scheme: Solving Problems and Decrypting Messages and more Exercises Advanced Algorithms in PDF only on Docsity! Euclid’s Algorithm and the RSA Encryption Scheme 1. For each pair of integers (a, b) below, find d = gcd(a, b) and find a pair of numbers j and k such that d = j · a + k · b. (Use the Extended Euclidean algorithm.) (a) 90 and 56 (b) 91 and 89 (c) 711 and 75 (d) 815 and 75 (e) 112 and 196 (f) 366 and 150 2. For each triple of integers p, q, e given below, find the (smallest positive) value of d that will work for the private key of the RSA algorithm (where, as usual, n = p · q and φ(n) = (p− 1)(q − 1), and ed ≡ 1 (mod φ(n)). Then suppose that the message (integer) M = 25 is to be encoded using the public key n, e pair. Find the ciphertext that corresponds to that message M . (a) p = 7, q = 13, e = 5 (b) p = 5, q = 11, e = 3 (c) p = 7, q = 17, e = 7 3. Finally, suppose that p = 5, q = 17, and e = 13. First find the private key d for the RSA method with these parameters. Then decrypt the ciphertext messages, C, below to find the original (plaintext) messages. (a) 12 (b) 9 (c) 27 (d) 84 1 docsity.com