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6. Let H be the subgroup of (S3, o) generated by p= ( 1 3 2 ) - 1 2 3 (a) List the right cosets of H. (b) Are the right cosets of H equal to the left cosets of H? 7. R+ = {x € R I T > 0} is the set of positive real numbers. (a) Show that (R+, x) is a group, where X is the usual multiplication of real numbers. (b) Show that f : (R,+) (R+,x) where = is a group homomorphism. Is f an isomorphism? 8. Prove that if f : G1 G2 and g : G2 G3 are group homomorphisms then g° f : G1 G3 is also a group homomorphism. Make sure to justify each step. 9. (a) A message was created using the integers from 11 to 36 to encode the letters A to Z, and 10 to encode a space. It was then broken into chunks of 3 digits to make it easier to encode. Translate the message: M (b) Given an RSA Public Key encryption code with 77 1517.6 17. encrypt the message M by calculating R, - E(Mr) = MM mod 1517. for M1 = 231, = 130, Mg = 124. You need to show complete working for how you would figure out Ri with a simple calculator. not a computer, for at least one set of 3 digits. (c) Given that P = 37 and q = 41 are prime numbers and 72 : use the Euclidean algorithm to find the decryption key d. 10. The ASCH code uses integers from 65 to 90 to encode the letters from A to Z consecutively. A blank is encoded as 32. Alice has set up an encryption scheme with the keys 72 | 10.057 and C = 475. Bob sends Alice the following message by translating his message into ASCII. breaking it into chunks of 4 digits, and encoding each of these E - (a) Given that 71 - 10.057 - 89 X 113 is a product of prime factors. lise the Euclidean algorithm to find the decryption kev d. (b) Decrypt the message that Bob sent to Alice.

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