M417 Homework 5 Spring 2004

Instructions : Be prepared to discuss and present in class on Friday, February 20. Written solutions are due Monday, February 23.

1. The RSA cipher, with public key n = 7822643 and encryption exponent e = 17, was used to encrypt a message. The ciphertext is: 5785045 6445108 3550040 475858 5843081. Determine the decryption exponent and the original plaintext message. [Hint: you may find the web forms on our class web site useful.]
2. If f: A -> B is a surjective function, show that f -1: 2B -> 2A is injective.
3. What can you say if f in the previous problem is injective? Write down and prove a statement.
4. Let G be the set of all maps f : R -> R of the form f(x) = ax + b, where |a| = 1, and b is an integer.
• Show that G is a group under composition of functions.
• For each g in G, find CG(g). [Hint: consider separately the case that g(x) = x, g(x) = x + b, and the case that g(x) = -x + b.]
• Find Z(G).
5. Let A be a subset of a subset B of a group G. Prove that CG(B) is a subgroup of CG(A).