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.

- 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.]
- If f: A -> B is a surjective function, show
that f
^{ -1}: 2^{B} -> 2^{A}
is injective.
- What can you say if f in the previous problem is
injective? Write down and prove a statement.
- 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 C
_{G}(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).

- Let A be a subset of a subset B of a group G.
Prove that C
_{G}(B) is a subgroup of C_{G}(A).