Problem Set 3: 

Course:		CS 407 -- Codes and Internet Security
Posting Date:   November 2, 1998
Due Date:       November 16, 1998
Builds On:

   For an explanation of most of the pseudo-TeX used here, please consult 
problem set 1.  By contrast with actual Tex, * denotes multiplication.

   There are six problems in this problem set.  The theorem -- proof problems 
are quite short.

1. Theorem 1: If x|n and x > 1, then there exists E_a \in Z_n ~ {E_0} such that 
      E_a*E_x = E_0.

   Prove this theorem.

2. Corollary: If n is a composite number, then (Z_n ~ {E_0}, *) is not an 
     Abelian group.

   Prove this corollary using theorem 1, without reference to the Euclidean 
algorithm.

3. Theorem 2: If gcd(x, n) = d and x is not equal to 0, then there exists E_a 
      \in Z_n ~ {E_0} such that E_a*E_x = E_d.

   Prove this theorem using the Euclidean algorithm and/or its consequences.

4. Corollary: If gcd(x, n) > 1, then there exists E_a \in Z_n ~ {E_0} such that 
     E_a*E_x = E_0.

   Prove this corollary using theorems 1 and 2.

5. Use the Euclidean algorithm to find integers a and b such that 
     a*60 + b*7 = 1

6. Use problem 5 to find the inverse of E_7 in Z_60.  (Hint: E_(-k) = E_(n-k).)