Problem Set 4: 

Course:		CS 407 -- Codes and Internet Security
Posting Date:   November 2, 1998
Due Date:       November 23, 1998
Builds On:

   For an explanation of most of the pseudo-TeX used here, please consult 
problem set 1.  Superscripts (exponents) are denoted by ^ and <> denotes 
inequality.  The lower case Greek letter phi is denoted by \phi.

   There are 6 problems in this problem set.  Only one is a theorem -- proof 
problem.

1. Give an example where y^p <> y^q mod n, even though p = q mod n.

2. Give an example where y^p <> y^q mod n, even though p = q mod \phi(n).

3. Give an example where y^\phi(n) <> 1 mod n.

4. Find the inverses of E_2 in Z_11 and E_5 in Z_13.

5. Use Fermat's Little Theorem to compute the remainder when 3^47 is divided by 
      23.

6. Suppose a and b are integers and p is prime.  Use the binomial theorem to 
      prove that (a + b)^p = a^p + b^p mod p.