#### CSCI 7000 - Crypto Seminar -
Spring 2003

### Assignment #3

##### Due: Mar 13th, 2003 at 11am MST

1. At one point I claimed in class that every row and column of a
group's "multiplication table" was a permutation of the group elements.
Please prove this property.
2. At one point we discussed if the property in question #1 was also
a sufficient condition for building groups. In other words, if we
draw an *n* by *n* multiplication table where each row
and each column is a permutation of the group elements, does that imply
that this table represents a group?

(**Hint:** The answer is no; prove this by providing a counterexample.)

3. Prove that a finite group of even order always has some element *i*
different from the identity, where *i* is its own inverse.

4. Consider the finite field GF(16) with irreducible polynomial
m(x) = x^{4} + x^{3} + 1. We will, as usual, represent
group elements as either polynomials in *x*, or hex number, or
binary strings, with the usual mapping between these representations.

**(a)** What polynomial corresponds to the hex number 'a'?

**(b)** What is 'a' + '7' in this field?

**(c)** What is 'a' * '7' in this field?

**(d)** What is the additive inverse of 'e' in this field?

**(e)** What is the multiplicative inverse of 'e' in this field? (Please
use the **xtime()** method and show your work.)