TABLE OF CONTENTS In this section I only briefly go through some basic definitions and results about finite fields and rings that are required later.
Let G be a nonempty set and ? be a binary operation defined on G. The pair ( G, ?) is a group if the following three properties hold:
( a ? b) ? c = a ? ( b ? c) for all a, b, c ? G.
There is an identity element e such that e ? a = a ? e = a for all a ? G.
For every a ? G there exists an inverse a ? 1 ? G such that a ? a ?1 = a ?1 ? a = e.
If, furthermore,
a ? b = b ? a for all a, b ? G,
the group is called abelian or commutative.
We often use the notation of ordinary multiplication or addition for the group operation. Using the multiplicative notation, we write a n = a a a ( n factors a), and in the additive notation na = a + a + + a ( n summands a). If n is negative, we define a n = ( a ? 1) ? n and
