ABSTRACT
For example, the addition table and multiplication table for finite field
GF (2) and GF (5) where p = 2 and p = 5 are shown in Tables A.1 and A.2
respectively.
Table A.1: The Computations in GF (3)
+ 0 1 2 · 0 1 2 0 0 1 2 0 0 0 0
1 1 2 0 1 0 1 2
2 2 0 1 2 0 2 1
Table A.2: The Computations in GF (5)
+ 0 1 2 3 4 · 0 1 2 3 4 0 0 1 2 3 4 0 0 0 0 0 0
1 1 2 3 4 0 1 0 1 2 3 4
2 2 3 4 0 1 2 0 2 4 1 3
3 3 4 0 1 2 3 0 3 1 4 2
4 4 0 1 2 3 4 0 4 3 2 1
Let f(x) be an irreducible polynomial over GF (2) with degree n, and a finite
field with 2n elements is defined as
GF (2n) = {x0 + x1α+ · · ·+ xn−1αn−1 | xi ∈ GF (2)}
where the addition and multiplication of GF (2n) are the ordinary polynomial
addition and multiplication and the results are reduced by modulo f(x), which
is called a defining polynomial of GF (2n). Sometimes we say that GF (2n) is a
binary (extension) field. We write GF (q) where q = p for the prime field case
or q = 2n for the binary extension field. (Note that GF (p) can be extended
to GF (pn) in a similar way as to extend GF (2) to GF (2n).) For small fields,
the computations are usually done through the look-up table.
