Cryptography Reference
In-Depth Information
Rijndael and Serpent are the candidates with the best performance in
hardware implementations, with a slight advantage going to Rijndael due to
its better performance in output and cipher feedback modes.
The report offers further criteria that contributed to the decision in favor of
Rijndael, which are collected into a closing summary (see [NIST], Section 7):
There are many unknowns regarding future computing platforms
and the wide range of environments in which the AES will be
implemented. However, when considered together, Rijndael's
combination of security, performance, efficiency, implementability,
and flexibility make it an appropriate selection for the AES for use in
the technology of today and in the future.
Given the openness of the selection process and the politically interesting fact that
with Rijndael an algorithm of European vintage was selected, one might expect
future speculation about secret properties, hidden trap doors, and deliberately
built-in weaknesses to be silenced, which never quite succeeded with DES.
Before we get involved with the functionality of Rijndael, we would like as
preparation to go on a brief excursion into the arithmetic of polynomials over
finite fields, which leans heavily on the presentation in [DaRi], Section 2.
11.1 Arithmetic with Polynomials
F
2
n
, the finite field with
2
n
We start by looking at arithmetic in the field
2
n
is represented as a polynomial
f
(
x
)=
a
n
−
1
x
n
−
1
+
a
n
−
2
x
n
−
2
+
···
+
a
1
x
+
a
0
with coefficients
a
i
in
F
elements, where an element of
F
2
(which
F
2
n
can be represented
simply as an
n
-tuple of polynomial coefficients, each representation offering
its own advantages. The polynomial representation is well suited for manual
calculation, while the representation as a tuple of coefficients corresponds well to
a computer's binary representation of numbers. To demonstrate this, we notate
F
2
3
as a sequence of eight polynomials and again as eight 3-tuples with their
associated numerical values (see Table 11-1).
Addition of polynomials proceeds by adding the coefficients in
Z
2
). Equivalently, an element of
is isomorphic to
F
2
:If
f
(
x
):=
x
2
+
x
and
g
(
x
):=
x
2
+
x
+1
, then
f
(
x
)+
g
(
x
)=2
x
2
+2
x
+1=1
,
since
1+1=0
in
F
2
3
column by
column. We see, then, for example, that the sum of
(1 1 0)
and
(111)
is
(001)
:
F
2
. We can carry out addition of 3-tuples in
110
⊕
111
001
