Cryptography Reference
In-Depth Information
Problems
4.1.
Since May 26, 2002, the AES (Advanced Encryption Standard) describes the
official standard of the US government.
1. The evolutionary history of AES differs from that of DES. Briefly describe the
differences of the AES history in comparison to DES.
2. Outline the fundamental events of the developing process.
3. What is the name of the algorithm that is known as AES?
4. Who developed this algorithm?
5. Which block sizes and key lengths are supported by this algorithm?
4.2.
For the AES algorithm, some computations are done by Galois Fields (GF).
With the following problems, we practice some basic computations.
Compute the multiplication and addition table for the prime field
GF
(7).Amul-
tiplication table is a square (here: 7
7) table which has as its rows and columns all
field elements. Its entries are the products of the field element at the corresponding
row and column. Note that the table is symmetric along the diagonal. The addition
table is completely analogous but contains the sums of field elements as entries.
×
4.3.
Generate the multiplication table for the extension field
GF
(2
3
) for the case
that the irreducible polynomial is
P
(
x
)=
x
3
+
x
+ 1. The multiplication table is in
this case a 8
×
8 table. (Remark: You can do this manually or write a program for
it.)
4.4.
Addition in
GF
(2
4
): Compute
A
(
x
)+
B
(
x
) mod
P
(
x
) in
GF
(2
4
) using the ir-
reducible polynomial
P
(
x
)=
x
4
+
x
+ 1. What is the influence of the choice of the
reduction polynomial on the computation?
1.
A
(
x
)=
x
2
+ 1,
B
(
x
)=
x
3
+
x
2
+ 1
2.
A
(
x
)=
x
2
+ 1,
B
(
x
)=
x
+ 1
4.5.
Multiplication in
GF
(2
4
): Compute
A
(
x
)
B
(
x
) mod
P
(
x
) in
GF
(2
4
) using the
irreducible polynomial
P
(
x
)=
x
4
+
x
+ 1. What is the influence of the choice of the
reduction polynomial on the computation?
1.
A
(
x
)=
x
2
+ 1,
B
(
x
)=
x
3
+
x
2
+ 1
2.
A
(
x
)=
x
2
+ 1,
B
(
x
)=
x
+ 1
·
4.6.
Compute in
GF
(2
8
):
(
x
4
+
x
+ 1)
/
(
x
7
+
x
6
+
x
3
+
x
2
)
,
where the irreducible polynomial is the one used by AES,
P
(
x
)=
x
8
+
x
4
+
x
3
+
x
+1.
Note that Table 4.2 contains a list of all multiplicative inverses for this field.
4.7.
We consider the field
GF
(2
4
), with
P
(
x
)=
x
4
+
x
+1 being the irreducible poly-
nomial. Find the inverses of
A
(
x
)=
x
and
B
(
x
)=
x
2
+
x
. You can find the inverses
