Cryptography Reference
In-Depth Information
either by trial and error, i.e., brute-force search, or by applying the Euclidean algo-
rithm for polynomials. (However, the Euclidean algorithm is only sketched in this
chapter.) Verify your answer by multiplying the inverses you determined by
A
and
B
, respectively.
4.8.
Find all irreducible polynomials
1. of degree 3 over
GF
(2),
2. of degree 4 over
GF
(2).
The best approach for doing this is to consider all polynomials of lower degree and
check whether they are factors. Please note that we only consider monic irreducible
polynomials, i.e., polynomials with the highest coefficient equal to one.
4.9.
We consider AES with 128-bit block length and 128-bit key length. What is the
output of the first round of AES if the plaintext consists of 128 ones, and the first
subkey (i.e., the first subkey) also consists of 128 ones? You can write your final
results in a rectangular array format if you wish.
4.10.
In the following, we check the
diffusion properties
of AES after a sin-
gle round. Let
W
=(
w
0
,
w
1
,
w
2
,
w
3
)=(
0x01000000
,
0x00000000
,
0x00000000
,
0x00000000
) be the input in 32-bit chunks to a 128-bit AES. The subkeys for the
computation of the result of the first round of AES are
W
0
,...,
W
7
with 32 bits each
are given by
W
0
=(
0x2B7E1516
)
,
W
1
=(
0x28AED2A6
)
,
W
2
=(
0xABF71588
)
,
W
3
=(
0x09CF4F3C
)
,
W
4
=(
0xA0FAFE17
)
,
W
5
=(
0x88542CB1
)
,
W
6
=(
0x23A33939
)
,
W
7
=(
0x2A6C7605
)
.
Use this topic to figure out how the input is processed in the first round (e.g., S-
Boxes). For the solution, you might also want to write a short computer program or
use an existing one. In any case, indicate all intermediate steps for the computation
of
ShiftRows
,
SubBytes
and
MixColumns
!
1. Compute the output of the first round of AES to the input
W
and the subkeys
W
0
,...,
W
7
.
2. Compute the output of the first round of AES for the case that
all
input bits are
zero.
3. How many output bits have changed? Remark that we only consider a single
round — after every further round, more output bits will be affected (
avalanche
effect
).
