Dr. A. Betten
Fall 2005
M360 Mathematics of Information Security
practice exam 10/28/05
not graded!
Exercise # 1
( points)
a) What is X 4 + X 3 + X + 1 divided by X 2 + X + 1 over Z2 ?
b) What is X 6 + X 3 + X + 1 divided by X 2 + X + 1 over Z2 ?
c) What is X 4 + 2X 2 + X + 2 divided by X 2 + 2 over Z3 ?
Exercise # 2
( points)
a) What is X 4 + X 3 + X + 1 times X 2 + X + 1 modulo X 4 + X 3 + 1 over Z2 ?
b) What is X 4 + 2X 2 + X + 2 times X 2 + 2 modulo X 3 + 2 over Z3 ?
Exercise # 3
( points)
Compute the gcd of the two polynomials over Z2 p(X) = X 3 + X 2 + 1 and
q(X) = X 4 + X and express it as a linear combination of the two.
Exercise # 4
( points)
2
Show that X + 1 is irreducible in Z3 [X]. Find the multiplicative inverse of
1 + 2X in Z3 [X] mod X 2 + 1.
Exercise # 5
( points)
a) Find all four solutions to x2 ≡ 133 mod 143
b) Find all two solutions to x2 ≡ 77 mod 143
Exercise # 6
( points)
A group of people are arranging themselves for a parade. If they line up three
to a row, one person is left over. If they line up four to a row, two people are
left over., and if they line up five to a row, three people are left over. What is
the smallest possible number of people? What is the next smallest number?
Exercise # 7
Find the last two digits of 999666 .
( points)
Exercise # 8
( points)
a) List the two differences between DES decryption and DES encryption.
b) Decrypt the simple-DES ciphertext
AQMn
using the key 011010011. Use the table below to code symbols to 6-bit
integers. Use the simple-DES machines on the attached sheets (note that
you have to change something for decryption). The S-boxes are:
101 010 001 110 011 100 111 000
S1 :
001 100 110 010 000 111 101 011
100 000 110 101 111 001 011 010
S2 :
101 011 000 111 110 010 001 100
a
b
c
d
e
f
g
h
i
j
k
l
m
n
o
p
dec.
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
binary
000000
000001
000010
000011
000100
000101
000110
000111
001000
001001
001010
001011
001100
001101
001110
001111
q
r
s
t
u
v
w
x
y
z
’’
0
1
2
3
4
dec.
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
binary
010000
010001
010010
010011
010100
010101
010110
010111
011000
011001
011010
011011
011100
011101
011110
011111
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
dec.
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
binary
100000
100001
100010
100011
100100
100101
100110
100111
101000
101001
101010
101011
101100
101101
101110
101111
Q
R
S
T
U
V
W
X
Y
Z
’.’
5
6
7
8
9
dec.
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
binary
110000
110001
110010
110011
110100
110101
110110
110111
111000
111001
111010
111011
111100
111101
111110
111111