Dr. A. Betten
Fall 2009
MATH 360 Mathematics of Information Security
Assignment # 3
Problem # 12
a) Find a primitive root g modulo 13.
b) Using your primitive root g, compute the integer x such that g x ≡ y mod 13. Here,
y = 0, 1, . . . , 12. Make a table of all (x, y) pairs.
Problem # 13
a) Find all four solutions to x2 ≡ 133 mod 143.
b) Find all two solutions to x2 ≡ 77 mod 143.
Problem # 14
Find a Fermat witness for the fact that 29353 is composite. What about 19267?
Problem # 15
After reading the chapter on DES (page 113 on), 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