The University of British Columbia
MATH 335 (201)
Midterm 2
16 March 2010
Time: 50 minutes
FULL NAME:
STUDENT # :
SIGNATURE:
This Examination paper consists of 5 pages (including this one). Make sure you have all 5.
instructions:
No memory aids allowed. Only one calculator allowed. No communication devices allowed.
marking:
Q1
Q2
Q3
Q4
TOTAL
/10
/10
/10
/10
/40
Name of Instructor: Mac Lean
MATH 335 (201) Midterm 2 — 16 March 2010 — p. 2 of 5
Q1
[10 marks]
Short answers: give one or two sentences as responses to the following. The use of examples is
encouraged.
(a) What is public key encryption?
(b) What is clock (or modular) arithmetic?
(c) What is the hypotenuse of a triangle?
(d) What is a Pythagorean triple ?
(e) What is a proof?
MATH 335 (201) Midterm 2 — 16 March 2010 — p. 3 of 5
Q2
[10 marks]
You have received the following encrypted message sequence from your friend, who has used
your public key to encrypt it by the RSA algorithm: (c1 , c2 , c3 ) = (20, 18, 14). Your private
key is (n, d) = (33, 7). Decrypt the message. The following table will be useful in turning the
message sequence (m1 , m2 , m3 ) into a word.
A
B
C
D
E
F
G
H
I
J
K
L
M
2
3
4
5
6
7
8
9
10
11
12
13
14
N
O
P
Q
R
S
T
U
V
W
X
Y
M
15
16
17
18
19
20
21
22
23
24
25
26
27
MATH 335 (201) Midterm 2 — 16 March 2010 — p. 4 of 5
Q3
[10 marks]
(a) Today is Tuesday, March 16, 2010. What day of the week is March 16, 2011?
(b) Use the multiplication table for Z5 to solve (i) 3x = 2 and (ii) 2x = 3:
·
0
1
2
3
4
0
0
0
0
0
0
1
0
1
2
3
4
2
0
2
4
1
3
3
0
3
1
4
2
4
0
4
3
2
1
(c) Consider the multiplication table for Z4 . Use it to explain why you can solve 3x = 2, and
why you cannot solve 2x = 3.
·
0
1
2
3
0
0
0
0
0
1
0
1
2
3
2
0
2
0
2
3
0
3
2
1
MATH 335 (201) Midterm 2 — 16 March 2010 — p. 5 of 5
Q4
[10 marks]
(a) State Pythagoras’ Theorem (you may use a diagram to help you) and give two examples
of Pythagorean triples.
(b) An equilateral triangle is drawn so that it has vertices at (0,0) and (8,0) in the coordinate
plane. What are the coordinates of the third vertex? Remember that the sides of an
equilateral triangle are all the same length. (Hint: You may use Pythagoras’ theorem to
solve this problem.) It will be useful to draw a diagram.