Math 342, Fall Term 2011
Final Exam
December 9th ,2011
Student number:
LAST name:
First name:
Signature:
Instructions
• Do not turn this page over. You will have 150 minutes for the exam.
• You may not use books, notes or electronic devices of any kind.
• Solutions should be written clearly, in complete English sentences, showing all your work.
• If you are using a result from the textbook, the lectures or the problem sets, state it properly.
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/10
/100
Math 342, Fall 2011
Final, page 2/11
Problem 1
a. Dene a is invertible in the commutative ring
points)
R and exhibit a unit in Z/30Z. (10
1
(15 points)
b. Use Euclid's algorithm to calculate
gcd(120, 14). (5 points)
Math 342, Fall 2011
2
Final, page 3/11
Problem 2
(10 points)
In this problem we consider the map
f : Z/30Z → Z/5Z given by
f ([n]30 ) = [n + 2]5 .
a. Assume that
[n]30 = [m]30 . Show that [n + 2]5 = [m + 2]5 . (5 points)
b. Is f a group homomorphism (for the addition operation)? Why or why not? (5
points)
Math 342, Fall 2011
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Final, page 4/11
Problem 3
A Linear Code (25 points)
In this problem we work over the eld with 7 elements, denoted F7 or Z/7Z.
Let H ∈ M3×7 (F7 ) be the following matrix:

1 1 0 1
H= 1 0 1 1
0 1 1 1
−1
0
0
and let CH = v ∈ F77 | Hv = 0 .
a. Show that
CH is a subspace of F77 . (5 points)
b. Show that
CH has weight 3. (7 points)
0
−1
0

0
0 
−1
Math 342, Fall 2011
c. Let
Final, page 5/11
Problem 3
v 0 ∈ F77 . Show that x ∈ F77 | Hx = Hv 0 is the coset CH + v 0 . (5 points)
e. For v 0 = (1, 2, 3, 6, 0, 1, 2)
from part c. (5 points)
mod 7 evaluate Hv 0 and nd the coset leader of the coset
Math 342, Fall 2011
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f. Find the v ∈ CH which is closest in Hamming distance to the
justify your answer. (3 points)
Problem 3
v 0 given in part e.;
Math 342, Fall 2011
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Problem 4
Reed-Solomon Codes (10 points)
RS
Let C
⊂ F77 be the Reed-Solomon code obtained by evaluating polynomials of degree
at most 3 at all 7 points of F7 . Find the weight of C . How does this code compare
with CH ?
RS
Math 342, Fall 2011
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Problem 5
Polynomials (15 points)
a. Calculate
points)
gcd(x3 + x + [1]3 , x2 + [2]3 ) in the ring of polynomials over F3 = Z/3Z. (5
b. Show that there are no
f, g ∈ F3 [x] so that
2
f
g
= x2 + [2]3 . (10 points)
Math 342, Fall 2011
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Final, page 9/11
Problem 6
RSA (15 points)
Bob advertises a public RSA key with modulus m = 33 and encoding exponent e = 7. You will
play the role of Eve, the eavesdropper.
a. Find the order
×
ϕ(m) of the group (Z/mZ)
b. Find the decoding exponent
c. Decode the messages
(4 pts).
d (4 pts).
[2]33 , [7]33 sent by Alice (7 pts).
Math 342, Fall 2011
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Final, page 10/11
Problem 7
Order of elements (10 points)
Let (G, e, ·) be a group, and let g ∈ G.
a. Show that
{n ∈ Z | g n = e} is an ideal of Z. (3 points)
b. Assume that
g 37 = e but g 6= e. Show that g n = e if and only if 37|n. (2 points)
Math 342, Fall 2011
c. Let m ≥ 1 and let
ab. Show: (5 points)
Final, page 11/11
a, b ∈ (Z/mZ)
×
have orders
rs 2 t
(r, s)
and
Problem 7
r, s respectively. Let t be the order of
rs
t
.
(r, s)