CS 1501 Summer 2006 Quiz 2
SOLUTIONS
1)
Fill in the Blanks and True/False (20 points -- 2
points each).
Complete the statements below with the MOST APPROPRIATE words/phrases.
a)
An algorithm that
is linear in the value of an integer,
X, is _____exponential___________ in
the number of bits, N, in X.
b)
Using
c)
Given a substitution
cipher on an alphabet with S characters, there are _______S!______________ possible keys for the cipher.
d)
A graph with V vertices
and E edges is considered to be sparse if _____E <= VlgV ______
e)
A graph G is said to be biconnected
if ____there are two distinct
paths between all vertex pairs (or, it has no articulation points)_____.
Indicate whether each of the
following is TRUE or FALSE, explaining
why in an informative way for false answers.
f)
If a file contains completely random
data it cannot be effectively
compressed by any general compression algorithm. True
g)
A Vernam cipher is
provably secure for one-time use. True
h)
If someone comes
up with an efficient, polynomial-time factoring algorithm, the RSA encryption
scheme will no longer be useful. True
i)
The Miller-Rabin
Witness algorithm is used to verify the authenticity of sent
messages. False – it is used to test primality.
j)
An advantage of an adjacency
matrix representation of a graph is that all neighbors of a vertex
can be found in time Theta(V) (where V
is the number of vertices in the graph).
False
– the time is correct but it is a DISadvantage.
2)
(10 points) Consider the simple
divide and conquer algorithm (NOT Karatsuba's) for
N-bit integer multiplication. Show
precisely how each number is divided and how the multiplication is done on the
numbers (i.e. show the formula for XY based on how X and Y are divided). Also state and justify the recurrence
relation for this process.
Answer: From lecture
slides 140-145
X = 2N/2(XH) + XL Y = 2N/2(YH)
+ YL
XY =
(2N/2(XH) + XL)*(2N/2(YH)
+ YL)
XY = 2NXHYH + 2N/2(XHYL
+ XLYH) + XLYL
Examining the last
equation we see 4 multiplications of size N/2, some ~N bit additions and some
shifting (multiplication by a power of 2).
This leads to the recurrence: T(N) = 4T(N/2) +
Theta(N)
3) (10 points – 5
+ 5)
Consider RSA encryption
a)
Show (using pictures and explanation in detail) how a digital
signature works, and why it is used.
See
Slides 189-191 of online notes
b)
Explain the direct
method for breaking RSA that we discussed in lecture. Be specific (i.e. mathematical) about the
details. Also explain why this is a
difficult task for cryptanalysts.
See
Slides 185-186 of online notes. Breaking
RSA directly involves factoring the value N, which is part of both the private
and public keys. Once N is factored into
its primes (N = XY) we can then determine PHI (= (X-1)(Y-1)). The public key E is already known. Thus the cryptanalyst will have E and PHI and
can determine D (the decryption key) by solving the formula ED mod PHI = 1.
Factoring
is believed to be an exponential problem, which, for large bit sizes is
infeasible to do in a reasonable period of time. Thus, if the keys are large and are not kept
for two long, they are quite secure from this attack.
4) (10
points – 8 + 2)
Consider
the graph below. Assume the vertices are
stored in alphabetical order, and that the edges
are stored in alphabetical order for each vertex.
a)
Complete
the table below, as it would look after a Depth-First Search Spanning
Tree (starting from vertex A) were created for the graph. val[] is the DFS visit order for the vertex, and dad[] is the parent vertex
in the DFS tree. Show your work above or
in the space below the table for partial credit.
|
|
A |
B |
C |
D |
E |
F |
G |
H |
I |
J |
|
val |
1 |
2 |
3 |
4 |
5 |
8 |
7 |
9 |
10 |
6 |
|
dad |
0 |
A 1 |
B 2 |
C 3 |
D 4 |
A 1 |
J 10 |
F 6 |
F 6 |
D 4 |
b)
Identify
all of the articulation points in the graph.
A,
B, D, F