Spring 2014
ECE 597AB/697AB Security Engineering
Prof. Christof Paar
Homework Assignment # 4
Due Date: March 11, 2014
You have to show all intermediate results for all problems.
1. Assume your task is to compute xe where e is the exponent e = (87 654 321)
466
Advanced concepts in random processes
Notes
11.4: Specication of random processes
Note 1. Comments analogous to Note 1 in Chapter 7 apply here. Specically, the set B
must be restricted to a suitable -eld B of subsets of IR . Typically, B is taken to b
1
Problems
1. A coee shop has 4 dierent types of coee. You can order your coee in a small,
medium, or large cup. You can also choose whether you want to add cream, sugar,
or milk. In how many ways can you order your coee?
Solution:
We can use the multipli
1. Let X Unif orm(1, 3). Suppose that we know
Y | X=x
Geometric(x).
Find the posterior density of X given Y = 3, fX|Y (x|3).
Solution: Using Bayes rule we have
PY |X (3|x)fX (x)
.
PY (3)
fX|Y (x|3) =
We know Y | X = x
Geometric(x), so
PY |X (y|x) = x(1 x)
1
Problems
1. Suppose that the universal set S is dened as S = cfw_1, 2, , 10 and A = cfw_1, 2, 3,
B = cfw_x S : 2 x 7, and C = cfw_7, 8, 9, 10.
(a) Find A B
(b) Find (A C ) B
(c) Find A (B C )
(d) Do A, B, and C form a partition of S ?
Solution:
(a)
A B
1
Problems
1. Let X be a discrete random variable with the following PMF
1
for x = 0
2
1
for x = 1
3
1
PX (x) =
for x = 2
6
0
otherwise
(a) Find RX , the range of the random variable X .
(b) Find P (X 1.5).
(c) Find P (0 < X < 2).
(d) Find P (X = 0|X < 2)
1. Let Xi be i.i.d Unif orm(0, 1). We dene the sample mean as
Mn =
X1 + X2 + . + Xn
.
n
(a) Find E [Mn ] and Var(Mn ) as a function of n.
(b) Using the Chebyshevs inequality, nd an upper bound on
P
Mn
1
1
2
100
Mn
1
1
2
100
.
(c) Using your bound, show
1. Let X, Y and Z be three jointly continuous random variables with joint PDF
0 x, y, z 1
x+y
fXY Z (x, y, z ) =
0
otherwise
(a) Find the joint PDF of X and Y .
(b) Find the marginal PDF of X .
(c) Find the conditional PDF of fXY |Z (x, y |z ) using
fXY
1. I choose a real number uniformly at random in the interval [2, 6], and call it X .
(a) Find the CDF of X, FX (x).
(b) Find EX .
Solution:
(a)
We saw that all individual points have probability 0; i.e.,P (X = x) = 0 for all
x in uniform distribution. Al
1. Let X be the weight of a randomly chosen individual from a population of adult
men. In order to estimate the mean and variance of X , we observe a random sample
X1 ,X2 , ,X10 . Thus, Xi s are i.i.d and have the same distribution as X . We obtain
the fo
1. Consider two random variables X and Y with joint PMF given in Table 1.
Table 1: Joint PMF of X and Y in Problem 1
Y =1 Y =2
X=1
1
3
1
12
X=2
1
6
0
X=4
1
12
1
3
(a) Find P (X 2, Y > 1).
(b) Find the marginal PMFs of X and Y .
(c) Find P (Y = 2|X = 1).
(
Spring 2014
ECE 597AB/697AB Security Engineering
Prof. Christof Paar
Homework Assignment # 6
Due Date: April 22, 2014
You have to show all intermediate results for all problems.
1. Perform a Differential Fault Attack on the AES that computes the rst byte
Spring 2014
ECE 597AB/697AB Security Engineering
Prof. Christof Paar
Homework Assignment # 5
Due Date: April 8, 2014
You have to show all intermediate results for all problems.
0. Reading Assignment: Read Chapter 2 as well as Sections 5.1 and 5.2 from the
Spring 2014
ECE 597AB/697AB Security Engineering
Prof. Christof Paar
Homework Assignment # 5 Solution
Due Date: April 3, 2014
1.
(a) Transforming A and B:
A = (A mod 5, A mod 7, A mod 13) = (0, 1, 2)
B = (4, 1, 3)
Computing the CRT parameters:
M = (M/5, M
Spring 2014
ECE 597AB/697AB Security Engineering
Prof. Christof Paar
Homework Assignment # 4 Solution
Due Date: March 11, 2014
1. e = (87 654 321)10 = (101001110010111111110110001)2
Number of squarings: #SQ = 26 (length of binary representation -1)
Number
Spring 2014
ECE 597AB/697AB Security Engineering
Prof. Christof Paar
Homework Assignment # 3
Due Date: February 25, 2014
Unless indicated otherwise, all problems are from our textbook Understanding Cryptography.
1.
1. gcd(4314, 836) = 2
2. gcd(1155, 377)
Spring 2014
ECE 597AB/697AB Security Engineering
Prof. Christof Paar
Homework Assignment # 2
Due Date: February 18, 2014
Unless indicated otherwise, all problems are from our textbook Understanding Cryptography.
Note the that complete Chapter 4 of the boo
Spring 2014
ECE 597AB/697AB Security Engineering
Prof. Christof Paar
Homework Assignment # 1
Due Date: February 11, 2014
Unless indicated otherwise, all problems are from our textbook Understanding Cryptography.
1. Tyler Durden in "Fight Club".
PLAINTEXT:
Spring 2014
ECE 597AB/697AB Security Engineering
Prof. Christof Paar
Homework Assignment # 3
Due Date: February 25, 2014
Unless indicated otherwise, all problems are from our textbook Understanding Cryptography.
1. Problem 6.5, but use the numbers
(a) r0
Spring 2014
ECE 597AB/697AB Security Engineering
Prof. Christof Paar
Homework Assignment # 2
Due Date: February 18, 2014
Unless indicated otherwise, all problems are from our textbook Understanding Cryptography.
Note the that complete Chapter 4 of the boo