M ath 470/501 Spring 2011
Exam 1
H and work ONLY
Name:
U IN:
#1. A Linear Feedback Shift Register with initialization vector 00100 and
r ecursion
m od 2 is used to encrypt a binary string.
T he encrypted string is 011010001110. Find the plaintext.
We comp
M ath 470/501 Spring 2011
Exam 1
M aple allowed
Name:
U IN:
#1. The title of this 1966 musical has been encrypted by a shift cipher:
For $200, Watson, what is its name?
> restart:
with(LinearAlgebra):
with(numtheory):
with(plots):
numb:=table(["=0,"a"=1,"
MATH 470 Exam 2
You may use a calculator provided it only has the ability to add, multiply, subtract and divide.
Some facts you might nd useful
(mod 1271) relations
2635 993 , 3635 150 , 23635 1270 ,
(mod 347) relations
8339 334 8166 3
137.8 55 137.82 93
MATH 470 Homework 6 Solutions
[1] Use the p 1 factoring method to compute the factors of n = 17513 . Only a modest, single digit value of B
is needed here so you should show all the steps of the calculation.
Try B = 6 so that 6! = 720 and 2720 (mod n) is
p. 193 #11.
Suppose that
a nd
W e cannot have
=
, since
o therwise
b y unique factorization in the integers. Without loss of
g enerality assume that
= , t he case of
b eing similar. Then we can
q uickly compute
b y using the Euclidean algorithm on
a nd .
MATH 470 Homework 1 Solutions
[1] Naive ned always signs his messages by appending ned afterwards. Use this to break his afne
cipher and nd the message that he sent. You have intercepted the ciphertext kgghehryyrrgz.
This means that ned rgz or 13 17, 4 6,
MATH 470 Homework 2 Solutions
[1] Show that gcd(1729, 801) = 1 and compute numbers s and t such 1729 + 801 = 1.
You should construct many similar examples on your own (using, for example, the code provided,
to check your work).
A by now routine run of the
SOLUTIONS MANUAL
for
INTRODUCTION TO
CRYPTOGRAPHY
with Coding Theory, 2nd edition
Wade Trappe
Wireless Information Network Laboratory
and the Electrical and Computer Engineering Department
Rutgers University
Lawrence C. Washington
Department of Mathematic
Math 470
Answers to Exam 2 Sample Problems
October 30, 2011
Please note: Not all of the work for solving these problems is given in the solutions below, but mainly
the solutions are intended to give you enough information about how to solve the problems c
Math 470
Exam 1 Sample Problems
September 22, 2011
Please note: These problems by no means make an exhaustive list of the types of problems that we have
seen in class or on the homework or that you can expect to see on the exam.
1. Use the Euclidean Algor
Math 470
Final Exam Sample Problems
December 2, 2011
Please note: These problems only cover material since the second mid-term exam, and even then they do
not cover all possible topics or types of problems. Mainly they can serve to provide you with ideas
Math 470
Answers to Final Exam Sample Problems
December 6, 2011
Please note: These problems only cover material since the second mid-term exam, and even then they do
not cover all possible topics or types of problems. Mainly they can serve to provide you
SCORE/xx:
Math 470
Communications and Cryptography
NAME:
PRACTICE MIDTERM I
(more problems and solutions to be posted soon.)
Please show your work and write only in pen. Notes are forbidden. Calculators, and all
other electronic devices, are forbidden. Br
Math 470
Answers to Exam 1 Sample Problems
September 27, 2011
1. Use the Euclidean Algorithm to nd gcd(1333, 1591) and to nd one solution of 1333x + 1591y =
gcd(1333, 1591) with x, y Z.
Solution: First we apply the Euclidean algorithm:
1591 = 1333 + 258,
Math 470
Exam 2 Sample Problems
October 25, 2011
Please note: These problems by no means make an exhaustive list of the types of problems that we have
seen in class or on the homework or that you can expect to see on the exam.
1. David and Eleanor are usi
MATH 470 Homework 8 Solutions
[1] Suppose you have discovered that 36 44 (mod 137) and 310 2 (mod 137) . Can you compute
x such that 3x 11 (mod 137) ?
A good deal of the work is already done: L3 (44) = 6 and L3 (2) = 10 so L3 (4) = 20 and we want
L3 (11)
When we look at ciphers systems between two (or more) parties to preserve secrecy
of information we have to consider several things.
When we look at ciphers systems between two (or more) parties to preserve secrecy
of information we have to consider sever
MATH 470 Homework 7 Solutions
[1] Use the p 1 factoring method to compute the factors of n = 17513 . Depending on how you
code this you may have to compute b = aB! (mod n) by using the sequence b1 = a (mod n) ,
bj = bj
j1 (mod n) for j = 2, . . . B . Simp
f (x) = x2 n (or f (x) = (x + a)2 n, x = 0, 1, . . .)
Look at f (a), f (a + 1), f (a + 2), f (a + x), . . . for each x 0.
Which numbers in the above list are divisible by p ?
Look at f (x) = x2 n 0 (mod p). This is a quadratic equation and for each p > 2
MATH 470 - Spring 2015
INSTRUCTOR:
OFFICE:
E-MAIL:
CLASS HOME PAGE-MAIL:
OFFICE HOURS:
William Rundell
Blocker 614C
[email protected]
http:/calclab.math.tamu.edu/rundell/m470/
Tuesday 8:30-9:30; Wednesday 10-12;Thursday 1-2pm;
I am often on-line and can pr
Communications and Cryptography
MATH 470 - 504, Fall 2015
Assignment 6
due Oct 19, 2015
Exercise 1 (10 points).
Alice sets up the RSA cipher as follows: p = 3877, q = 4253, e = 5. What is her public key and
what is her private key?
Bob wants to send her t
The Quadratic Sieve
Factoring a number n using the quadratic sieve depends on the familiar principle: If there exists
x , y with x y (mod n) such that x2 y 2 (mod n) then n is composite and gcd(x y, n)
factors n . This is the quadratic part of the name.
T
MATH 470 Homework 4 Solutions
[1] If x 1 (mod 4) , x 3 (mod 5) and x 4 (mod 7) , what is x (mod 140) ?
The rst congruence shows that x = 1 + 4s for some integer s . Using this in the second gives
1+4s 3 (mod 5) or 4s 2 (mod 5) . Finding the inverse of 4 (
MATH 470 Homework 3 Solutions
[1] Given that x 17 (mod 1729) and x 29 (mod 801) nd the value of x (mod 1729.801).
We must use the Chinese remainder Theorem: if x a (mod m), x b (mod n), with
gcd(m, n) = 1 (as we have here from the problem above) then the
MATH 470 Homework 5 Solutions
[1] Sally Silly makes up her own version of RSA. She doesnt want to mess with two primes so she just
takes the modulus n to be a prime p but everything else is the same. Why is this not a good idea?
Because now she has to cho
MATH 470 Homework 6 Solutions
[1] For each of the following use, the Jacobi symbol to determine which of the quadratic problems have
a square root giving reasons.
[a] x2 23 (mod 173)
[b] x2 28 (mod 99)
The Jacobi symbols have value +1 and 1 respectively.
MATH 470 Homework 1
Solutions
[1] If you make up an afne cipher with the function y 15x + 11 (mod 26) what would be the
decryption function?
Then 15x y 15 (mod 26). Also 7.15 1 (mod 26), (that is, 7 is the multiplicative inverse
for 15 (mod 26) so that mu
MATH 470 Homework 1
Solutions
[1] If you make up an afne cipher with the function y 15x + 11 (mod 26) what would be the
decryption function?
Then 15x y 11 (mod 26). Also 7.15 1 (mod 26), (that is, 7 is the multiplicative inverse
for 15 (mod 26) so that mu
MATH 470 Some Practice Problems
(try to work them before looking at the solution)
1. What are the last two digits of 123456 ?
We want to know the remainder after we divide this number by 100. That is, calculate
123456 (mod 100). Now (100) = 100(1 1 )(1 1
4/17/2008
Digital Signatures
Murat Kantarcioglu
Based on Prof. Lis Slides
Digital Signatures: The Problem
Consider the real-life example where a person
pays by credit card and signs a bill; the seller
verifies that the signature on the bill is the same
w
Review Exam
October 29, 2016
1.
a) (10 pts.) Describe the ElGamal cryptosystem in terms of a communication between Alice and
Bob.
b) (5 pts.) What mathematical problem does the security of the ElGamal protocole against eavesdroppers depend on.
Oct 29, 201
Review Exam
September 22, 2016
1. You want to share two keys (numbers) with Alice using Diffie Hellman
key exchange protocol. You and Alice have agreed to use modulus p = 17
and the base g = 2. Alice sends you A1 = 13 and A2 = 15.
a) Choose two secret num
Final Review
December 5, 2016
1.
Answer the following common questions
a) (5 pts.) In modern cryptography what two things should you assume to be known to an adversary.
b) (5 pts.) Explain, through a conversation between Alice and Bob,
a public key encryp
MATH 470 Homework 1
Natalie Cluck
September 7, 2016
1
Chapter 1, #5
a.) If c|a then c|am where m is any arbitrary number. Since m can be any
number, let m b. Therefore c|ab. For c|b, the same method applies if any
number a is multiplied with b, it is stil
MATH 470 Homework 2
Natalie Cluck
September 14, 2016
1
Chapter 2, #1
a.) 54 32
b.) 5625 is a square.
2
Chapter 2, #2
a.) 26 33
b.) It is squarefree, and it is a cube.
3
Chapter 2, #6
a.) Let a and b be integers.
a2 p2a2 3a3 5a5 .q2
b2 p2b2 3b3 5b5 .q2
dp1