Introduction to Cryptography Summer 01 Homework #2 Assigned: Tuesday, 5/22/01 Due: Tuesday, 6/5/01
Do the following problems from the textbook: ppg. 3943: 1.1, 1.4, 1.5, 1.11 Extra questions: For questions a and b, let a, b, c, d and n be arbitrary posit
Sample Cryptography Questions and Answers
1) (10 pts) Consider a substitution cipher where 52 symbols were used instead of 26. In
particular, each symbol in the cipher text is for either a lowercase English letter, or an
uppercase English letter. (For exa
Cryptography Lecture One Overview
We first talked about the shift cipher and then split up into three groups. Each group
would have a message for another using the shift cipher. The group the message was
intended for would know the key. Once each group de
2004 Summer COT 5937 Program #1
Advanced Encryption Standard
Assigned: 5/27/04 (Thursday)
Due: 6/16/04 (Wednesdsay)
(Note: Work on this project in pairs. If you work alone, you'll still have to do the
whole thing.)
You will implement AES encryption and de
Lecture 14: Factoring Algorithms
(Taken from Stinson section 4.8 and Elementary Number Theory, Rosen)
Fermat Factorization
Note that if n (odd)
n = ab, then we can rewrite n = s2  t2, for some value of s and t, since n = (st)(s+t),
where s = (a+b)/2 and
Lecture 13: Elliptic Curve Cryptography
In general, an elliptic curve can be defined by the equation y2+axy+by = x3+cx2+dx+e.
For our purposes, we'll only consider those of the form: y2 = x3 + bx + c.
Consider the following example: Let a=1, b=1. We defin
Lecture 12: Discrete Logarithms, RSA, DiffieHellman Key Exchange
Calculating Modular Exponents
Before we move on, one quick note about computation. Let's consider the amount of
work involved in exponentiation:
Consider calculating an1 (mod n), for a lar
Lecture 11: Phi function, Euler's formula, Probabilistic Primality Testing
Euler Phi Function
First, lets define the Euler (phi) function:
(n) = the number of integers in the set cfw_1, 2, ., n1 that are relatively prime to n.
(p) = p 1 , for all prime
Lecture 10: Blowfish and Intro to Num Theory
Uses a key anywhere from 32 bits to 448 bits, in increments of 32 bits.
Let the key be n*32 bits, then the key can be split into n parts: K1, K2, . Kn, where n is an
int in between 1 and 14, inclusive.
Now, you
2004 Summer COT 5937 Program #2
El Gamal and the Discrete Log Problem
Assigned: 7/13/04 (Tuesday)
Due: 7/21/04 (Wednesdsay)
The heart of the security of the El Gamal cryptosystem is the Discrete Log problem. Here
is the specific problem statement:
Given p
COT 5937 Quick Test Review Questions
1) Assuming you want no fixed points, (these are letters that encrypt to themselves) how
different valid keys are there for the affine cipher?
When a=1, only key that gives rise to a fixed point is b=0. So there are 25
ASSIGNMENT #1 1) Amount of data in the box = (1000GB = 1000*109 bytes = 8000*109 bits) = 8*1012 bits Speed of Fast Rider = 30km/hr = 8.33m/s If the Fast Rider analogy is compared to a network link carrying data, we see that in the case of the Fast Rider,
Assignment 4 Solution
1. Caesar Cipher k = 20 C = E(P + k) mod (26), C = Cipher Text, P = Plain Text k = shift "I am sending a secret message" i (8 + 20) mod 26 = 2 = c a (0 + 20) mod 26 = 20 = u m (12 + 20) mod 26 = 6 = g c ug myhxcha u mywlyn gymmuay 2.
Solution Assignment 9
7.3
a. A sends a connection request to B, with an event marker or nonce (Na) encrypted with the key that A shares with the KDC. If B is prepared to accept the connection, it sends a request to the KDC for a session key, including A's
Debugging

Most of the coding went quite smoothly. The main problem encountered was that
the % operator in C+ returns a negative number when the first operand is
negative. However, x0, when x is negative will always return a value in
between n+1 and 0.
Group: Set of elements with a binary operation cfw_G, that adheres to these properties:
A1) Closure
A2) Associative
A3) Identity Element (Additive), ae=ea=a
A4) Inverse Element, aa'=a'a=e
EXAMPLE: Permutation, On Paper
A group is abelian if the following
Block Cipher Principles
Two types: stream, block
A stream cipher encrypts one bit or byte at a time, often times adapting the encrypting
key based upon the previous bit or byte encrypted.
A block cipher breaks the plaintext into blocks of equal size and u
I. Breaking the Vigenere Cipher
The first key piece of information we would need in order to break this cipher is the
length of the keyword. Once we know this, we can group the letters in accordance to
which were shifted with the same key.
Two ways to get
Cryptography Lecture Two Overview
We started the lecture continuing to talk about the affine cipher. We first showed how to
find an inverse of a number a (mod n). Next, we used this information to show how to
find the keys to decrypt an affine cipher give
Cryptography Lecture 6/19/01 Overview
Public Key cryptography: The basic idea is to do away with the necessity of a secure key
exchange, which is necessary for all private key encryption schemes. The idea is as
follows:
1) Bob creates two keys, a public k
Cryptography(COT 5937) Lecture 13
6/21/01
Last time we summarized RSA encryption. One of the main reasons that RSA works is
Eulers Theorem, which was simply stated in the last lecture. This time, well prove it:
(n)
Eulers Theorem: If gcd(a,n) = 1, then a
Cryptography(COT 5937) Lecture 16
7/10/01
Data Encryption Standard(DES)
Here is the basic algorithm used for DES:
To encrypt a plaintext x of 64 bits and a secret key K of 56 bits do the following:
1) Compute x0 = IP(x), a fixed permutation of the bits in
COT 5937 Summer 2004 Exam #1 Solutions
1) The given information leads to setting up the two following equations:
f(17) = 17a + b = 40 (mod 52)
f(50) = 50a + b = 19 (mod 52)
Subtracting the top equation from the bottom we get 33a = 21 (mod 52)
Since gcd(5
Introduction to Cryptography Summer 01
Homework #1
Assigned: Thursday, 5/10/01
Due: Friday, 5/18/01 (5 pm)
In this class, I will have you all implement several cryptographic schemes. For each
scheme, you will be required to write a function that takes in
COT 5937 2004 Summer Homework #1 Solutions
1)
Encryption Key:
CRYPT
OGAHB
DEFIK
LMNQS
UVWXZ
Divide plaintext into pairs:
PL EA SE ME ET ME AT TH EP AR KT OM OR RO WA TX TW OP MI FI TI SR AI NI
NG IW IL LB EB EH IN DT HE RE DB EN CH
Encryption Process:
Eg.
Introduction to Cryptography Summer 01
Homework #3
Assigned: Tuesday, 6/19/01
Due: Tuesday, 6/26/01
Create a BigNum class that allows for integer operations on large values. In particular,
your class show be able to handle integers of up to 50 digits. Eac
Introduction to Cryptography Summer 01
Homework #4
Assigned: Tuesday, 6/26/01
Due: Thursday, 7/12/01
Do the following problems in the textbook(ppg. 157161): 4.3, 4.4, 4.6(both), 4.7, 4.8
1) Given that gcd(m,n) = 1, prove that (mn) = (m) (n).
2) Using the