Theorem 1 Let cfw_pi and cfw_qi be two sequences of length n 1 such that p1 + + pn = qi + + qn = 1
and pi , qi 0 for all i. Then the following inequality must hold:
n
X
pi ln qi
i=1
n
X
pi ln pi .
(0.1)
i=1
Proof
Draw a polygon with vertices (xi , xi l
Breaking Rectangular
ma187s: Cryptography
October 16, 2005
Ciphertext Only Attack of Rectangular Transposition
We shall view rectangular transposition in a slightly different manner than we did originally.
We assume that the sender and receiver choose a s
The Canonical Roulette and the game of Craps
1
The Canonical Roulette
To have a random number generator we shall imagine an ideal device that will be responsible
for generating random numbers.
Imagine that it consists of a solid wheel whose outer circumfr
Trees, codes, information and entropy
1. Codes and English
A code is an alphabet or a list of words which are used in place of another alphabet or
group of letters. All of our encryption systems are examples codes whose purpose is to hide
the meaning of a
Information Theory
ma187s: Cryptography
June 10, 2005
OF BASIC FACTS
INFORMATION THEORY: RESUME
Entropy and Information
If A = cfw_A1 , A2 , ., Ak is a partition of the probability space we set
H(A) =
k
X
P (Ai ) log2 1/P (Ai )
(1)
P [X = a] log2 1/P [X
The Markov Chain Monte Carlo Revolution
Persi Diaconis
Abstract
The use of simulation for high dimensional intractable computations has revolutionized applied mathematics. Designing, improving and understanding the new tools leads to (and leans on) fascin
Math 4161
January 5, 2015
Cryptography is the science of secret writing, enabling the condentiality
of communication through an insecure channel.
Basic objectives of cryptography:
1. Condentiality - keep information secret from unauthorized parties.
2. Da
Math 4161
January 14, 2015
Letter Frequency Analysis
Estimated probabilities of occurrence of the letters:
letter
A
B
C
D
E
F
G
probability
0.082
0.015
0.028
0.043
0.127
0.022
0.020
letter
H
I
J
K
L
M
N
probability
0.061
0.070
0.002
0.008
0.040
0.024
0.06
Math 4161
January 09, 2015
Numbers
1 Denition. Let a, b Z.
We say a divides b if b = ac, for some integer c. We write a | b.
The greatest common divisor of a and b, denoted by gcd(a, b), is the largest
positive integer that divides both a and b.
We say
Math 187 - Cryptography
Garsia - Spring 2013
QUIZ 7
Name:
PID:
You may use your notes. You may not use your neighbors. You will need a calculator. Show all your work.
(1) Consider the random cryptosystem pictured here:
Suppose that m1 is chosen with proba
Breaking Vigenere
Cryptography
February 6, 2017
Index of Coincidence
Suppose that the ciphertext is
C = C1 , C2 , . . . , CN
which is known to have been encrypted by a Vigen`ere substitution with keyword of length p. For
convenience we assume that N (the
Index of Conincidence
A. Garsia - 1988
The estimation of period in a substitution cipher.
Let us suppose that we are given a ciphertext
C1 , C2 , . . . , CN
which is known to have been encrypted by a Vigenere substitution with keyword of length
p. For con
Caesar
ma187s: Cryptography
October 16, 2003
Caesar and Vigenere Substitutions
Given the plaintext
A penny saved is a penny earned
let us use the following substitution
ABCD EFGH I J K L MNOPQR S T UVWXY Z
DE FGH I J KLMNO P QR S TUVWXY Z ABC
(1)
to enc
Rectangular Transposition
ma187s: Cryptography
October 16, 2003
Complete Rectangular Transposition with Keyword
This cipher consists of a single key k 1 : a word of length n. To give an example of this method
we will encode the following plaintext
Challen
ADFGVX
ma187s: Cryptography
September 18, 2007
The ADFGVX System
The ADFGVX cipher system consists of two keys:
k1 : A square consisting of 6 rows and 6 columns. The rows and columns are labeled from top
to bottom and left to right by A, D, F, G, V and X
Vernam
ma187s: Cryptography
October 16, 2003
The Vernam Two Tape System
In an early (1926) paper on secret communication by wire and telegraph G. S. Vernam proposed
an encryption system based on a pseudo random one time pad. The pad itself was to be const
Homophonic
ma187s: Cryptography
October 16, 2003
Homophonic Substitution
Homophonic substitution is another method that is used to fool frequency analysis. In this
cipher, we will replace each letter of the alphabet with a particular number based on a ran
Hill
ma187s:Cryptography
October 29, 2004
Hill Encipherment
In the Hill Encipherment, the key consists of the following ingredients:
1. The BLOCK SIZE: An integer k;
2. The HILL MATRIX: A is an k k matrix of integers 0, 1, . . . , 28 whose determinant is
Playfair
ma187s: Cryptography
September 18, 2007
The Playfair Encipherment System
This is an encipherment system devised by the Baron Playfair of St Andrews for the purpose
of secret communication. It uses the Latin alphabet and it enciphers based upon a
The Jargon of Probability
EXPERIMENT, RANDOM VARIABLES: This refers to an activity, not necessarily
scientific, which involves the production of data some of which are random. We
denote an experiment by E and the data by X, Y, Z, . . . The latter are usua
Information Theory
ma187s: Cryptography
October 26, 2005
CRYPTOGRAPHY AND INFORMATION
We shall work here with a Random Cryptographic System. More precisely, the ingredients are
as follows:
a) A message space
M = cfw_m1 , m2 , . . . , mn .
These are the pl
Monkey Words
ma187s: Cryptography
October 16, 2003
Probability as a Tool for Codebreaking
In our exploration of cryptography, we want to exploit the structure of plaintext messages.
To this end, let us first consider the statistical structure of english p
Identities and Inequalities
ma187s: Cryptography
October 26, 2005
The Entropy Function: Identities and Inequalities
Theorem 1 H(X, Y ) = H(X) + H(Y |X) = H(Y ) + H(X|Y )
Proof. We prove only the first equality since the second is proved in an entirely ana
Math 4161 Assignment 3
Due on Monday April 13 at 11:30 a.m.
Submit your solution in class.
1. Denition: Let n be a positive integer. A Latin square of order n is
an n n array L whose entries are the integers 1, . . . , n such that each
number appears exac