thesis cryptology

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SINGULAR VALUE APPROXIMATIONS IN CRYPTOGRAPHY

by

Joshua Boone

Bachelor of Science in Mathematics, Southern Illinois University, 2006

A Research Report

Submitted in Partial Fulfillment of the Requirements for theMaster of Science Degree

Department of Mathematicsin the Graduate School

Southern Illinois University CarbondaleJuly, 2008


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RESEARCH REPORT APPROVAL

SINGULAR VALUE APPROXIMATION IN CRYPTOGRAMS

By

Joshua Boone

A Research Report Submitted in Partial

Fulfillment of the Requirements

for the Degree of

Master of Science

in the field of Mathematics

Approved by:

Dr. Robert Fitzgerald, Chair

Dr. Jerzy Kocik

Dr. David W. Kammler

Graduate SchoolSouthern Illinois University Carbondale

June 30, 2008


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AN ABSTRACT OF THE RESEARCH REPORT OF

Joshua Boone, for the Master Of Science degree in Mathematics, presented on

August 2, 2008, at Southern Illinois University Carbondale.

Singular Value Approximations in Cryptography

MAJOR PROFESSOR: Dr. Robert Fitzgerald

The English Language is a vfc language, that is, vowels follow consonants

a majority of the time. We will use this fact, along with some creative linear

algebra, to separate the letters of a cryptogram into three major groups: vowels,

consonants, and neuter. First, an adjacency matrix is created, which shows how

many times a given letter follows another. Then the Singular Value Analysis of

this matrix begins, which will use the Perron-Frobenius Theorem. After some sign

comparisons of certain singular vectors are done, the letters are separated into the

aforementioned groups. This process is easily implemented as a computer program.

After partitioning, it is relatively simple to decode the message with some elementary

frequency analysis.

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TABLE OF CONTENTS

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

1 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Elementary Cryptography . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 The Singular Value Decompostion . . . . . . . . . . . . . . . . . . . 2

1.3 The Perron-Frobenius Theorem . . . . . . . . . . . . . . . . . . . . 5

2 Consonant/Vowel Partitioning via Singular Value Analysis . . . . . . . . 8

2.1 The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 The SVD and its rank-two approximation . . . . . . . . . . . . . . 9

3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Java and MATLAB Code used for computation . . . . . . . . . . . . . . 19

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

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LIST OF FIGURES

3.1 The Adjacency MatrixA . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 The second singular vectors. . . . . . . . . . . . . . . . . . . . . . . . . 13

3.3 The partitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.4 The Adjacency MatrixA . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.5 The second singular vectors. . . . . . . . . . . . . . . . . . . . . . . . . 16

3.6 The partitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.7 The Letter Frequency of the English Language. [7] . . . . . . . . . . . 18

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CHAPTER 1

PREREQUISITES

1.1 ELEMENTARY CRYPTOGRAPHY

In simplest terms, cryptography can be thought of as the study of hiding

information from unwanted viewers. One of the most ancient applications of cryp-

tography is transmission of secret messages. A cryptogram is an example of this

application.

Definition. A substitution cryptogramis a message in which each letter is replaced

by another in a one-to-one manner. The method by which this is done is called the

substitution cipher. Whitespace and punctuation are removed entirely.

Definition. Ashift cipheris a substitution cipher Csin which each letter is mapped

to another letter by a constant shift 1 s 26. If we assign a number to eachletter, beginning with 1 for A, 2 for B, all the way to 26 for Z, then we can write

Cs() = + s(mod26), where 1

26 corresponds to a letter in our alphabet.

Example 1.1.1. Austin wants to send Bob the message Kenny is great using

the C2 shift cipher. Notice that the letter Y will be mapped to A. The encrypted

message would be MGPPAKUITGCV. To decrypt, Bob would simply use the

inverse (here, C24) cipher on the message.

Breaking the substitution cipher of a cryptogram is usually done by analysis

of the relative frequency of certain letters, as well as certain combinations of letters.

This process can be simplified by first separating the letters into vowels and con-

sonants, which we explore in this paper. In order to understand the mathematics

behind this idea, we must first introduce a special matrix.

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Definition. Theadjacency matrix of a substitution cryptogram is a 2626 matrixwith integer entries such that the i, j entry is the number of times the j-th letter

follows the i-th letter in the cryptogram.

Notice that the sum of the components of the i-th row equals the sum of the

components of the i-th column, which equals the number of times the i-th letter

occurs in the cryptogram. Since a combination of two letters is called a digraph,

this matrix is also sometimes called the digraph frequency matrix.

1.2 THE SINGULAR VALUE DECOMPOSTION

In this section, we introduce some definitions and results from linear algebra.

This material is taken from [2].

Definition. The hermitian conjugate, or adjoint, of an m n matrix A (writtenA) is the n m matrix whose i, j entry is the complex conjugate of the j, i entryofA. IfA= A, A is called hermitian.

Definition. A square matrix Q Cnn

is unitary ifQ

=Q

1

.

Definition. Let A Cmn. A singular value decomposition (SVD) ofA is a fac-torization

A= UW

where

U

Cmm is unitary,

W Cnn is unitary, and

Rmn is diagonal with entriesi such that1 2 . . . k 0, wherek= min(m, n)

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The components of this factorization have special names. The columns of U

and Ware called the left and right singular vectors, respectively. The i are called

thesingular values.

Of course, this factorization is only useful when it exists. The following theo-

rem proves thateverycomplex matrix has a singular value decomposition. In order

to understand the theorem, we must first understand the vector and matrix norms.

Definition. The2-normof a column vector v Cm is the length ofv, given by

v2

=

vv

wherev is the adjoint ofv.

Definition. The induced 2-norm of a matrix A Cmn is the smallest numberC Csuch that the inequality

Av2 Cv

2

holds for every v Cn.

Theorem 1.2.1. Every matrixA Cmn has an SVD.

Proof. Set 1 =A2. We can assume 1 > 0 since otherwise A = 0 and thedecompostion 0I0 (where 0 is the zero matrix) is an SVD. Since{Az: z

2= 1}

is compact,Az2

achieves its maximum. So, there must be vectorsw1 Cn andu1 Cm with w12= u12= 1 andAw1= 1u1. Consider any extensions ofw1to

an orthonormal basis {wj} ofCn

and ofu1 to an orthonormal basis {uj} ofCm

, andlet U1 and W1 denote the unitary matrices with columns uj and wj, respectively.

Then

U1AW1= S=

1 x

0 B

, (1.1)

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where 0 is a column vector of all zeros with dimension m 1, x is a row vector ofdimensionn 1, and B has dimensions (m 1) (n 1). Furthermore,

1 x

0 B

1x

2

21+ xx= (21+ xx)1/21

x

2

, (1.2)

which impliesS2 (21+ xx)1/2. Since U1 and W1 are unitary, we can see that

S2

=A2

= 1, so x = 0. If n = 1 and m = 1, were done. Otherwise, B

describes the action of A on the subspace orthogonal to w1. By induction, B has

an SVD: B = U22W

2 . Since the product of diagonal matrices is diagonal, it is

obvious that

A= U1

1 0

0 U2

1 0

0 2

1 0

0 W2

W1 (1.3)

is an SVD ofA.

An important piece of information is the rankof the matrix A, which can be

determined as follows:

Theorem 1.2.2. The rank ofA isr, the number of nonzero singular values.

Proof. The rank of a diagonal matrix is the number of nonzero entries. Since, in

A = UW, U and W are of full rank, the rank of A is the rank of , which is

r.

For the sake of this particular paper, it is more useful to consider the SVD as

a sum of rank-one matrices, as follows:

Theorem 1.2.3. The matrix A Cmn can be written as the sum of r rank-onematrices, wherer is the rank ofA. In other words,

A=r

j=1

jujw

j . (1.4)

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Proof. If we consider the matrix as the sum of r matrices i, where i =

diag(0, ..., 0, i, 0, ..., 0), then we can see this follows directly from (1.1).

We will use arank two approximationof the SVD, by taking only the first two

terms of the sum.

1.3 THE PERRON-FROBENIUS THEOREM

One last piece of information we need is the behavior of positive matrices. The

Perron-Frobenius theorem gives us all the information we need. We will assume the

following topological fact in order to prove this theorem. This material is taken

from [3].

Theorem 1.3.1. (Brouwers Fixed Point Theorem). LetS be a nonempty, closed,

bounded, convex set inRn. Letf : S Sbe a continuous map. Then there existsanx Ssuch thatf(x) =x.

The following theorem is one of the most important theorems in matrix the-

ory. There are many versions of this theorem, but the following simple version is

sufficient. Here, the notation A >0 means the matrix Ahas only positive entries.

Theorem 1.3.2. (Perron-Frobenius Theorem). Let A > 0 be an n n matrix.Then:

(i) Ay= 0y for some0 >0, y >0

(ii) The eigenvalue0 is maximal in modulus among all the eigenvalues ofA. In

other words, for any eigenvauee ofA,|e| 0.

(iii) The eigenvalue0 is geometrically simple, so that any two eigenvectors corre-

sponding to 0 are linearly dependent.

(iv) Any positive eigenvector ofA (of any eigenvalue) is a scalar multiple ofy.

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Proof.

(i) Let

S=

(x1, x2,...,xn)

T

:i x

i = 1 and xi 0 for all i .Define the map f :S Sas follows:

f(x) =

i

(Ax)i

1

Ax

Here (Ax)i is the i-th coordinate of Ax. Since A > 0, x S implies thatthe vector Ax is nonzero and the map f is well-defined. It is easy to check

thatf is continuous and maps S into S. By Brouwers Fixed Point Theorem,

f(y) =y for some y S. Thusi

(Ay)i

1

Ay= y.

If we set

i(Ay)i = 0, then 0 > 0 and Ay = 0y. Since A > 0, it follows

that y >0.

(ii) If we apply (i) to AT, we see that ATz=1zfor some 1 >0, z >0. Now

1yTz= yTATz= 0y

Tz,

and since yTz > 0, we have 0 = 1. Thus ATz = 0z. Let Au = eu for

some eigenvalue e C. Let u+ be defined by u+ = (|u1|, |u2|, ..., |un|)T, whereu= (u1, u2,...,un)

T. Without loss of generality, let u+ be a probability vector.

We have

j

aij |uj| j

aijuj

= |eui| = |e||ui|.Thus Au+ |e|u+. If we premultiply this by zT, we get|e| 0

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(iii) SupposeAv= 0v for some real, nonzero vector v . We want to show that v is

a scalar multiple ofy. Ifv and y are linearly independent, then R suchthaty v is a nonnegative, nonzero vector with at least one zero coordinate.Since

A(y v) =0(y v),

we see that y v is an eigenvector of A. However, since A > 0, any non-negative eigenvector must be postive. This is a contradiction. Thus, v is a

scalar multiple of y. By considering the real and imaginary part separately,

we can show that any complex eigenvector ofA corresponding to0is a scalar

multiple ofy.

(iv) Suppose Au= eu for u > 0. Then ezTu= zTAu= zTu. Since zTu >0, we

have e= 0. The rest follows from (iii).

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CHAPTER 2

CONSONANT/VOWEL PARTITIONING VIA SINGULAR VALUE

ANALYSIS

2.1 THE PROBLEM

Now we are ready to discuss the problem of breaking a given cryptogram. The

idea that follows is from [4]. Our entire scheme relies on the fact that, in the English

language, vowels usually follow consonants and vice versa. This is why English is

referred to as a vfc language. Formally, if

number of vowel-vowel pairs

number of vowels = A && ch = a && ch


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This MATLAB program returns the singular values, second left and rightsingular vectors, the frequency vector, and the partitions of the letters into vowels,consonants, and neuter elements.

%thesis.m%Josh Boone, Department of Mathematics%Southern Illinois University Carbondale%Coded for use in MATLAB, Mathworks Inc.%%Returns the singular values, the second right singular vector, the%second left singular vector, the vowel, consonant, and neuter%elements of the cryptogram, and the letter frequency vectorfunction thesis(A);[U, S, W] = svd(A);singular_values = diag(S)second_left_singular_vector = U(:,2)second_right_singular_vector = W(:,2)

letter_frequency_vector = sum(A)l = roundn(second_left_singular_vector, -5);r = roundn(second_right_singular_vector, -5);

k = max(size(A));alpha = ABCDEFGHIJKLMNOPQRSTUVWXYZ;

disp(Vowels:);for i=1:k

if sign(r(i)) ~= sign(l(i))if r(i) > 0

disp(alpha(i));end;

end;end;

disp(Consonants:);for i=1:k

if sign(r(i)) ~= sign(l(i))if r(i) < 0

disp(alpha(i));end;

end;end;

disp(Neuter Elements:);for i=1:k

if sign(r(i)) == sign(l(i))disp(alpha(i));

end;end;

Java is a trademark of Sun Microsystems, Inc.MATLAB is a trademark of The MathWorks, Inc.Maple is a trademark of Maplesoft, Inc.

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REFERENCES

[1] Bapat, R.B. and Raghavan, T.E.S., Nonnegative Matrices and Applications,

Cambridge University Press, 1997.

[2] Bau, D. and Trefethen, L.,Numerical Linear Algebra, Society for Industrial and

Applied Mathematics, Philadelphia, PA, 1997.

[3] Gaines, H.F.,Cryptanalysis, Dover, New York, 1956.

[4] Moler, C. and Morrison, D., Singular Value Analysis of Cryptograms, The

American Mathematical Monthly, Vol. 90, No. 2, (Feb., 1983), pp. 78-87.

[5] Famous Speeches by Presidents, Prime Ministers and

Martin Luther King. Famous Quotes. May-June 2008

.

[6] Sinkov, A.,Elementary Cryptanalysis, A Mathematical Approach, New Math-

ematical Library, vol 22, Mathematical Association of America, Washington,

D.C., 1968.

[7] Frequency Analysis. Wikipedia. 29 Apr. 2008. May-June 2008

.

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APPENDIX


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GEORGE W. BUSHS 9/11 ADDRESS

Good evening. Today, our fellow citizens, our way of life, our very freedomcame under attack in a series of deliberate and deadly terrorist acts. The victimswere in airplanes, or in their offices; secretaries, businessmen and women, military

and federal workers; moms and dads, friends and neighbours. Thousands of liveswere suddenly ended by evil, despicable acts of terror. The pictures of airplanesflying into buildings, fires burning, huge structures collapsing, have filled us withdisbelief, terrible sadness, and a quiet, unyielding anger. These acts of mass murderwere intended to frighten our nation into chaos and retreat. But they have failed; ourcountry is strong. A great people has been moved to defend a great nation. Terroristattacks can shake the foundations of our biggest buildings, but they cannot touchthe foundation of America. These acts shattered steel, but they cannot dent the steelof American resolve. America was targeted for attack because were the brightestbeacon for freedom and opportunity in the world. And no one will keep that lightfrom shining. Today, our nation saw evil, the very worst of human nature. Andwe responded with the best of America with the daring of our rescue workers,with the caring for strangers and neighbours who came to give blood and help

in any way they could. Immediately following the first attack, I implemented ourgovernments emergency response plans. Our military is powerful, and its prepared.Our emergency teams are working in New York City and Washington, D.C. to helpwith local rescue efforts. Our first priority is to get help to those who have beeninjured, and to take every precaution to protect our citizens at home and aroundthe world from further attacks. The functions of our government continue withoutinterruption. Federal agencies in Washington which had to be evacuated today arereopening for essential personnel tonight, and will be open for business tomorrow.Our financial institutions remain strong, and the American economy will be open forbusiness, as well. The search is underway for those who are behind these evil acts.Ive directed the full resources of our intelligence and law enforcement communitiesto find those responsible and to bring them to justice. We will make no distinctionbetween the terrorists who committed these acts and those who harbour them. Iappreciate so very much the members of Congress who have joined me in stronglycondemning these attacks. And on behalf of the American people, I thank the manyworld leaders who have called to offer their condolences and assistance. America andour friends and allies join with all those who want peace and security in the world,and we stand together to win the war against terrorism. Tonight, I ask for yourprayers for all those who grieve, for the children whose worlds have been shattered,for all whose sense of safety and security has been threatened. And I pray they willbe comforted by a power greater than any of us, spoken through the ages in Psalm23: Even though I walk through the valley of the shadow of death, I fear no evil,for You are with me. This is a day when all Americans from every walk of life unitein our resolve for justice and peace. America has stood down enemies before, andwe will do so this time. None of us will ever forget this day. Yet, we go forward to

defend freedom and all that is good and just in our world.

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DR. MARTIN LUTHER KINGS I HAVE A DREAM SPEECH

Five score years ago, a great American, in whose symbolic shadow we standsigned the Emancipation Proclamation. This momentous decree came as a greatbeacon light of hope to millions of Negro slaves who had been seared in the flames of

withering injustice. It came as a joyous daybreak to end the long night of captivity.But one hundred years later, we must face the tragic fact that the Negro is stillnot free. One hundred years later, the life of the Negro is still sadly crippled bythe manacles of segregation and the chains of discrimination. One hundred yearslater, the Negro lives on a lonely island of poverty in the midst of a vast ocean ofmaterial prosperity. One hundred years later, the Negro is still languishing in thecorners of American society and finds himself an exile in his own land. So we havecome here today to dramatize an appalling condition. In a sense we have come toour nations capital to cash a check. When the architects of our republic wrote themagnificent words of the Constitution and the declaration of Independence, theywere signing a promissory note to which every American was to fall heir. Thisnote was a promise that all men would be guaranteed the inalienable rights of life,liberty, and the pursuit of happiness. It is obvious today that America has defaulted

on this promissory note insofar as her citizens of colour are concerned. Instead ofhonouring this sacred obligation, America has given the Negro people a bad checkwhich has come back marked insufficient funds. But we refuse to believe thatthe bank of justice is bankrupt. We refuse to believe that there are insufficientfunds in the great vaults of opportunity of this nation. So we have come to cashthis check a check that will give us upon demand the riches of freedom and thesecurity of justice. We have also come to this hallowed spot to remind America ofthe fierce urgency of now. This is no time to engage in the luxury of cooling offor to take the tranquillising drug of gradualism. Now is the time to rise from thedark and desolate valley of segregation to the sunlit path of racial justice. Nowis the time to open the doors of opportunity to all of Gods children. Now is thetime to lift our nation from the quick-sands of racial injustice to the solid rock ofbrotherhood. It would be fatal for the nation to overlook the urgency of the momentand to underestimate the determination of the Negro. This sweltering summer ofthe Negros legitimate discontent will not pass until there is an invigorating autumnof freedom and equality. Nineteen sixty-three is not an end, but a beginning. Thosewho hope that the Negro needed to blow off steam and will now be content will havea rude awakening if the nation returns to business as usual. There will be neitherrest nor tranquillity in America until the Negro is granted his citizenship rights. Thewhirlwinds of revolt will continue to shake the foundations of our nation until thebright day of justice emerges. But there is something that I must say to my peoplewho stand on the warm threshold which leads into the palace of justice. In theprocess of gaining our rightful place we must not be guilty of wrongful deeds. Letus not seek to satisfy our thirst for freedom by drinking from the cup of bitternessand hatred. We must forever conduct our struggle on the high plane of dignity

and discipline. We must not allow our creative protest to degenerate into physicalviolence. Again and again we must rise to the majestic heights of meeting physicalforce with soul force. The marvellous new militancy which has engulfed the Negrocommunity must not lead us to distrust of all white people, for many of our whitebrothers, as evidenced by their presence here today, have come to realize that theirdestiny is tied up with our destiny and their freedom is inextricably bound to ourfreedom. We cannot walk alone. And as we walk, we must make the pledge thatwe shall march ahead. We cannot turn back. There are those who are asking thedevotees of civil rights, When will you be satisfied? We can never be satisfiedas long as our bodies, heavy with the fatigue of travel, cannot gain lodging in the

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motels of the highways and the hotels of the cities. We cannot be satisfied as longas the Negros basic mobility is from a smaller ghetto to a larger one. We cannever be satisfied as long as a Negro in Mississippi cannot vote and a Negro in NewYork believes he has nothing for which to vote. No, no, we are not satisfied, andwe will not be satisfied until justice rolls down like waters and righteousness like a

mighty stream. I am not unmindful that some of you have come here out of greattrials and tribulations. Some of you have come fresh from narrow cells. Some ofyou have come from areas where your quest for freedom left you battered by thestorms of persecution and staggered by the winds of police brutality. You have beenthe veterans of creative suffering. Continue to work with the faith that unearnedsuffering is redemptive. Go back to Mississippi, go back to Alabama, go back toGeorgia, go back to Louisiana, go back to the slums and ghettos of our northerncities, knowing that somehow this situation can and will be changed. Let us notwallow in the valley of despair. I say to you today, my friends, that in spite ofthe difficulties and frustrations of the moment, I still have a dream. It is a dreamdeeply rooted in the American dream. I have a dream that one day this nationwill rise up and live out the true meaning of its creed: We hold these truths to beself-evident: that all men are created equal. I have a dream that one day on the

red hills of Georgia the sons of former slaves and the sons of former slave-ownerswill be able to sit down together at a table of brotherhood. I have a dream that oneday even the state of Mississippi, a desert state, sweltering with the heat of injusticeand oppression, will be transformed into an oasis of freedom and justice. I have adream that my four children will one day live in a nation where they will not be

judged by the colour of their skin but by the content of their character. I have adream today. I have a dream that one day the state of Alabama, whose governorslips are presently dripping with the words of interposition and nullification, will betransformed into a situation where little black boys and black girls will be able to

join hands with little white boys and white girls and walk together as sisters andbrothers. I have a dream today. I have a dream that one day every valley shall beexalted, every hill and mountain shall be made low, the rough places will be madeplain, and the crooked places will be made straight, and the glory of the Lord shall

be revealed, and all flesh shall see it together. This is our hope. This is the faithwith which I return to the South. With this faith we will be able to hew out of themountain of despair a stone of hope. With this faith we will be able to transform the

jangling discords of our nation into a beautiful symphony of brotherhood. With thisfaith we will be able to work together, to pray together, to struggle together, to goto jail together, to stand up for freedom together, knowing that we will be free oneday. This will be the day when all of Gods children will be able to sing with a newmeaning, My country, tis of thee, sweet land of liberty, of thee I sing. Land wheremy fathers died, land of the pilgrims pride, from every mountainside, let freedomring. And if America is to be a great nation this must become true. So let freedomring from the prodigious hilltops of New Hampshire. Let freedom ring from themighty mountains of New York. Let freedom ring from the heightening Allegheniesof Pennsylvania! Let freedom ring from the snow-capped Rockies of Colorado! Letfreedom ring from the curvaceous peaks of California! But not only that; let freedomring from Stone Mountain of Georgia! Let freedom ring from Lookout Mountain ofTennessee! Let freedom ring from every hill and every molehill of Mississippi. Fromevery mountainside, let freedom ring. When we let freedom ring, when we let it ringfrom every village and every hamlet, from every state and every city, we will be ableto speed up that day when all of Gods children, black men and white men, Jewsand Gentiles, Protestants and Catholics, will be able to join hands and sing in thewords of the old Negro spiritual,Free at last! free at last! thank God Almighty, weare free at last!

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VITA

Graduate SchoolSouthern Illinois University

JOSHUA BOONE Date of Birth: February 6, 1984

6685 OLD HIGHWAY 13, APT. C2, CARBONDALE, ILLINOIS 62901

Southern Illinois University at CarbondaleBachelor of Science, Mathematics, May 2006

Special Honors and Awards:SIUC Masters Fellowship

Research Paper Title:Singular Value Approximations in Cryptography

Major Professor: Dr. Robert Fitzgerald

thesis cryptology

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