7/25/2019 Moler

1/11

Singular Value Analysis of CryptogramsAuthor(s): Cleve Moler and Donald MorrisonSource: The American Mathematical Monthly, Vol. 90, No. 2 (Feb., 1983), pp. 78-87Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2975804.

Accessed: 14/03/2011 13:52

Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at.

http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unlessyou have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you

may use content in the JSTOR archive only for your personal, non-commercial use.

Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at .http://www.jstor.org/action/showPublisher?publisherCode=maa..

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed

page of such transmission.

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of

content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms

of scholarship. For more information about JSTOR, please contact support@jstor.org.

Mathematical Association of Americais collaborating with JSTOR to digitize, preserve and extend access to

The American Mathematical Monthly.

http://www.jstor.org

http://www.jstor.org/action/showPublisher?publisherCode=maahttp://www.jstor.org/stable/2975804?origin=JSTOR-pdfhttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/action/showPublisher?publisherCode=maahttp://www.jstor.org/action/showPublisher?publisherCode=maahttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/stable/2975804?origin=JSTOR-pdfhttp://www.jstor.org/action/showPublisher?publisherCode=maa

7/25/2019 Moler

2/11

78

CLEVEMOLERAND DONALD

MORRISON [February

newsletter culminated

in the

creation of our very successful,

very well-received

newsletter,

FOCUS. Its editor, Marcia P. Sward,calls Ed the Father

of FOCUS .

Let me conclude by emphasizingother service to the Association. Ed was elected chairmanof

the Texas Section, but

he left for

Michigan

before he took

office. He was a visiting lecturer or the

MAA, and served a five-yearterm as editor

of

the MathematicalNotes Section of the MONTHLY.

As chairmanof the Committeeon Publicationssince 1971,he guided the unprecedentedgrowthof

ourjournalsand our series

of books and

monographs

with

skill, determination,and enthusiasm.

In all his activities,Ed Beckenbachenlisted the cooperationof his colleagues by his skill at

negotiation,his unfailing courtesyand considerationtoward others, and his common sense and

good humor.

But Ed's

cooperative

and

accommodatingspirit

at the committee table completely

disappeared

in

another of

his

roles. On the tennis court Ed was a hard contender, a tough

adversary

who

showed

'em

no

mercy. Captain

of

the tennis team at Rice

University

back

in

the

twenties, and later the

coach of the

team,

he

spanned

six decades with his favorite

sport.

Even

recently

Ed and

his wife Alice

competed

in

national

tournaments

of

superseniors ;

n

more

peaceful

moments

they indulged

their

lifelong hobby,

tending

their

hillside acre

of

plants ranging

from

apricots

to orchids.

Incidentally,

Alice

surely

holds a record for faithful attendance at

meetings of mathematicians,having

started at the

age

of eleven as daughter of a one-time

president

of the Association.

Ed Beckenbachclearly served the mathematicalcommunityvery well. We are all indebted to

him for his

preeminent eadership.

Ed

Beckenbach

s indeed

a most worthy recipient

of the Award

for

Distinguished

Service

to Mathematics.

* * *

Edwin F. Beckenbachdied on September5,

1982.

SINGULAR

VALUE

ANALYSIS

OF

CRYPTOGRAMS

CLEVE

MOLER AND

DONALD

MORRISON*

Department

of Computer

Science,

Universityof

New Mexico,

Albuquerque,

NM

87131

1. Singular

Value Analysis

and

Cryptanalysis.

The singular

value decomposition

is a matrix

factorization

which can produce approximations

to large arrays. Cryptanalysis

s the task

of

breaking

coded messages.

In

this paper,

we

present an unusual

mergerof the

two in which the

singular

value

decomposition

may

aid the cryptanalyst

n discovering

vowels and consonants

in

messages

coded

in certain

variations

of simple

substitution

ciphers.

Texts in

many

languages,

including

English, have

the

property

that vowels

are frequently

followed by

consonants,and

consonants

are

frequently

followed

by vowels.

We

say

a text

is

a

Cleve

B. Moler received his Ph.D.

at Stanford

under

George Forsythe.

He

has

taught

both mathematics

and

computer

science

at

Stanford,

the

University

of

Michigan,

and

the

University

of New Mexico. He recently

succeeded

Morrison as

Chairman

of the

Department

of

Computer

Science at New Mexico.

His

academic

and research

interests

include:

numerical analysis,

mathematical software,

and scientific computing.

Don Morrison earned his Ph.D. in mathematics at the University of Wisconsin in 1950, under C. C. MacDuffee.

He taught

mathematics

at Tulane,

1950-55,

and computer

science

at the University

of

New Mexico, 1967 to

the

present.

He

was a staff member,

division supervisor

and

department

manager

at

Sandia Corporation,

1955-71.

He

has

one wife,

three

children

and numerous

grandchildren.

His present

research interests

include cryptology,

data

structures,

and

design

of

algorithms.

*This work was

partially

supported

by

a

grant

from the National Science

Foundation.

7/25/2019 Moler

3/11

1983]

SINGULARVALUE ANALYSISOF CRYPTOGRAMS 79

vowel-follows-consonant text, or more briefly, a vfc text if the proportion of vowels

following vowels is less than the proportionof vowels following consonants.That is,

if

numberof vowel-vowelpairs

an

>

0

xiTx=

uij

(the Kronecker

delta)

Yii

ij

The coefficients

aj

are known as the

singularvalues and

xj,

and

yj

are the left and right singular

vectors,

respectively.They can

also be characterized

n terms

of the

solutions

to the

symmetric

eigenvalue

problem:

tT o} y}

It is not hard to see that therankof a

matrix s the numberof nonzero singularvalues. (In

fact,

this is

a

particularly

useful

way

to define

rank.)

There is a

fast, reliable algorithm

for

computing the SVD [3],

[6]. For a 26-by-26 matrix, the

computation

of

the

singular

values and

corresponding

eft and

right

vectors

takes only a few

seconds

on a medium

speed moderm omputer.

The

normalization

of

the vectors

xj

and

yj

insures that all of

the rank one

matrices

xjyT

have

the same norm

(that is,

the sum

of

the

squares

of

the

elements).

Consequently,

the

numerical

importance

of each term

in

the sum

(3)

can be

measured

by

the

size of the

coefficient

aJ..

f

the

a

decrease

fairly rapidly

as

j increases,

then an accurate

approximation

to

A

can be obtained

by

truncating

he series after

only

a few

terms.

Truncating

he

series after

k nonzero terms

provides a

rank

k approximation

o

A. Such approximationsare related

to

those

obtained by factor analysis

and have a wide

variety

of

applications.

One unusual

application

s

to

digital image processing[1].

4.

Rank

One

Approximation.

A

rough approximation

to a

digram

frequency

matrix can

be

obtained

by taking only

the first

term

in

its

SVD:

A

~

1 =

a1xXlyT.

Let e

be the

vector of all l's

and

let

f

be the vector with

f

=

the

numberof occurrencesof the i

th

letter

in

the text. Then

Ae

=

ATe

=

f

and so

a1

(yTe)xl

a-(-xaTe)x

fy

.

Consequently,

f

we were to

assume

that the

digram frequency

matrix

were

only

rank

one,

we

would

conclude that it

was

symmetric

and that

each

row and column

was

proportional

to the

frequency

vector

f.

Of

course,

in

practice,

A

is not rank one.

Nevertheless,

the first left

and

right singular

vectors

tend to

be

approximately qual

and reflect

the

frequencies

of

the letters

in

the text.

5.

Rank

Two Approximation.

The

rank two

approximation

obtained

from

the

first two terms

of the

SVD

is the

simplest approximation

which

takes

into account

the correlationbetween

pairs

of

letters

in the text. The second

left and

right singular

vectors

contain the

key

to

the

solution of

the

ciyptanalyst'sproblem.

Assume that A is

approximatedby

a matrix of rank

two,

so that

(4)

A

A2

=

a xTyf

+

a2x2yT.

We

propose

to use the

signs

of

the

components

of

x2

and

Y2

to

partition

the

alphabet

as follows:

7/25/2019 Moler

5/11

1983]

SINGULAR

VALUE

ANALYSISOF CRYPTOGRAMS

81

I

ifx2

>

O

andy,2