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WAB DEI'ARTl\lENT

O.FI'ICE OF THE CHIEI' SIGNAL OFFICER

w.\5HINGTON

~ARY

PART II

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mctea

WAR DEPARTMENT

OFFICE OF THE CHIEF SIGNAL OFFICER

WASHINGTON

MILITARY CRYPTANALYSIS

Part II

SIMPLER VARIETIES

OF POLYALPHABETIC SUBSTITUTION SYSTEMS

By

WILLIAM F FRIEDMAN

Principal Cryptanalyst

Chiet of Szgnal Intellzgence Section

War Plans and Trammg Dunston

PREPARED UNDER THE nIIlECTIO"'l OF THE

CHIEF SIGNAL OFFICER

UNITED STATES

GOVERNMENT PRINTING OFFICE

WASHINGTON: 1938
MILITARY CRYPTANALYSIS. PART II. SIMPLER VARIETIES OF

POLYALPHABETIC SUBSTITUTION SYSTEMS

Section Paragraphs Pages I Introductory remarks _ __ 1-4 1-3

II Cipher alphabets for polyalphabetrc substrtution, _ _ 5-7 4-9

III Theory of solution of repeating-key systems __ _ __ _ 8-12 10-16

IV Repeatmg-key systems wrth standard cipher alphabets 13-15 17-23

V Repeating-key systems with mixed cipher alphabets, L____ __ _ 16-26 24-48

VI Repeatmg-key systems with mixed cipher alphabets, II. _ __ _ _ 27-30 49-51

VII Theory of mdirect S;}IDIDetry of posrtion III secondary alphabets _ __ 31 52-59

VIII Application of prmoiples of indirect symmetry of posrtion 32-36 60-77

IX Repeating-key systems with mixed cipher alphabets III _ _ 37-40 78-83

X Repeating-key systems wrth mrxed cipher alphabets, IV 41-46 84-95

Appendix 1_____ __ _____ ___ ___ _____ _______ __ ____ ___ _ ______ __ __ __ __ __ _ __ 96-107

Appendix 2_____ _ __ _ _ __ 108-118

Index __ __ _ _ __ _ _ __ _ 119-120

(IU)
SECTION I

INTRODUCTORY REMARKS

PlII'a/P'8ph

The essential difference between monoalphebetre and polyalphabetae substitution __ 1

Pnmary elasstfieataon of polyalphabetre systems _ __ _ _ 2

Pnmary classification of penodic systems______ 3

Sequence of study of polvalphabetie systems __ 4

1 The essential dIfference between monoalphabetie and polyalphabetlc subsntuncn-c-e

In the subsntutron methods thus far discussed It has been pomted out that their baSIC feature

18 that of monoalphabetrcrty From the cryptanalytic standpoint, neither the nature of the

cipher symbols, nor their method of production IS an essential feature, although these may be

differentiatmg charactenstios from the cryptographic standpomt It IS true that m those cases

designated as monoalphabetic substitution with variants or multiple equivalents, there IS a

departure, more or le c; considerable, from strict monoalphabetacity In some of those cases,

indeed, there may be nvailable two or more "holly mdependent sets of equrvalents, "Inch,

moreover, may even be arranged m the form of completely separate alphabet'! Thus, while a

loose termmology rmght permit one to designate such systems as polyalphabetic, It IS better to

reserve this nomenclature for those cases wherein polyalphabeticitv JS the essence of the method,

specifically mtroduced with the purpose of imparting a pomtwnal vananon in the substitutrve

equivalents for plum-text letters, m accordance WIth some rule directly or indirectly connected

with the absolute positions the plain-text letters occupy m the message This point calls for

amplification

b In monoalph-rbetic -ubstrtutron with vnnants the object of having different or multiple

equivalents IS to suppress, SO far as possible by SImple methods, the charactenstic frequencies

of the letters occurring m plum text As has been noted, It IS by means of these cheractensnc

frequencies th-it the CIpher equivalents can usually be Identified In these systems the varymg

equivalents for plam-text letters are subject to the free choice and caprice of the enciphenng

clerk, If he Ie:; careful find conSCIentIOUS in the work, he "ill really make use of all the different

equivalents afforded by the system, but If he 1" shp-shod and hurried m his work, he will use the

same equrv-ilents repeatedly rather than take pains and time to refer to the charts, tables, or

diagrams to find the vanants Moreover, and this IS a crucial pomt, even If the individual

encrphermg clerks are extremely careful, when many of them employ the same system It ISentirely

unpossible to insure a complete drversrty m the encipherments produced by two or more clerks

working at different message centers The result IS mevitably to produce plenty of repetruons

in the texts emanating from several stations, and when texts such as these are all available for

study they are open to solution, by a comparison of their smnlanues and differences

e In true poly alphabetic systems, on the other hand, there IS estabhshed a rather defimte

procedure which automatacally determmes the shifts or changes in equivalents or m the manner

m WhICh they are mtroduced, so that these changes are beyond the momentary wInm or choice of

the enciphering clerk When the method of sluftmg or changing the equivalents 18 screntifically

sound and sufflcientlv complex, the research necessary to establish the values of the CIpher

characters Ie:; much more prolonged and difficult than IS the case even m complicated mono alphabetic substrtution WIth variants, as WIll later be seen These are the objects of true polyalphabetic substatution systems The number of such systems IS quite large, and It WIll be possible to

(1)
2

describe m detail the cryptanalysis of only a few of thc more common or typical examples of

methods encountered m practical rmhtary commumcanons

d The three methods, (1) smgle-equrvalent monoalphabetic subsututron, (2) monoalphabetic aubstrtution With variants, and (3) true polyalphabetic substitutron, <;h0W the following

relanonships as regards the equivalency between plam-text and cipher-text umts

A In method (1), there IS a set of 26 symbols, a plain-text letter IS always represented by

one and only one of these symbols, conversely, a symbol always represents the same plain-text

letter The equrvalence between the plain-to....t and the CIpher letters IS constant m both encipherment and decipherment

B In method (2), there IS a set of n symbols, "here n may be any number greater than 26

and often IS a multiple of that number, a plain-text letter may be represented by 1 2 3

different symbols, conversely, a symbol always represents the same plain-text letter the'same as

IS the case m method (1) The equivalence between the plain-text and the cipher letters IS

variable In encipherment but constant m decipherment I

C In method (3) there IS, as m the first method, II set of 26 symbols II plem-text letter

may be represented by 1, 2, 3, 26 different symbols, conversely, a symbol may represent

1, 2, 3, 26 dIfierent plain tevt letters, depending upon the system and the specific key

The equivalence between the plum-text and the cipher letters IS vanable in both encipherment

and decioherment

2 Pumary classlll.catlon of polyalphabetlc systems -a A primary classification of polyalphabetIc systems into two rather distinct types may be made (1) penodie systems and (2)

apenodic systems When the enciphering process involves a cryptograplnc treatment which IS

repctitrve m character, and which results m the production of c'l/cltc phenomena m the cryptographic text, the' system IS termed periodic When the enciphering process 10; not of the type

descnbed ITI the foregoing general terms, the svstem 1<; termed apenodu The substrtution In

both C8'3eS involves the use of two 01 more cipher alphabets

b The cyclic phenomena mherent ill a periodic system may be exhibited externally In

which case they are "aid to be patent, or they may not be exhibited externally, and must be 'uncovered by a prehmmary step ill the analysis, ill which case thcy are saad to be latent The

periodicity may be quite definite in nature, and therefore deternunable with mathematical

exactrtude allowing for no vanabihty, in WhICh case the penodicity IS said to bejixed In other

mstances the penodicity IS more or less flexible m character and even though It may be deterI There ISa monoalphabetie mcthod m which the mverse rcsult obtains, the correspondcnce bemg constant

m enclpherme lt but vanable m deCipherment, tins IS a method not found m the usual books on cryptography

but m an essay on that subJect by Edgar Allan Poe, entitled, In some edItIOns of hiS works, A Jew word8 on 8ecret

wnhng and, m other edItIOns, Cryptography The mcthod IS to draw up an enClphermg alphabet such as the

followmg (usmg Poe's example)

Plam _ -_ __ ABC D E F G H I J K L M Nap Q R STU V W x y Z

Clpher - -- SUA V I T F R I N MaD a FaR TIT E R I N R E

In such an alphabet, because of repetitIOns m the CIpher component, the plam-te"{t eqUlvalcnts are subJect to a

con81derable degrec of vanabllIty, aE w111 be seen m the deClphenng alphabet

Clpher__ ABC D E F G H I J K L M Nap Q R STU V Wx y z

{

c MGa E KJL HAFBD

Plam___ __ x

W Y

Tlus type of variability gIves noo to amblgllltlell m deCIpherment A clpner group such as TIE. would yield

such plaln-te"{t sequences as REG, FIG, TEU, REU, etc, which could be read only by contut No system of such a

character would be practIcal for serIOus usage For a further diSCUSSion of thiS type of Cipher alphabet see

Friedman, Wilham F, Edgar Allan Poe, Cryptographer, Signal Corps BulletInS NOll 97 and 98, 1937-38

3

mmable mathematically, allowance must be made for a degree of vanabIhty subject to hmits

controlled by the specific system under mvestiganon The penodicity 1<;m this case said to be

f{,eXtble, or umable 'IlJI,th'/,n l'/,m'/,ts

3 Primary classlll.catlon of periodic systems -a Penodic polyalphabetrc substitutron

systems mny pnmanly be classIfied into two kmds

(1) Those in which only a few of a whole set of CIpher alphabets are used in enciphermg

mdrvidual messages, these alphabets bemg employed repeatedlym a fixedsequence throughout each

message Because It IS usual to employ a secret word, phrase, or number as a key to determme

the number, Identity, and sequence With which the CIpher alphabets are employed, and this

key IS used over and over agam in encipherment, this method IS often called the repeattng-key

system, or the repeattng-alphabet system It IS also sometames referred to as the m'lllhple-alphabet system because If the keymg of the enure message be considered as a whole It IS composed

of multiples of a short key used repetitively 2 In this text the designation "repeatIng-key

system" will be used

(2) Those m which all the CIpher alphabets comprising the complete set for the system are

employed one after the other successively m the encipherment of a message, and when the

last alphabet of the senes has been used, the encipherer begins over again With the first alphabet

'I'his IS commonly referred to as a progres81-ve-alphabet system because the Cipher alphabets are

used m progression

4 Sequence of study of polyalphabetlc systems -a In the studies to be followed m connectIon With polyalphabetic systems, the order in which the work will proceed conforms very

closely to the claseificataone made in paragraphs 2 and 3 Penodie polyalphabetlc substitutron

Ciphers will come first, because they are, as a rule, the SImpler and because a thorough understanding of the pnnciples of their analysis IS prerequisrte to a comprehension of how apenodie

systems are solved But in the final enalysie the solution of examples of both types rests upon

the conversion or reduction of polyalphabetacity mto monoalphabencrty If this IS possible,

solunon can always be achieved, granted there are sufficient data m the final monoalphabetic

distnbunons to permit of solution by recourse to the ordmary pnnerples of frequency

b FIrst in the order of study of periodic systems will come the analysis of repeatmg-key

systems Some of the more SImple vanetIes will be discussed in detail, With examples Subsequently, Ciphers of the progressive type will be discussed There will then follow a more or less

detaI1ed treatment of apenodic systems

a French termmology calls tins the "double-key method", but there IS no lOgIC m such nomenclature
SECTION II

CIPHER ALPHABETS FOR POLYALPHABETIC SUBSTITUTION

Paragraph

ClassIficatIon of cipher alphabets upon the basis of their denv ation _ 5

PrImary components and secondary alphabets - - -- -_ __ __ _ __ __ _ ~_~~~_~~ __~~~_ -~_~~_ -~~__.,__ -_~_~-., 6

PrImary components, CIpher disks, and square tables - _ 7

I) Classification of CIpher alphabets upon the baSIS of their denvanon -a The substatunon processes in polyalphebstae methods involve the use of a plurality of CIpher alphabets

The latter may be denved by various schemes, the exact nature of which determmes the principal

charactensncs of the CIpher alphabets and plays a very important role in the preparation and

solution of polyalphabetic cryptograms For these reasons It IS advisable, before proceeding to a

discussion of the pnnciples and methods of analysis, to point out these varIOUS types of CIpher

alphabets, show how they are produced, and how the method of their production or denvation

may be made to YIeldImportant clues and short-cuts in analysis

b A primary classification of CIpher alphabets for polyalphabetic substrtutaon may be made

into the two following types

(1) Independent or unrelated CIpher alphabets

(2) Denved or mterrelated CIpher alphabets

c Independent CIpher alphabets may be disposed of m a very few words They are merely

separate and distinct alphabets showing no relationshrp to one another many way They may

be compiled by the VRUOUS methods discussed in Section IX of Elementa1"lJ MU1tary Cryptography

The solution of cryptograms written by means of such alpha bets IS rendered more difficult bv

reason of the absence of any relationshrp between the equrvalonts of one cipher alphabet and

those of any of the other alphabets of the same cryptogram On the other band, from the POIntof

VIew of pracncabrhty In their production and then handhng In oryptographmg and decryptographmg, they present some diffioulnes which make them less favored by cryptographers than CIpher

alphabets of the second type

d Denved or mterrelated alphabets, as their name mdicates, are most commonly produced

by the mteractwn of two primary components, which when Juxtaposed at the varIOUS POInts of

eomcidencs can be made to YIeld secondary alphabets 1

6 PrImary components and secondary alphabets -'1'Vlo basic, ..hdable sequences or components of n characters each will YIeld 'TI secondary alphabets The components may be elassified according to varIOUS schemes For cryptanalytic purposes the following classificanon 'WIll be

found useful

CASE A The primary components are both normal sequences

(1) The sequences proceed in the same direction (The secondary alphabets are direct

standard alphabets) (Pars 13-15)

(2) The sequences proceed in opposite directrons (The secondary alphabets are reversed

standard alphabets, they are also reciprocal CIpher alphabets) (Par 131, 14g)

CASE B The pnmary components are not both normal sequences

(1) The plain component IS normal, the CIpher component IS a mixed sequence (The

secondary alphabets are mixed alphabets) (Par 16-25)

1 See Sec. VIII and IX, ~mentar71 MIl,ear71 Crypeography

(4)

5

(2) The plam component IS a mixed sequence, the CIpher component IS normal (The

secondary alphabets are mixed alphabets) (Par 26)

(3) Both components are mixed sequences

(a) Components are rdentical mixed sequences

I Sequences proceed In the same direction (The secondary alphabets are

mixed alphabets) (Par 28)

II Sequences proceed In opposite direcnons (The secondary alphabets nro

reciprocal mixed alphabets) (Par 38)

(b) Components are different mixed sequences (The secondary ulphabcts are mixed

alpha bets) (Par 39)

7. Primary components, Cipher disks, and square tables -a In preceding texts It has

been shown that the equivalents obtainable from the use of quadncular or square tables may be

duplicated by the use of revolvmg CIpher disks or of shdmg primary components It was also

stated that there are vanous ways of employing such tables, disks, and shdmg components

Cryptograplncally the results may be quite diverse from different methods of using such paraphernaha, smce the specific equivalents obtained from one method may be altogether different

from those obtained from another method But from the cryptannlytac pomt of VIew the

diversity referred to 18 of httle significance, only in one or two cases does the specific method of

employmg these cryptographic matrumentahnes have an Important boanng upon the procedure

in cryptanalysis However, It IS advisable that the student learn something about these different

methods before proceeding WIth further work

b There are, not t'1JlO, butfourletters Involved in every case of finding equrvalents bv means

of sliding pnmary components, furthermore, the deterrmnation of an equivalent for a given

plam-tevt letter IS representable by two equations mvolving four elements, usually letters

Three of these letters fire by this time well-known to and understood by the student, VI7, Sk, Sp,

and So The fourth element or letter has been passed over WIthout much comment, but ciyptographically It IS just as Important a factor as the other three Its Junction m..ty best be mdu ated

by noung what happens when two primary components are Juxtaposed, fOI the purpose of finding

equivalents Suppose these components are the Iollowmg sequences

(1) ABC 0 E F G H I J K L M N 0 P Q R STU V W X Y Z

(2) F B P Y R C Q ZIG S E H T D J U MK V A L WN 0 X

Now suppose one IS merely asked to find the equrvalent of Pp when the key letter IS K WIthout

further specificanon, the CIpher equivalent cannot be stated, fOI It IS necessary to know not only

which K WIll be used as the key letter, the one in the component labeled (1) 01 the one in the

component labeled (2), but also what letter the Kk WIll be set against, in order to juxtapose the

two components Most of the tame, in preceding texts, these two factors have been tncitly

assumed to be fixed and well understood the Kk IS sought In the mixed, or Cipher component,

and this KIS set agaInst A m the normal, or plain component Thus

Plam Index

(1) Plam.i., __ ABCDEFGHIJKLMNOPQRSTUVWXYZABCDEFGHIJKLMNOPQRSTUVWXYZ

(2) Clphcr_______ FBPYRCQZIGSEHTDJUMKVALWNOX

i i

CIpher Key

WIth tlus settmg Pp=Zo
6

1~~~I~~~~~~~~I~ H T D J U M K V A L W~I~ x

B P Y R C Q 2 I G S E H T D J U M K V A L W N a x F ----------------

P Y R C Q ZIG S E H T D J U M K V A L W N a x F B

YRcQzIGSEHTD J U M K V A L W N a x F B P --------------

R C Q 2 I G S E H T D J U M K V A L W N a x F B P Y --------------

C Q 2 I G S E H T D J U M K V A L W N a x F B P Y R

-QZIGS-EHTDJUM K v A L W N a x F B P Y R C --------------

2 I G S E H T D J U M K V A L W N a x F B P Y R C Q -------------

I G S E H T D J U M K V A L W N a x F B P Y R C Q 2 -------------

G S E H T D J U M K V A L W N a x F B P Y R C Q 2 I --------------

I~~~~~~~~~~~~ W N a x F B P Y R C Q 2 I G

:~~~~~~~~~~~!i_ N a x F B P Y R C Q 2 I G S

H T D J U M K V A L W N a x F B P Y R C Q 2 I G S E -------------

T D J U M K V A L W N a x F B P Y R C Q 2 I G S E H --------------

D J U M K V A L W N a x F B P Y R C Q 2 I G S E H T --------------

J U M K V A L W N a x F B P Y R C Q 2 I G S E H T D ---------------

U M K V A L W N a x F B P Y R C Q 2 I G S E H T D J --------------

M K V A L W N a x F B P Y R C Q 2 I G S E H T D J U --------------

K V A L W N a x F B P Y R C Q 2 I G S E H T D J U M ------------

V A L W N a x F B P Y R C Q 2 I G S E H T D J U M K ----------------

A L W N a x F B P Y R C Q 2 I G S E H T D J U M K V ---------------

L W N a x F B P Y R C Q 2 I G S E H T D J U M K V A ----------------

W N 0 X F B P Y R C Q 2 I G S E H T D J U M K V A L ----------------

---------------

N a x F B P Y R C Q 2 I G S E H T D J U M K V A L W

a x F B P Y R C Q 2 I G S E H T D J U M K V A L W N --------------

x F B P Y R C Q 2 I G S E H T D J U M K V A L W N a

ABC D E F G H I J K L M Nap Q R STU V W x Y 2

7

For example

(1) For enciphering equations 8k/2=81/l, 8p/I = 8 0 /2

Locate 8 p m top sequence, locate 8k in first column,

8 e IS letter within the square at mtersection of the two hnes thus determined

Thus.

g The twelve resultant" obtainable from juxtaposmg shdmg components as indicated under

the precedmg subparagraph may also be obtained either from one square table, III which case

twelve difiercnt methode; of finding equivnlents must be applied, or from twelve difImcnt square

tables, In which case one standard method of findmg equivalents WIll serve all purposes

h If but one table such as that shown below as Table I-A IS employed, the various methods

of finding equivalents are difficult to keep in mind

TABLE I-A

(7) 8k/2=8p/I, 81/2=80/1

(8) 8k/2=80/1' 8 1/2=8p /1

(9) 8k/l=8p/2, 8 1/1=8012

(10) 8k/1 = 8 012 , 81/1 = 8p/2

(11) 8k /1 = 8p /2 ; 81/2=80/1

(12) 8k /l = 80 /2 , 81/2= 8 p/l

8k/2=81/I , 8p/l=80/2

8k/2=81/l' 8P/2=80/1

8k/l=81/2, 8p/I=80/2

8k/l=81/2, 8p/2=80/1

8k/2=8p/I, 81/1=80/2

8k/2= 8 0111 8 1/1= 8p/2

c The letter A m tills case may be termed the index letter, symbohzod AI The index letter

constitutes the fourth element mvolved in the two equations apphcable to the findmg of equrvalents by shding components The four elements are therefore these

(1) The key letter, 8k

(2) The index letter, 8 1

(3) The plain-text letter, 8p

(4) The CIpher letter, 8 0

The index letter IS commonly the mrtial letter of the component, but this, too, IS only a convention It m~ght be any letter of the sequence constrtutmg the component, as agreed upon by

the correspondents However, vn. the subsequent diecuesuni ~t unll be assumed that the index letter

zs the ~mtwlletter oj the component ~n uhach. ~t ~s located, unless oiherunee stated

d In the foregoing case the enciphermg equations are as follows

(I) Kk=A I, Pp=20

But there Iq nothmg about the use of slIdmg components which excludes other methods of finding

equivalents than that shown above For instance, despite the labehng of the two components

as shown ahova, there IS nothmg to prevent one from seeking the plain-text letter m the component laboled (2), that IS, the CIpher component, and takmg as Its CIpher equivalent the letter

opposite It in the other component labeled (1) Thus

CIpher Index

(1) ABCDEFGHIJKLMNOPQRSTUVWXY2ABCDEFGHIJKLMNOPQRSTUVWXY2

(2) FBPYRCQ2IGSEHTDJUMKVALWNOX

t t

Plam Kev

Thus

(II) Kk=A 1, Pp=Ko

e Smco equations (I) and (II) yield different resultants, even WIth the "arne index, key,

and plain-text letters, It 1" ObVIOl'S that an accurate formula to cover a specific pair of enciphermg

equations must melt-de data showing m what component each of the four letters comprismg the

equations IS located Thus, equations (I) and (II) should read

(I) Kk in component (2)=Al m component (1), Ppm component (1)=20 m component (2)

(II) Kk m component (2) =AIin component (1), Ppm component (2) = K, m component (1)

For the sake of brevity, the followmg notation wIII be used

(1) Kk/2=AI/I, Pp/l=20/2

(2) Kk/2=AI/I, Pp/2= K o/1

j Employmg tw0 shdmg components and the four letters entering into an enCIphenng

equation, there are, in all, twelve different resultants possible for the same set of components

and the same set of four basic elements These twelve differences m resultants anse from a set

of twelve dlffelent enciphenng condrttons, as set forth below (the notation adopted m subparagraph e IS used)

(1)

(2)

(3)

(4)

(5)

(6)
8

(2) For cnciphenng equations ~12=81/1' 8p12=8clI

Locate 8k In first column, follow hne to right to ep, proceed up tlus column 8 0 IS

letter at top ,

Thus

Kk 12= AI/1 , PP/2=KO/ 1

(3) FOI cnciphennsr equations 8 k /

1=81/

2 , 8 plI = e C

/

2

Locate 8k m top sequence and proceed down column to r> L \71,

ocate Op m top sequence, ec IS letter at other corner of rcctansrlo thus formed

Thus

Kk /I=A1/ 2 , PplI=XC / 2

Only three different methods have been shown and the student no doubt already has encountered

difficulty III keepmg them segregated III Ius mmd It would obviously be very confusmg to try

to remember all twelve methods But If one standard or fixed method of finding equivalents IS

follov ed with several different tables, then this difficulty disappears Suppose that the following

method IS adopted Arrange the square so that the plain-text letter may be sought In a separate

sequence, mnmged alphabetically, above the square and so that the key letter mtly be sought

In a separate sequence, also arranged alphabetically, to the left of the square, look for the plaintext letter III the top row, locate the key letter In the 1st column to the left, find the letter standmg withm the square at the intersection of the verticnl and horizontal hnes thus determined

Then tueloe squares, equivalent to the twelve different conditions hsted III subparagraph f can

readily be constructed They are all shown In Appendix 1, pp 96-107 '

'VIWll these square tables are exammed c-irefullv, certain mtcrestmg points ale noted

In the first place, the tilbles may be paired <JO th-it one of a pair n av serve for enciphenng and the

other of the pall mav serve for deciphcrmg, 01 vice versa For c'\:ftmrle, tables I and II hear thi-,

r~clprocal relntionslnp to each other, III and IV, V and VI, VlI and VIII, IX and X, XI and

XII In the second place, the Internal dtsposrtions of the lCLtpr<J, although the tables are derrvcd

from the same pair of component'S, are quite drverss For example, III table 1-B the horizontal

sequences are identical, but are merely chsplaced to the right and to the left different mtervals

accordmg to the successive key letters Hence tlus square show'S what may be termed a horIzontally-dIsplaced, direct symmetry of the CIpher component Vertrcally It shows no symmetrv

or If there IS symmetry, It IS not VISIble 2 nut \\ hen Table 1-B IS more 'carefullv exarrnned ~~

mvisible, or mdirsct, vcrtreal symmetry may he discerned where at first glance It I~ not apparent

If one tahl'<J any two columns of the table, It IS found that the interval between the members of

any pair of Iet.ter-, in one column IS the same as the mterval between the members of the homologous pair of letters in the other column, if the dnstance zs measured on the cipher component For

example, consldel the 2d and 15th columns (headed by L and I, respectrvsly), take the letters P

and G m the 2d column, and J and WIn the 15th column The distance between P and G on the

CIpher component IS 7 Intervals, the distance between J and Won the same component IS also

7 intervals ThIS phenomenon implies a kind of hidden, or latent, or mduect symmetry within

the CIpher square In fact, It may be stated that every ta ble which sets forth m systematic fashion

the various <Jecondarv alphabets denvaole by slIdmg two pnmary sequences through all POInts of

comCIdence to find CIpher eqUIvalents mU<Jt show some kmd of <Jymmctry, both hOrizontally and

2 It IS true that the first column "l\lthlll the table ~ho"l\s the plam-component sequence, but thiS IS merel v

because the method of findmg the eqUivalents III thiS case IS such that thiS sequence IS bound to appear III that

column, smce the successive key letters are A. B. C. Z. and tIllS sequence happens to be IdentICal With

~:bi~:I;;~z;:x~ent III thiS case The same IS true of Tables V and XI, It IS also applIcable to the first row of

9

verttcally The symmetry may be termed ousble or direct, If the sequences of letters m the lOWS

(or columns) are the same throughout and are identical WIth that of one of the primary components, It may be termed liuideti or uulireci If the sequences of letters ll1 the rows or columns

are different, apparently not related to either of the components, but are in reality decimations

of one of the primary components

J When the twelve tables of Appendix 1 are exammed ll1 the light of the foregoing remarks,

the type of symmetry found In each may be summarized ll1 the following manner

Honzontal Vertical

Visible or direct Invisible or mduect Vrsible or direct Invisible or Indirect

1 able

Follow" Follows Follows FOllows Follows Follows Follow" Follows

plain Cipher plain CIpher plain Cipher pl-un Cipher

component component component component component COIIlIJOnent component component

L ____ - - - - - -- - - - - -- - -------- X - - -- -- - - - - - --- - - -- --------- --------- -- - -- - - -- X

11.-________________ --------- ------ - -- X - - --- ---- --------- --------- X --------- IIL__ - - ----------- --------- X - -------- --------- --------- X -------- - - - ------- 1V_________________ --------- --------- X --------- X - -------- --------- - - ------- V_________________ - - ... _------ X --------- --------- --------- --------- --------- X

VI. ---------------- --------- --------- X --------- ------- - - --------- X - -- ------ VII________________ X --------- --------- - -------- - - ------- --------- X --------- VIII.-------------- X --------- - ------ -- ------- -- --------- --------- X --------- IX_________ -------- --------- --------- ------- - - A --------- --------- --------- X

X___ - - ------------- --------- - - -- - -- - ------- - - X --------- --------- --------- X

XL_- - - - - -- -------- - -- - --- - - ------- - - X --------- X --------- --------- - -------- X11 ________________ - - ------- X - -- ---- - - --------- --------- X --------- ---------

Of these twelve types of CIpher squares, corresponding to the twelve different ways of using a

parr of sliding primary components to danve secondary alphabets, the ones best known and

most often encountered In cryptographic studies are Tables I-B and II, referred to as being of

the VIgenere type, Tables V and VI, referred to as being of the Beaufort type, and Tables IX

and X, referred to as bemg of the Delastelle type It will be noted that the tables of the Delastelle type show no direct or VISIble symmetry, either horizontally or vertioally and because of

tlus are supposed to yield more security than do any of the other types of tables But It WIll

presently be shown that the supposed increase m security IS more Illusory than real

k 1 he foregoing facts concerning the varIOUS types of quadncular tables generated by diverse

methods of usmg shdmg primary components or their equivalent rotating CIpher disks will be

employed to good advantage, when the studies presently to be undertaken MIl bring the student

to the place where he can comprehend them m the analysis of polyalphabetic systems But m

order not to confuse him WIth a multiplicity of details which have no direct beanng upon baSIC

prmciples, one and only one standard method of finding equivalents by means of shding components WIll be selected from among the twelve available, as set forth m the precedmg subparagraphs Unless otherwise stated, this method WIll be the one denoted by the first of the formulae

lIsted III subpar j, VIZ

8J</2=el/l,8 p /1=eC!2

CallIng the pIam component "1" and the CIpher component "2", thIS WIll mean that the keyletter

on the CIpher component WIll be set OppOSIte the mdex, WhICh WIll be the first letter of the plam

component, the plam-text letter to be enCIphered WIll then be sought on the plam component and

Its eqUIvalent will be the letter OppOSIte It on the CIpher component
11

CRYPTOGRAM

E C P M P L C C L N X B WC S o X U V D S C R H T U S YES

H X I P L I B C I J U S Y E E G U R D P A Y B C X o F P J W

X V E U E L E J Y Q MUS C X J Y M S G L LET A J EM G P

LED E C G B MF I FIGURK 1

b a

ABLE

S P R A

C TIC

FAR A

T HIT

INS 0

T T R A

I N WI

o MBA

ITS C

E E P S

a

C E G U

A R 0 F

DVAN

THE A

CHI N

GIN T

A L I 0

HER E

N MAR

BAT T

LERY

R TIL

[Key BLUE. using direct standard alphabets]

CIPHER ALPHABETS

ABC D E F G H I J K L M N 0 P Q R STU V WX Y Z Plltm________________ STU V WX Y Z A

\

(1) BCD E F G H I J K L MX

N yO zP A

Q R

B C D E F G H I J K

C h (2) L M N 0 P Q R STU V W

IP er --------- F G H I J K L M N 0 P Q R S T (3) U V WX Y Z ABC D E --------- Q R STU V WX Y Z ABC D (4) E F G H I J K L M N 0 P B L U E

B L U E B L U E B L U E

THEA ARDK THEA ARDK

U S Y E B C X 0

R TIL E E P S

SEC P F P J W

L E R Y ITS C

M P L C J E MG

BATT OMBA

C L N X P X V E

ALIO TTRA

B WC SUE L E

N MAR I N WI

o X U V J Y Q M

CHI NTH I T

D S C R USC X

GIN TIN S 0

HTHX JYMS

HERE FARA

I P L I G L L E

A R 0 F S P R A

B C I J TAL E

THE ACT I C

U S Y E DEC G

DVAN ABLE

EGUR BMFI

C E G U

D PAY

b

MESSAGE

SECTION III

THEORY OF SOLUTION OF REPEATING-KEY SYSTEMS

Paragraph The three steps m the analysis of repeatmg-key systems __ __ _ __ _ _ 8

FIrst step finding the length of the period__ ------__ _ __ _ 9

General remarks On factoring --------------______________________ 10

Second step distnbutmg the CIpher text into the component monoalphabeta, __ _ __ __ 11

Third step solving the monoalphabene dIstnbutlOns___________________________________ 12

8 The three steps m the analysis of repeatmg-key systems -a The method of enciphermg

according to the pnncrple of the repeating key, or repeating alphabets IS adequately explained in

Section XI of Elementary Mthtary Cryptography, and no further reference need be made at this

time The analysis of a cryptogram of this type, regardless of the kind of CIpher alphabets

employed, or their method of production, resolves Itself into three distinct and successive steps

(1) Determmation of the length of the repeating key, whioh IS the same as the determmataon

of the exact number of alphabets mvolved m the cryptogram,

(2) Allocation or distribution of the letters of the CIpher text into the respective CIpher alphabets to wluch they belong ThIS IS the step WhICh reduces the polyalphabetio text to monoalphabetic terms,

(3) AnalYSIS of the individual monoalphabetie distributions to determine plain-text values of

the CIpher letters In each distribution or alphabet

b The foregoing steps WIll be treated lU the order In wluch mentioned The first step may

be described bnefly as that of deterrmmng the period The second step may be described briefly

as that of reduciuni to monoalphabetie terms The third step may be designated as uiennficaium. oj

cvpher-texi values

9 First step. findmg the length of the period -a The determmataon of the penod, that

IS, the length of the key or the number of CIpher alphabets involved m a cryptogram enciphered

by the repeatmg-key method IS, as a rule, a relatrvely SImple matter The cryptogram Itself

usually manifests externally certam phenomena which are the direct result of the use of a repeatmg key The pnnciples Involved are, however, so fundamental III cryptanalysts that their

elucidation warrants a somewhat detarled treatment 'I'his Will be done III connection WIth a

short example of encipherment, shown In FIg 1

THE ARTILLERY BATTALION MARCHING IN THE REAR OF THE ADVANCE GUARD KEEPS

ITS COMBAT TRAIN WITH IT INSOFAR AS PRACTICABLE

(10)
12

b Regardless of what system IS used, identical pIam-text letters enciphered by the same

cipher alphabet I must yield identical CIpher letters Referrmg to FIg 1, such a condition I:>

brought about every time that identical plain-text letters happen to be enciphered with the same

key-letter, or every time identical plain-text letters fall into the same column in the encipherment 2 Now SlIH'e the number of columns or pOSItIOns WIth respect to the key IS very hrru ted

(except m the case of very long key words), and since the repetation of letters IS an inevitable

condition m pIam text, It follows that there \VIII be in a message of fair length many cases where

identicnl plum-text letters must fall mto the same column They \VIII thus be enciphered by the

same crphei alphabet, resultmg, therefore, in 1,1 L pi oduction of many identical letters m the

CIpher text and these \VIII represent identical letters in the plain text When identical plain-text

polygraphs fall mto identical columns the result IS the formation of identical CIpher-text polygraphs, that IS, repetiuons of groups of 2, 3, 4, letters are exhibited m the cryptogram

Repettuons of this type WIll hereafter be called causal repetuions, because they are produced by

a defimte, traceable cause, tn.z, the encipherment of identical letters by the same CIpher alphabets

c It WIll also happen, however, that dnfferen; plain-text letters falling m dtfferent columns

WIll, by mere accident, produce identical CIpher letters Note, for example, m FIg 1 that in

Column 1,Rp becomes S, and that in Column 2,H, also becomes S; The production of an identical

CIpher text letter in these two cases (that IS, a repetation where the plam-text letters are different

and enciphered by different alphabets) IS merely fortuitous It IS, in every day language, "a

mere coincidence", or "an accident " FOl this reason repetitaons of this type WIll hereafter be

called accidental repeiuums

d A considerataon of the phenomenon pomted out m c makes It ObVIOUS that m polyalphabetic ciphers It IS Important that the cryptanalyst be able to tell whether the repetations he finds

in a specific case are causal or accidental in their origin, that IS, whether they represent actual

encipherments of identical plam-text letters by identical keymg elements, or mere coincidences

brought about purely fortuitously

e Now accidental repetrttons WIll, of course, happen fairly frequently WIth mdividual letters,

but less frequently WIth digraphs, because m this case the same kmd of an "accident" must take

place twice in succession Intuitively one feels that the chances that such a purely fortuitous

coincidence will happen two times m succession must be much less than that It WIll happen evet y

once m a winle m the case of smgle letters SImIlarly, mtUItIOn makes one feel that the chances

of such aCCIdents happenmg m the case of three or more consecutIve letters are stIll less than 1ll

the case of dIgraphs, decreasmg very rapIdly as the IepetitIOn mcreases m length

j The phenomena of cryptographlC IepetItIOn may, fortunately, be dealt WIth statlstlC'1.lly,

thus takmg the matter outSIde the realm of mtmtIOn and puttmg It on a firm mathematIcal or

objectIve h<lsrs Moreover, often the statIstIcal analYSIS WIll tell the cryptanalyst when he has

arram!,ed 01 H'arranged Ins teAt properly, that IS, when he IS approachmg or has reached monoalphabetlcity m hIS efforts to reduce polyalphabetlc text to ItS SImplest terms HoweveI, m

order to preserve contmUIty of thought It IS deemed madvlsable to mJect these statIstIcal conSIderatIOns at thIS place m the text proper, they have been mcorporated m AppendIX 2 hereof

The student IS adVIsed to study the AppendIx very carefully after he has fimshed readmg thIS

sectiOn of the text

g At tIns pomt It WIll merely be mdlCated that If a cryptanalyst were to have at hand only

the cryptogram of FIg 1, WIth the repetItIOns underhned as below, a statIstIcal study of the

1 It IS to be understood, of course, that Cipher alphabets With smgle eqUIvalents are meant 111 tIns ease

2 The frequency With WhICh thIS condItIOn may be expected to oecur can be defimtely calculated A dISCUSSIOn of tIllS pomt falls beyond the SLOpe of the present text

13

number and length of the repetatrons wrtlnn the message (Par 5 of Appendix 2) would tell him

that \V hile some of the digi aphic repetrtions 111.1y be ucciuental, tho chances that they all are

accidental are small In the case of the tctrngraphic repetrtion he would realize that the

chances of ItS bemg accrdcntnl are vNy small mdeed

A U S YES E C P M E---1 eeL N X B W C S o X U V D

B S C R H T H X I P L I Bel J U S Y E E G U R D P

C A Y B C X o F P J W J E M G P X V E U E L E J Y Q

D M USC X J--.Y M S G L LET A LED ~....Q G B M F I

h A consideration of the facts therefore le.ids to but one conclusion, mz, that the repetitaons

exhibited by the cryptogram under mvestag inon are not acculerdol but are causal m their ongm,

and the cause IS m this case not diffir ult to find repctations in the plain text were actually enCIphered by identic al alpha bets In order for this to O«.. ur, It was necessary that the tetragraph

USYE, for example, fall both times In eractly the s-une relativ e positiou WIth respect to the key

Note, for example, that UYSE m Fl~ 1 represents m both cases the plain-text polygraph THEA

The first tame It occurred It fell m pOSItIOns 1~2-3-4 WIth respect to the key, the second trme It

OCL urred It happened to fall ill the very same relative positions, although It might Just as well

have happened to fall in any of the other three pos-ible relatrve pOSItIOns WIth respect to the

key, mz, 2-3-4-1, 3-4-1-2, or 4~1-:l 3

t Lest the student be n.isled, how ever, a few more words are necessary on this subject

In the preceding subparagraph the word "happened" was used , this word correctly expresses

the Idea m mmd, because the mscrtron or deletion of a single plum-text letter between the two

occurrences would have throw n the second OCL urrence one letter forward or backward, respectively, and thus caused the polygraph to be enciphered by a sequence of alphabets such as can

no longer produce the cipher polygraph USYE from the plam-text polygraph THEA On the

other hand, the insertion or deletion of tlus one letter might bnng the letters of some other

polygraph into similar columns so that some other repetiuon would be exhibited m ease the

USYE repetmon had thus been suppressed

J The encipherment of smnlar letters by similar CIpher alphabets IS therefore the cause of

the production of i cpetinons m the CIpher text in the case of repoatmg-kcy CIphers What

prmclples can be derrved from tIns fact, and how can they be employed m the solutIOn of cryptograms of thIs type?

k If a count IS made of the numbel of letters from and mcludmg the first USYE to, but not

mcludmg, the second occurreIlce of U::3Yt~, a toLlI of 40 ktters lb found to mtervene between the

two occurrences ThIS number, 40, must, of (ourse, be an eAact multIple of the length of the key

Havmg the plam-text before one, It IS easIly seen that It IS the lOth multi!)le, that IS, the 4-letter

key has repeated Itself 10 tImes between the fin,t and the second occurrence of USYE It follows,

therefore, that If the length of the key \V ele not known, the number 40 could safely be taken to

be an exact multiple of the length of the key, mother 'vords, one of the juctors of the number

40 would be equal to the length of the key The word "safely" IS used m the precedmg sentence

to mean that the mterval 40 apphes to a rC}ietitlOn of 4 letters and It has been shown that the

chances that thIS repetItIOn IS aCCIdental are smnJI The factors of 40 are 2, 4, ti, 8, 10, and 20

So far as thIs smgle repetItIOn of USYE IS concernpd, If the length of the key \V ere not known, all

that could be saId about the latter would be that It IS equal to one of these factors The repetItIOn by Itself gIves no further mdicatIOns How can the exact factor be selected from among a

hst of several pOSSIble factors?

152018-38-2
-~- - I~~~~~~--~-I

m Arc all these repetruons caUSalll'petItIOns 7 It can be shown (Appendl:\. 2, par 1c) that

the odds against a theory that the UYSE repetition IS a, cidcntal are about 09 to 1 (since the

probability for ItS occmrence IS 01) It can also be shown that the odds azmnst a theory that the

10 digraphs wlnch occur two or more times are accidental repetrtrons ,11; over 4 to 1 (AppendIX

2, par 5c), the odds agamst a theory tl,at tl,e two digraphs which occur 3 times are accidental

repetitions are quite large (Probabihty IS calculated to be about 06) The chances are very

great, therefore, that all or nearly all these repetrtions are causal Certamly the chances acumst

the two occurrences of the tetragraph UYSEand the three occurrences of the two dIfferent digraphs

(LE and US) bemg nccidontal are qurte hrgh, and It IS therefore not astomshmg that the mtervals

between all the varIOUS repetrtrons, except m one case, contam the factors 2 and 4

n ThIS means that If the CIpher IS wrrtten out in either 2 columns or 4 columns all these

repetrtions (except the ex repetrnon) would fall into the same columns From this 'It follows

that the length of the key IS either 2 or 4, the latter, on practical grounds, bemg more probable

than the former Doubts concermng the matter of choosing between a 2-letter and a 4-letter

key WIll be dissolved when the CIpher text IS distributed mto ItS component umhteral frequency

distribu tions

o The repeated digraph ex in the foregoing message IS an aceniental repetition, as will be

apparent by referring to FIg 1 Had the message been longer there would have been more

such accidental repetmons, but, on the other hand, there would be a proportionately greater

number of causal repetitaons ThIS IS because the phenomenon of repetrtion m plain text IS

so all-pel vadmg

p SometImes It happens that the cryptanalyst qUIckly notes a repetItIOn of a polygraph of

four or mOle letters, the rnterval bet\\ een the fIrst and second occurrences of whIch has only

two factor'>, of wluch one IS a relatIvely smull nl,lmber, the other a relatIvely hIgh mcommensurable number He may therefore assume at once that the length of the key IS equal to the

smaller factor \\Ithout searchmg for addItIonal recurrences upon wlnclt to corroborate hlS

assumptIOn Suppose, for example, that m a relatIvely "hort cryptogram the mterval between

the first and second OGcurrences of a polygraph of five letters happms to be a number such as

203, the tlCtors of whlCh are 7 and 29 EVIdently the number of alphabets may at once be

14 15

assumed to be 7, unless one IS deahng with messages exchanged among correspondents known

to usc long keys In the latter case one could assume the number of alphabets to be 29

q The foregoing method of detenmmng the period m a polyalphabetao cipher IS commonly

referred to the hteraturo as "Iactormg the mtervals between repetations'", or more often It IS

simply called "factorrng " Because the latter IS an apt term and IS bnef, It WIll be employed

hereafter m this text to designate the process

10 General remarks on factormg -a The statement made in Par 2 WIth respect to the

cychc phenomena said to be exhibrted m cryptograms of the penodic type now becomes clear

The use of a ShOI t repeating key produces a periodicity of recurrences or repetations collectively

termed "cyclic phenomena", an analysis of which leads to a determmauon of the length of the

period or cycle, and this gives the length of the key Only in the case of ralatrvely short cryptograms enciphered by a relatrvely long key docs factoi mg fall to lead to the correct datermmataon

of the number of CIpher alphabets m a repeating-key cipher, and of course, the fact that a cryptogram contains repetitions whose factors show constancy IS m Itself an mdioation and test of ItS

periodic nature It also follows that If the cryptogram IS not a repeating-key CIpher, then

factonng WIll show no dehmte results, and conversely the fact that It docs not yield definite

results at once indicates that the cryptogram IS not a periodic, repeatmg-key CIpher

b There are two cases m which factormg leads to no defimte results One IS in the case of

monoalphabetic suhstrtutaon ciphers Here recurrences are very plentiful as a rule, and the

mtervals separating these recurrences may be factored, but the factors unli show no constancy,

there WIll be several factors common to many or most of the recurrences ThIS in Itself IS an

mdication of a monoulphubetac si.bstrtution CIpher, If tile" ery fact of the presence of many

recurrences falls to Impress Itself upon the moxperienced cryptanalyst The other case In which

the process of factormg IS nonsignificant involves certam types of nonpenodic, polyalphabetic

CIphers In certain of these CIphers recurrences of dicraphs, tngraphs, and even polygraphs

may be plentiful in a long message, but the intervals between such recurrences bear no defunte

multiple relation to the length of the key, such as m the case of the true penodic, repeating-key

CIpher, m which the alphabets change WIth SUCCCSSIve letters and repeat themselves over and

over agam

c Factormg IS not the only method of determmmg the length of the penod of a penodic,

polyalphabetic subsutution cipher, although It IS by far the most common and easily applied

At this pomt It WIll merely be stated that when the message under study IS relatively short m

comparison WIth the length of the key, so that there are only a few cycles of CIpher text and no

long repetations affording a baSIS for factormg, there are several other methods available

However, It being deemed inadvisable to interject the data concermng those other methods

at this pomt, they WIll be explamed subsequently It IS desirable at this Juncture merely to

indicate that methods other than Iactormg do exist and are used in practical work

d Fundamentally, the factoring process I'> merely a more or less SImple mathematical method

of studymg the phenomena of periodrcrty m cryptograms It WIll usually enable the cryptanalyst to ascei tam defimtely whethE'r or not a gIVell cryptogram IS penochc rn nature, and If

so, the length of the penod, stated ~"l terms of the cryptographw umt ~1UJohed By the latter

statement IS meant that the factOrIng plccess may be applIed not only 1Il analyzmg the perIOdICIty

malllfested by cryptograms III wInch the plam-tevt umts subJected to cryptograplnc treatment

arc monographIC m naieIlt' (1 e are smgl,' letters) but ~lso 111 studYlll~ the peuodlclty exlublted

by those occaslOral clyptoglams \\heleI,l the rllllh-text umts me ciIgrapluc, tngrapluc, or

n-graphlc III character The student should he,lr tIns pomt III mmd when he camps to the study

of SubstItutIOn systems of the latter sort However, the plesent text \\111 dral solely WIth cases

of the former type, whelem the plam-tr:\.t umb sllb]ected to clyptograplnc treatmeut are "wgle

letters

They are as

Factors

2, 4, 5, 8, 10, 20

2, 4, 8

5

2, 4, 11, 22, 44

2, 4, 8

2,4

2, 4, 5, 10

2,4

2 3, 4, 6, 8 10, 12

2, 4, 1&, 2u

2, 3, 4, 6, 9, 18

40

16

25

88

16

4

20

8

24

52

36

Hpl'ctItlOn Interv al

1st USYE to 2d USYE_ _ _ _ _ _ _ __ _ _ _

lbt BC to 2d BC___ _ _ _ __ __ _ _ _ _

1st CX to 2d CX___ _ _ _ _ _ _ _ __ _ _

1st EC to 2d EC _ _ _ _ _ _

1st LE to 2d LE _ _ _ _ _

2d LE to 3d LE_ __ _ _ _ _ _ _

1st LE to 3d LE _ __ _ _ _ _ __

1st Jy to 2d JY _ _ __ _ _ _ _ _

1st PL to 2d PL _ _ _ _ _ _ _ _ __ _ _

1st SC to 2d SC_ __ _ _

(1st SY to 2d SY, already mcluded lll-USYE)- -

(1st US to 2d US, already includcd m USYE )

2d US to 3d uS _ _ _ __ _ _ _ _ _ _

(1st uS to 3d US, already included III USYE )

(1st YE to 2d YE, already included III USYE )

1 Let the intervals between all the i epetataons m the cryptogram be Iisted

follows
(17)

SECTION IV

REPEATING-KEY SYSTEMS WITH STANDARD CIPHER ALPHABETS

RepetitIOn LocatIOn Interval Factors

LUFMPZJNVC 01. K3 160 2,4,5,8, 10

JZXIG - El, H4 90 2, 3, 5, 6, 9, 10

EJK B4, L2 215 5 -- - PTE - E3, G3 50 2,5,10 - QGK 04, HI 85 5

UKH Al. C2 55 5

ZLA Jl. L4 65 5 - - AS - D3, L3 175 3,5,7,

EJ B4. L2 115 5 -

FM A5, Dl 57 3 - - FM A5, J2 185 5 - - - FM -- --"..- J2, J4 12 2,3,4,6

FM - --- J4, K3 20 2,4,5,10 - FM - -- K3, 14 30 2,3,5,6, 10 -- 2, 3, 4, 5, 6, 10 JA A2, C4 60

LA - FI, JI 75 3,5

LA Jl. L4 65 5

LL G5, H2 10 2, 5

NL - 01, H2 105 3,5,7 - 3,5,9 NL - H2. Kl 45 ------ 2,4,5,10 VX - - -~- Cl. C5 20 -- - -- YM A3, B3 25 5 - - -

A

B

C

Paragraph ________________ . 13

Solution by applying prmciples of frequency__ -------- - ---- -- ------- - -- ---- -- -- ._____ 14

Solution by r-ompleting the plaill-component sequen~e_______ __ _~._~~~~~~_~~_~~~~ ~~_~~ ~-~~~~~-~~~~~~_~~~~ 15

Solution by the "probable-word method"----

13 SolutIOn by applymg principles of frequency -- a In the hght of the foregomg prmcrples,

let the followmg cryptogram be stuched

MESSAGE 1 2 3 4 5

Z Y MWM J MI G X N F ML X A U K.J:! Y JAM K I ---

MZ M I L V ME J K U T G E TIM I Z H B H RAY--

D P V X K Q ~tl Q L H V R M J A Z N G G Z V X E

U A S H K Q G KIP L WP D N L U F M P Z J N V C H ---

E A J Z X I GUM T V D f-T E E C MY S Q Y B A V

N Y E E Y X E E U D P X R F A L A H Y POE X W P V

G B V Z V I Z I I V 0 S PTE G K U B B R Q L L X P

I K W D J Z X I G 0 I 0 I H WF Q..JLK N L L L E P T

N P L Z I 0 V V F M Z K T X G J Z LAM V K F M W F

J L U F M P Z J N V C A I G I K M-1 MD F A A E X I

L UAWPR NVIW~KZAS ZLALM HS

A h for repetItIOns discloses the follOWIng short list WIth the intervals and factors

searc tt d (for previous expPrlence may lead to the conclusion that It IS unlikely that the

above 10 orm -ol th 10 alphabets showing the number of recurrences which It does) cryptogram mvo ves more an ,

16

11 Second step drstributmg the cipher text into the component monoalphabets-a

After the number of cipher alphabets Involved in the cryptogram has been ascei tamed, the next

step IS to rewnte the message m groups corresponding to the length of the key, or in columnar

fashion, whichever IS more convenient, and this automo ucally divides up the text so that the

letters belongmg to the same CIpher alphabet occupv smular pocitions in the groups, or, If the

columnar method IS used, fall m tl.e same column The letters are thus allocated or distributed

Into the respective CIpher alphabets to which they belong ThIS reduces the polyalphabetic

text to monoalphabetic terms

b Then separate umhteral frequency distrtbutions for the thus Isolated individual alphabets

are compiled For example, m the case of the CIpher on page 13, navmg determmed that four

alphabets are involved, and having rewritten the message m four columns, a frequency distnbution IS made of the letters in Column 1, another IS made of the letters m Column 2, and so on for

the rest of the columns Each. oj tl.e reeuliunq disinbution» ~:s therefore a monoaiphabeiic frequency

d~8tnbutwn If these distnbutions do not give the charactenstic Irregular crest and trough

appearance of monoalphabetrc frequency distrrbuuons, then the analysis wlnch led to the

hypothesis as regards the number of alphabets urvolved IS fallacious In fact, the appearance of

these individual distnbuuons may be considered to be an mdex of the correctness of the factoring

process, for theoretically, and prncncally, the md.vidual distnbunons constructed upon the

correct hypothesis WIlltend to conform more closely to the irregular crest and trough appearacne

of a monoalphabetic frequency distnbunon than WIll the graphic tables constructed upon an

incorrect hypothesis These indivtdual dist.nbutrons may also be tested for monoalphabetictty

by statastical methods

12. Third step. solving the monoalphabetic distrrbutions -Thr difficulty experienced III

analyzmg the individual or Isolated frequency drstubutions depends mostly upon the type of

CIpher alpnabets that IE:> used It 1'3 app-irent that mixed alphabets may be used Just as easily as

standard alphabets, and, of course, the CIpher letters themselves give no indicauon as to which

IS the case However,Just as It was found that m the case of monoalphabetic subsutuuon CIphers,

a umhteral frequency drstubution gives clear mdicutions as to whether the CIpher alphabet IS a

standard or a mixed alphabet, by the relative pOSItIOns and exteneions of the crests and troughs

ill the table, so It IS found tbat m the case of repeating-key ciphers, unihteral frequency distnbunons for the Isolated or mdrvidual ulphabets will also give clear mdications as to whether these

alphabets are standard alphabets or mix..ed alphabets Only one or two such frequency distnbunons all' necessary for this determmauon , If they appear to be standard alphabets, similar distnbutions can be made for the rest of the alphabets, but If they appear to be mixed alphabets, then

It IS best to compile trihteral frequency drstributions for all the alphabet"! The analysis of the

values of the cipher letters 111 each table proceeds dong the same Imes as m the case of monoalphabetio CIphers The analysis IS more difficult only because of the reduced SIze of the tables, but

If the message be very long, then each frequency drstributron WIll con tam a sufficient number of

elements to enable a speedy solution to be achieved
18

b The factor 5 appears III all but two ca"1p<; each of h h I

almost certain that the number of all'hal>pt~ 0 fi S" ,IC IllVO ves only a digraph It seems

five letters, It IS unnecessary to rev II/e tl I'; n c '~!lce the text ulrcady appears III groups of ~u " L 18 meSs')'re I'he ne t ttl

quency distnbuuon for Alphabet 1 t f t 'L 1 x s ep 1', 0 rna co a umhteral frebets are Involved It IS as follows 0 see u n r an e c rtclll'mcd whether or not st-mdarr] alphaALPHABr;r 1

....... ::::~~ ::::~~i!~ ":7 /

ABC D E F G H I j K LMNaP QR STU V WX Y

c Although the mdlcatlOll'3 arr not vor (I, It f

small amount of data the assun t f 1:) c.u (l ,yet 1 ono takes into consideration the

test Aecore1mo-Iy a Sl~Il'''I el1s1tPriblOl~1tlO •,l ( 1r('c11"tflcn(ALllH1 alphabet v.ith We~Ap , IS "01 th further c» ( OJ! IS mnr e or phlll-jet 2

ALPHABFT 2

- BCD E FG i J MN a Q R STG WX

b t

d "j"herp IS every mdICatIOn of '1 Jlrect stand 1[<1 alnha ~,"t with H _ A

U lOT' I f 1 I s: .h , , r.. Let srrmlar drstri- s ie mac c or t 18 ast three alphrbet'l They "re '1,S follo" " e - p

AI PHAIn r -)

=:::~ :;;--:::;;1:. -:::--

ABC D E F G H I J KLMN0 PQR STU WXY

ALPHABET 4

"":::: --;:::: :::::: ::::- ABCDEFGHIJKL~~OPQRSTUV~~YZ

ALPHABET .5

::-::

~:::::~ ::::::

ABCDEFGHIJKLMNOPQ~~TU~WXYZ e After but httle expenm t t f } the nor 1 h th f 11 en 1 IS ounr that the drstnbutions can best be made to fit

ma w en e 0 owing values are assumed

Alphabet L Ap -= We

Alphabet 2 Ap=He

Alphabet 3 Ap = L,

Alphabet 4 Ap= Te

Alphabet 5 Ap=Ee

j Note the key word give b th

the correctness of the anal "1 n y e successrvo equivalents of Ap WHITE The real proof of

cryptogram Th f T y~l S 1"1, of course, to test the values of the solved alphabets on the

e 1\ e comp ete O1pllPr alphabets are as fellow b

PlaIll--

1

--i

------- -- ABC D E F G H I J K L M N 0 E..JLB STU V WX Y Z

2----------- WX Y Z ABC D E F G H I J K L M N 0 P Q R STU V

CI her 3------ ---- H I J K L M Nap Q R STU V ld X Y Z ABC D E F G

p 4----------- I J K L M N 0 P Q R STU V Wx Y Z ABC D E F G H

----------- T U V WX Y Z ABC D E F G H I J K L M N 0 P Q R S

5 .•__ .. .Ii; F G H J: J K L M N 0 P Q R STU V WX Y Z ABC D

FIGURE 2

19

g Applying these values to the first few groups of our message, the following 1<; found

123451234512345123451234'\

CIpher AUK H Y JAM K I Z Y M WM J MIG X N F M L X

Plalll__________ E NCO U N T ERE D RED I N FAN TRY EST

h Intelhgrole text at once results, and the solution can now be completed very quickly

The complete message IS as follows

ENCOUNTERED RED INFANTRY ESTIMATED AT ONE REGIMENT AND MACHINE GUN COMPANY IN TRUCKS NEAR EMMITSBURG AM HOLDING MIDDLE CREEK NEAR HILL 543 SOUTHWEST OF FAIRPLAY WHEN FORCED BACK WILL CONTINUE DELAYING REDS AT MARSH

CREEK HAVE DESTROYED BRIDGES ON MIDDLE CREEK BETWEEN EMMITSBURG-TANEYTOWN

ROAD AND RHODES MILL

In the foregomg example (whn h IS tvpical of the system erroneously attributed, In cryptographic literature, to the French cryptogi aphcr Vigenere, although to do him justice, he

made no claim of having "mvented" It), direc t standard alphabets were used, but It IS ObVIOUS

that reversed standard alphabets may be used and the solution accomplished III the same

manner In fact, the now obsolete cipher disk used by the Umted States Army for <1 number

of ypars yields exa-fly this type of cipher, whu h IS also known m the hterature as the Beaufort

CIpher, and by other names In fitting the Isolated frequent y distnbutions to the normal, the

directaon of "reading" the crests and troughs IS merely reversed

14 Solution by completmg the plain-component sequence -a There is another method

of solvmg this type of CIpher, which is worthwhile explaimng, because the underlying prmciples

WIll be found useful m many cases It is a modification of the method of solution by completmg

the plam-component sequence, already explained m jJ;hhtary Cryptanalys~s, Part I

b After all, the mdrvidual alphabets of a CIpher such as the one Just solved are merely

direct standard alphabets It has been seen that monoalphabenc CIphers m which standard

CIpher alphabets are employed may be solved almost mechanically by completmg the plaincomponent sequence The plain text reappears on only one generatnx and tlus generatnx IS the

same for the whole message It IS easy to pick this generatnx out of all the other generatrices

because It ISthe only one which yields mtchrgible text Is It not apparent that If the same process

IS applied to the CIpher letters of the uuiundual. alphabets of the CIpher Just solved that the plaintext equtvalents of these letters must all reappear on one and the same generatrrx? But how

WIll the generatnx wlnch actually con tam" the pl am-text letters be drstmguishable from the

other generatnces, smce these plain-text letters are not consecutive letters m the plain text but

only letters separated from one another by a constant interval? The answer is Simple The plamtext generatnx should be distmgutshable from the others because ~t unll show more and a better

assortment of h~gh1requency letters, and can thus be selected by the eye from the whole set of generatrices If this IS done WIth all the alphabets m the cryptogram, It WIll merely be necessary to

assemble the letters of the thus selected generatrrces m proper order, and the result sould be

consecutive letters formmg intelligible text

c An example WIll serve to make the process clear Let the same message be used as before

Factormg showed that It mvolves five alphabets Let the first ten Cipher lettf'rs ~n each alphabet

be set down m a horizontallme and let the normal alphabet sequences be completed Thus
21

Plrontext ENCOUNTERED RED INFANTRY ESTIMATED AT ONE

REGIMENT AND MAC

c Extraneous circumstcnces lead to the asaumption of the presence of the word AMMUNITION One may assume that this word begms the message Using shding normal components, one reversed, the other direct, the key letters are ascertained by notmg what the successive

equivalents of Ap are Thus

Crpher M D S T J L Q C X C

Plain text__________ A MMU NIT ION

"Key" .. _.. ._ M PEN W T J K L P

K Z A S A NY Y K 0 L P

MESSAGE

M D S T J L Q C X C

e Solution by this method can thus be achieved without the compilation of any frequency

tables whatever and IS very quickly attained The mexpenenced cryptanalyst may have difhcnlty at first in selectmg the generatrices which contam the most and the lwst assortment of

high-frequency letters, but with increased practrce, a high degree of prof ciency IS attamed

After all It IS only a matter of experiment, trial, and error to select and assemble the proper

generatnces so as to produce mtelhgible text

j If the letters on the shdmg strips were accompanied by numbers representmg their relntrve

freq uencies m plain text, and these numbers were added across each gcneratux, then that generatrix with the highest total frequency wor-ld theoreticollq always be the plain-teo..t generatrrx

Practically It "" III be among the generatnces "Inch show the first three or four greatest totals

Thus, an entirely mathematical solution for this type of cipher may be applied

g If the cipher alphabets are reversed standard alphabets, It IS only necessary to couvert

the cipher letters of each Isolated alphabet mto their normal, plum-component equivalents and

then proceed as in the case of direct standard alphabets

h It has been seen how the key word may be discovered m tlns type of cryptogram Usually

the key IS made up of those letters in the successive alphabets whose equivalents are Ap but other

conventions are of course possible Sometimes a key number IS used, such as 8-4-7-1-12,

wluch means merely that Ap IS represented by the eighth letter from A (m the normal alphabet)

m the first CIpher alphabet, by the fourth letter from A m the second cipher alphabet, and so on

ThIS modification IS known m the literature as the Gronsfeld cipher How ever, the method of

solution as Illustrated above, bemg independent of the nature of the key, IS the same as before

15 Solution by the "probable-word method "-0 The common use of key words m cryptograms such as the foregomg makes possible U method of solution that IS SImple and can be used

where the more detailed method of analysis usmg frequency distnbutions or by completmg the

plain-component sequence IS of no avail In the case of a. very short message which may show

no recurrences and grve no indrcations as to the number of alphabets involved, this modified

method will be found most useful

b Bnefly, the method consists m assummg the presence of a probable word m the message,

and referring to the alphabets to find the key letters applicable when this hypothetical word IS

assumed to be present m varIOUS pOSItIOns m the CIpher text If the assumed word happens to

be correct, and IS placed m the correct pOSItIOn in the message, the key letters produced by

referrmg to the alphabets WIll yield the key word In the followmg example It IS assumed that

reversed standard alphabets are known to be used by the enemy

20

ALPHABET I ALPHABET 2 ALPHABET 3 ALPHABET 4 ALPH\BET 6

1 AJZJNEZAIJ UAYMFTHYLK KMMIMIBMVU HKWGLMH.lMT YJMXXIRMEG 2 BKAKOFABJK VBZNGUIZML LNN'Jt\IJCNWV ILXHMNIJ\NU ZJNYYJSNFH 3 CLBLPGBCKL WCAOHVJAl'hll MOOKOKDOXW JMYINOJBOV Al{CZZKTOGI 4 DMCMQHCDLM XDBPIWKBON NPPLPlEPYX KNZJOPKCPW BLPAALUPHJ 5 ENDNRIDEMN YECQJXLCPO OQQMQiVlFQZY LOAKPQLDQX CHQBB1V1VQIK 6 FOEOSJEFNO ZFDRKYMDQ? PRRNRNGRAZ MPBLQRMERY DNRCCNWRJL 7 GPFPTKFGOP AGESLZNERQ QSSIJSOHSBA NQCMRSNFSZ EOSDDOXSKM 8 HQGQULGHPQ BHFTMAOFSR RTTPTPITCB Q.Fm'STQG'rA FPTE;EPYTLN 9 IRHRVMHIQR CIGUNBPGTS SUUQUQJUDC PSEOTUPHUB GQUFFQZUMO 10 JSISWNIJRS DJHVOCQHUT TVVRVRKVFD QTFPUVQIVC HRVGGRAVNP 1 t KTJTXOJKST EKIVvPDRIVU UWWSWSU\FE RUGQVWRJWD ISWHHSBWOQ 12 LUKUYPKLTU FLJXQESJWV VXXTXTMXGF SVHRWXSKXE JTXIITCXPR 13 MVLVZQLMUV GMKYRFTKXW WYYUYlJ, JYIIG TWISXYTL'iF KUYJJUDYQS 14 NWMWARMNVW HNLZSGLJLYX XZZVZVOZIH UXJTYZUMZG LVZKKVEZRT 15 OXNXBSNOWX IOMATHV1V,ZY YAAVlAWPAJl VYKUZAVNAH MWALLWFASU 16 PYO'tCTOPXY JPNBUIWN4.l ZBBlBXQBK,] WZLVABWOBI NXBMMXGBTV 17 QZPZDUPQYZ KQOCVJXOBA ACCYCYRClK XAMYVBCXPCJ OYCNNYHCUW 18 RAQAEVQRZA LRPDWKYPCB BDDlDZSDML YBNXCDYQDK PZDOOZIDVX 19 SBRBFWRSAB 1V1SQEXLZQDC CEEAEATF"JM ZCOYDEZREL QAEPPAJEWY 20 TCSCGXSTBC NTRFY~ARED DFFBFBVrON ADPZEFASFM RBFQQBKFXZ 21 UDTDHYTUCD OUSGZNBSFE EGGCGCVGPO BEQAFGBTGN SCGRRCLGYA 22 VEUEIZlJVDE PVTHAOCTGF FHHDl-IDWHOP CFRCG:-fCUHO TDHSSDMI-lZB 23 WFVFJAVWEF QWUIBPDUHG GU8ICXIRQ DGSCHID'JIP UEITTENIAC 24 XGWGKBWXFG RXVJCQEVIH HJJFJFYJSR EHTDIJEWJQ VFJlJUFOJBD 25 YHXHLCXYGH SYWKDRFWJI IIC(GI<GZKTS FIUEJKFXKR WGKVVGPKCE 26 ZIYIMDYZHI TZXLESGXKJ JLLHLHALUT GJVFKLGYLS XHLWWHQLDF FIGURF 3

FIGURE 4

d If the hIgh-frequency generatnces underlmed in FIgure 3 are selected and their letters

ale Juxtaposed ~n columns the conseoutrvr, letters of mtelligible plain text rmrnediatsly present themselves Thus

(

F Or Alphabet 1, generatnx 5 END N RID E M N

For Alphabet 2, genei atnx 20 N T R F Y MAR E D

Selected Generatuces For Alphabet 3, gcneratnx 19 C E E A EAT E N M

For Alphabet 4, generatrrx 8 0 R D N S TOG T A

For Alphabet 5, generatnx 23 U E ITT E N I A C

12345

E NCO U

N T ERE

D RED I

N FAN T

Columnar juxtaposition of letters RYE S T

from selected generatnces I MAT E

D A TON

ERE G I

MEN T A

N D MAC
22

The key does not spell any mtelhgible word One therefore sInfts the assumed word one letter

forward and another tnal is made

Cipl-er D S T J L Q C X C K

Plain text__________ A M M U NIT ION

"Key" D E F D Y Y V F Q X

ThIS also J ields no intelhgible key word One contmues to shift the assumed word forw ard

one space at a time until the follo\\mg point I" reached

Cipher L Q C X C K Z A S A

Plam text__________ A M M U NIT ION

"Key"__ L COR P S S I G N

The key now becomes evident It IS a cychc permutation of SIGNAL CORPS It should be

clear that since the key word or key phrase repeats Itself durmg the encipherment of such So

message, the plain-text word upon whose assi med presence In the message this test IS being

based may begin to be enciphered at any point In the kev, and continue over Into Its ne....t repennon If It IS longer than the key When this IS the case It IS merely necessary to shift the latter

part of the sequence of key letters to the first part, as m the case noted LCORPSSIGN IS transposed into SIGN LCORPS, and thus SIGNAL CORPS

d It will be seen m the foregoing method of solution that the length of the key IS of no

particular mterest or consequence In the step"! taken In eflcctmg the solution The determmation of the length and elements of the key comes after the solution rather than before It In this

case the length of the penod IS seen to be eleven, corresponding to the length of the key (SIGNAL

CORPS)

e The foregoing method IS one of the other methods of determmmg the length of the key

(besides factormg), referred to In Par lOe

j If the assumptron of reversed standmd alphabet" yields no good results, then direct

standard alphabets are assumed and the test made exactly In the same manner As will be

shown subsequently, the method can also be used as a last resort when mixed alphabets are

employed

g When the assumed word IS longer than the key, the sequence of recovered key letters will

show a periodicity equal to the length of the key, that IS, after a certain number of letters the

sequence of key letters "WIll repeat ThIS phenomenon would be most useful m the case of keys

that are not mtclhgible words but are composed of random letters or figures Of course, If such

a key IS longer than the assumed word, this method IS of no avail

h Till"! method of solution by searching for a word IS contingent upon the following CIrcumatance,

(1) That the word whose presence IS assumed actually OCCUN in the message, IS properly

spelled, and correctly enciphered

(2) That the shdmg components (or equivalent CIpher disks or squares) employed m the

search for the asvumcd word are actually the ones which were employed in the eneipherment,

or are sueh as to give identical results as the ones which were actually used

(3) 'Lhat the pair of enciphering equations used m the test IS actually the pan which was

employed m the encipherment, or If a CIpher square IS used m the test, the method of finding

equivalents gives results that correspond WIth those actually obtamed m the encipherment

(See par 9)

23

The foregomg appears to be quite an array of contmgencies and the student may thmk

that on tins account the method WIll often Iaal But exammmg these eontmgencies one by one,

It WIll be seen that successful application of the method may not be at all rare-after the <olution

of some messages has disclosed what sort of paraphcrnaha and methods of employing them are

favored by the enemy From the foregomg remark It IS to be mferred that the probable-word

method has ItS greatest usefulness not III an untial solunon of a system, but only after successful

study of enemy commumcataons by more difficult processes of analysis has told ItS story to the

alert eryptanalvst Although It 1"1 commonly attnbuted to Bazenes, the French cryptanalyst

of 1900, the probable-word method IS very old III cryptanalysis and goes buck several centuries

Its usefulness in practrcal work may best be mdicated by quoting from a competent observer 1

There IS another (method] WhICh IS to ttns first method" hat the geometne method IS to analysis m certam

SCiences,ami, according to the whims of mdtviduals, certain cryptanalysts prefer one to the other Ccrtam others,

incapable of getting the answer WIth one of the methods m the solution of a difficult problem, conquer It by means

of the other, WIth a disooneertmg masterly stroke Thia other method IS that of the probable word We may

have more or less defimte opmions eoncermng the subject of the cryptogram We may know something about Its

date, and the correspondents, who may have becn mdiscreet in the subject they have treated On this baSIS,the

hypothesis IS made that a certam word probably appears In the text In eertam classes of documents,

rmhtary or drplomatio telegrams, banking and mmmg affaIrs, etc, It IS not unpossible to make very Important

asaumptrons about the presence of certain words m the text After a cryptanalyst has worked for a long time

With the writmgs of certam correspondents, he gets used to their eXprCS&IOnS He gets a "hole load of words

to try out, then the changes of key, and sometimes of system, no longer throw mto his ,,&\ the difficultie« of an

ab'lolutely new study, WhICh might require the analvtreal method

1 Givierge, M , ('OIlT8 de ('Typtograph~e, Paris, 1925, p 30
Sl<C'I'ION V

REPEATING-KEY SYSTEMS WITH MIXED CIPHER ALPHABETS, I

Plll'lIflI'8Ph Reason for the use of mixed alphabets - __ 16

Interrelated mixed alphabets _ _ _ _ _ _ _ __ __ _ _ 17

Prmciples of direct symmetry of positron __ __ __ _ _ _ __ _ __ 18

Inrtial steps in the solution of a tYPical evample -- _ _ _ __ _ _ 19

Apphcation of pnnciples of direct symmetry of posrnon., __ _ _ _ _ _ • 20

Subsequent steps 10 solution - -- --------.---- - ----_ __ _ _ _ __ _ _ _ 21

Completing the soluuon - - --- --- ------------- - _ __ 22

Solution of subsequent messages enciphered by same CIpher component _ __ __ _ _ 23

Summation of relatn e frequeneies as an aid to the selection of the correct gelleratrlceB.__ __ __ __ 24

Solution by the probable-word method --- -- -- -______ __ __ __ _ _ _ __ __ _ __ 25

Solutton when plain component IS mixed, the Cipher component, the normal, __ _ __ __ __ _ 26

16. Reason for the use of nuxed alphabets -a It has been SN'n In the examples considered

thus far that the use of several alphabets In the same message does not greatly complicate the

analysis of such a cryptogram There are three reasons why this IS so FIrstly, only relatively

few alphabets" ere employed, secondly, these alphabets were employed In a periodic or repeatmg

manner, gIVIng rise to cyclic phenomena In the cryptogram, by weans of which the number of

alphabets could be determmsd, and, thirdlv, the CIpher ulphahpts "ere knous: alphabets, by

which ISmeant merely th'1t the sequences of letters In Loth components of the CIpher alphabets

were known sequences

b In the case of monoalphabotie ciphers It ,'as found that the use of a mixed alphabet

delayed the solution to a considerable degree, and It will now Le seen that the use of mixed alphabets In polyelphabenc ciphers renders the anulysrs much more difficult than the use of standard

alphabets, but the solution IS sull fauly easy to achieve

17 Interrelated mixed alphabets -a It was stated in Par 5 that the method of producing

the mixed alphabets In a polyalphabcnc cipher often affords clues wlnch are of great aseistenca

ill the analysis of the CIpher alphabets ThIS IS "0, of course, only when the CIpher alphabets

are mterrelated secondary alphabets produced by shdrag components or their equrvalents

Reference IS now made to tho classiflcation set forth In Par 6, m connecuon WIth the types of

alphabets which may be employed In polyelphabeuc substitutron It "Ill be seen that thus far

only Cases A (1) and (2) have been treated Case B (1) WIll now be discusaed

b Here one of the components, the plain component, IS the normal sequence, while the

CIpher component ISa mn..ed sequence, the vanous ruxtaposrtrons of the two components yieldmg'

mixed alphabets The mixed component may be a systematICallY-mIXed or a random-mixed

sequence If the 25 successrve dIsplacements of tho mixed component are recorded In separate

Imes, a symmetnoel CIpher square such as that shown In FIg 5 results therefrom It ISidentical

In form with the square table shown On p 7, labeled Table I-A

(24)

25

Plain ._ ABC D E F G H I J K L M N 0 P Q R STU V W X Y Z

LEAVNWORTHBCDFGIJKMPQSUXYZ

EAVNWORTHBCDFGIJKMPQSUXYZL

AVNWORTHBCDFGIJKMPQSUXYZLE

VNWORTHBCDFGIJKMPQSUXYZLEA

NWORTHBCDFGIJKMPQSUXYZLEAV

WORTHBCDFGIJKMPQSUXYZLEAVN

ORTHBCDFGIJKMPQSUXYZLEAVNW

RTHBCDFGIJKMPQSUXYZLEAVNWO

THBCDFGIJKMPQSUXYZLEAVNWOR

H BCD F G I J K M P Q S U X Y Z LEA V N W 0 R T

BCDFGIJKMPQSUXYZLEAVNWORTH

C D F G I J K M P Q S U X Y Z LEA V N W 0 R T H B

DFGIJKMPQSUXYZLEAVNWORTHBC

CIpher_____ F G I J K M P Q S U X Y Z LEA V N W 0 R T H BCD

GIJKMPQSUXYZLEAVNWORTHBCDF

IJKMPQSUXYZLEAVNWORTHBCDFG

J K M P Q S U X Y Z LEA V N W 0 R T H BCD F G I

K M P Q S U X Y Z LEA V N W 0 R T H BCD F G I J

MPQSUXYZLEAVNWORTHBCDFGIJK

PQSUXYZLEAVNWORTHBCDFGIJKM

Q S U X Y Z LEA V N W 0 R T H BCD F G I J K M P

S U X Y Z LEA V N W 0 R T H BCD F G I J K M P Q

UXYZLEAVNWORTHBCDFGIJKMPQS

X Y Z LEA V N W 0 R T H BCD F G I J K M P Q S U

YZLEAVNWORTHBCDFGIJKMPQSUX

ZLEAVNWORTHBCDFGIJKMPQSUXY

FIGURE S

c Such a CIpher square may be used In exactly thc same manner as the VIgenere square

WIth the key word BLUEand conformmg to the normal enciphenng equations (811:/2=81/ 1, 8p/ l =

8 c/2) , the following lmes of the square would be used

ABCDEFGHIJKLMNOPQRSTUVWXYZ

BCD F G I J K M P Q S U X Y Z LEA V N W 0 R T H

LEA V N W 0 R T H BCD F G I J K M P Q S U X Y Z

UXYZLEAVNWORTHBCDFGIJKMPQS

EAVNWORTHBCDFGIJKMPQSUXYZL

FlGVRIio6a

These hnes would, of course, yield the followmg CIpher ulphabets

Plalll________ ABC D E F G H I J K L M N 0 P Q R STU V W X Y Z

(1) CIpher BCD F G I J K M P Q S U X Y Z LEA V N W 0 R T H

Plron___________ ABC D E F G H I J K L M N 0 P Q R STU V W X Y Z

(2) CIpher_________ LEA V N W 0 R T H BCD F G I J K M P Q S U X Y Z

PlaIn ABC D E F G H I J K L M N 0 P Q R STU V W X Y Z

(3) CIph~;~~~~~~~~~ U X Y Z LEA V N W 0 R T H BCD F G I J K M P Q S

PlaIn ABC 0 E F G H I J K L M N 0 P Q R STU V W X Y Z

(4) CIph;;~~~~~~~~~ E A V N W 0 R T H BCD F G I J K M P Q S U X Y Z L

FIGURE Gb
26

4 5

-A B C DIEIF!GIHIII J K ~.!.,~ o PfQfRTSITjUIVjWIX

=== VIN I=I=I~I=I= --1- G ='=I=I=lpl-=I=I=I=

A

B

2 _

27

C

D

E

F

G

H

J

K

L

M

N

P

Q

R

S

This grves the new values Ve=Dp and Ye=Yp in Alphabet 2 Alphabets 3 and 4 have a common

letter I, which perrmts of the placement of Q and Wm Alphabet 3, and of Band L In Alphabet 4

e The new values thus found are 01 course immerhutely inserted throughout the cryptogram, thus leading to the assumption of further vulues 1Il the CIpher text ThI"! process, mz, the

reconbtructwn of the pnmary components, by the application of the pnnr iple-, of direct S)mmctry

of pOSItIOn to the cells of the reconstruction skeleton, thus facihtates and hestens <olution

f It must be clearly understood that before the pnncrples of direct <;\ mmetry of pOSItIOn

can be applied In cases such as the foregoing, '/,t '/,s necessary that the plazn component be a known

sequence Whether It IS the normal sequence or not 10 nnmatenal, so IOl1g as the sequence IS

known Obviously, If the sequence It'> unknown, symmetry, even If present, cannot be detected

by the cryptanalyst because he has no base upon which to try out hIS assumptions for

symmetry In other words, direct symmetry of pOSItIOn IS mumfested In the illustrative

example because the plam component IS a known sequence, and not because It IS the

normal alphabet The sigmficance of this POInt will become apparent later 011 In connection

wrth the problem discussed In Par 26b

19 Initial steps In the solution of a typical example --a In the light of the foregoing pnneiples let a tYPICal message now be studied

MESSAGE

3

I S P J L

MT Z E I

Thus, the values of two new letters m Alphabet 1, VIZ, Pc=Jp , and Ne=Up have beer, automatacally determmed, these values were obtained WIthout any analysis based upon the frequency of

P, and Nc Iakewise, m Alphabet 2, the letters Y and Vmay be inserted m thebe POSItIOllS

flam ~~

YI-

_ A IB IC D E F G H I J K L M N 0 P Q R S T UIV W X Y Z

=1==

G Y V - --- - - - - - - - - - - - - - - - - - - - - - - - - N G P ----- - - - - - - - - - - - - - - - - - - - - - - - - L B I - - - - -;1-- - - - -- - - --- - - - - - - - - - - - - - I Q

AlBiC D EIFIGjH 2.1~1~1~~'~I~PIQ ~1~12.~IV -- W X Y Z

-='=1= GI='=I=-/pl-I--I-IY -1-'--=---- I V N I 1 _ ----

PlaIn _

1 _

2 _

Cipher 3 _

4 _

18 Pnnciples of direct symmetry of position -a It was stated directly above that FIg 5

IS a symmetrical cipher square, by which IS meant that the letters in Ito successive horizontal

lmes show a symmetry of pos'/,twn with respect to one another They constitute, m reahty, one

and only one sequence or series of letters, the sequences bcmg merely displaced successively 1,

2, 3, intervals The symmetry e....lnbited IS ObVIOUS and IS said to be visible, or d'/,rect

This fact can be used to good advantage, as has already been alluded to in par 7}

b Consider, for example, the pair of letters Go and Voin CIpher alphabet (1) of FI~ 6b The

letter VO IS the 15th letter to the rIght of Ge In CIpher alphabet (2), VO I~ also the 15th letter to

the rIght of Ge, as IS the case In each of the four cipher alphabets in Ihg 6b, since the relatwe

posiuons they occupy are the same In each honzontal hne In FIg ba, that IS, 11l each of the suecessrve recordings of the Cipher component as the 1 tter IS shd to the right agamst the plum or

normal component If, therefore, the relative POSItIOllS occupied by two letters, 81 and 82, m

such a CIpher alphabet, CI, are known, and If the position of 81 In another CIpher alphabet, C2,

belonging to the same senesIS known, then 8J may at once be placed into It"! correct posrtron m C2

Suppose, for example, that as the result of an analysis Lased upon consrderations of frequency

the following values in four CIpher alphabets have been tentatively determined

(1) Plam ABC D E F G H I J K L M N 0 P Q R STU V W X Y Z

Clpher_________ G Y V

(2) Plam___________ ABC D E F G H I J K L M N 0 P Q R STU V W X Y Z

Clpher_________ N G P

(3) Plam___________ ABC D E F G H I J K L M N 0 P Q R STU V W X Y Z

Clpher_________ L B I

(4) PlaID ABC D E F G H I J K L M N 0 P Q R STU V W X Y Z

Clpher______ _ W I Q FIGURE 7a

C The CIpher components of these four secondary alphabets may, for convenience, be assembled into a cellular structure, hereinafter called a sequence reconsirucium. skeleton, M shown In

FIg 7b Regarding the top line of the reconstruction skeleton In lhg 7b as being common to all

four secondary CIpher alphabets hsted 11l FIg 7a, the successrv e lmes of the reconstruction skeleton

may now be termed CIpher alphabets, and may be referred to by the numbers at the left

FIC'URE 7b

d The letter GIS common to Alphabets 1 and 2 In Alphabet 2 It IS noted that N occupies

the 10th pOSItIOn to the left of G, and the letter P occupies the 5th pOSItIOn to the nght of G

One may therefore place these letters, Nand P, III their proper pOSItIOns III Alphubet 1, the letter N

being placed 10 letters before G, and the letter P, 51ettero after G Thus

Plam _
A B C 0 E F G H I J K L M N 0 P Q R S T U V W x Y Z

SN RZ IJ IM GG Me MB I'll QF WB BO ZH IP MZ IX QB GN MJ

TG VG QG GG VZ QG B7 BG 00 IG CG QF VY BO QA

IE VG IO SZ QI vw LZ NZ LV QY PF

MJ CB RG VO KL OJ MY IJ LM YN

SG IG KH MY MJ CJ EG QB LN

CY MJ RZ XN VJ AY VY

I'll AJ VY BN

IJ

BF

RN

YO

QB

KF

GF

QJ

ALPHABET 2

29

V W X Y Z

v W X Y Z

VC PM VC we BE

KJ WE TE

BR WI EY

TZ KZ

IZ TA

TE EZ

BZ RN

PH OY

GN FY GN ZG ZG

GG GO YT

GG BQ ZG

DQ DT

EU YQ

ZF GN

JQ YT

FL

P Q R STU

FU GH BI

GQ BI

JQ MU

GP GS

GU DU

JD

JR

IN

BI

WX

SJ

o P Q STU

WR ZQ

L M N 0

AlPHABET 4

ALPHABET 3

IC WG WB

DR WF

XF

XF

AC

XC

IB

K L M N

PB BY WE CQ RC IE CC

PK LC EX OC WK

OR VW IV YJ

WY XP TY CK

XB WZ CX PQ

FZ WJ OI PE

EC CX BJ

LQ TR

BR CR

00 VR

BZ PE

AO WY

RQ

VQ

C 0 E F G H I J

ABC 0 E F G H I J K

YH WR

IK

WR

CY

WI

NR

A B

ZO NQ YA GG ZY NL MW AQ YG PL BN

DL JI GN YU NW YL GG JY JA

ON XU ZI NG BI JF DA

NA FO FQ WQ JX

HN IW FQ

NO JL

HA JL

W YW

OQ

NL

VS

11)2018-38-'1

ALPHABET 1

A B C 0 E F G H I J K L M N 0 P Q R S T U V W X Y Z

QC GW NT TV AE AS UD UW IT UT QP NX -W LB LA LA IW NN QI UX QR PT OP TC AD WC FI QX II UP YW YW DE IW

GK TT LX HW FW LV OT NW QD RB UE

OW we LW NO LR SY QC QD LC

GL GV WC GI GP

GX WC GP QL QB

XD AB RI NW

GB JF YV QE IV DI NY IP

NR SW UP

AK QW

QB

FIGURE 8

-. i!: :::: :g

~~~ ~~~-~-~~~~ ~~- ABC D E F G H I J K L MN 0 P Q R STU V WX Y Z

28

b The pnncipal repentions of three or more letters have been underlined rn the message and

the factors (up to 20 only) of the mtervals between them are as follows

QWBRIVWY___ __ 45=3, 5, 9, 15

CGXGR__ 60=2, 3, 4, 5, 6, 10, 12, 15, 20

PJEL_ _ 95=5, 19

ZZGL 145=5

BRIV 285=3, 5, 15, 19

BRI __ 45=3, 5, 9, 15

KAG___ 75=3, 5, 15

QRD 165=3, 5, ]5

QWB __ ___ 45=3, 5, 9, 15

QWB 275=5, 11

WIC 130=2, 5, 10, 13

XNF_ _ 45=3, 5, 9, 15

YZT 225=3, 5, 15

ZTC 145=3, 5

The factor 5 IS common to all of these repetmons, and there seems to be every indication that

five alphabets are Involved SInce the message already appears In groups of five letters, It IS

unnecessary In this case to rewnte It In groups corresponding to the length of the key The

umhteral frequency distnbution for Alphabet 1 IS as follows

c Attempts to fit this distnbunon to the normal on the basis of a duect or reversed standard

alphabet do not grve positrve results, and It IS assumed that mixed alphabets are Involved

Individual tnhteral frequency distnbutions are then compiled and are shown In FIg 9 These

tables are similar to those made for single mixed alphabet CIphers, and are made in the same

way except that instead of taking the letters one after the other, the letters which belong to the

separate alphabets now must be assembled m separate tables FOI example, In Alphabet 1,

the tngraph QAC means that A occurs m Alphabet 1, Q, Its prefix, occurs In Alphabet 5, and C, ItS

suffix, occurs In Alphabet 2 All confusion may be avoided by placing numbers mdieatmg the

61:1

alphabets m which they belong above the letters, thus QAC
ALPHABET 5

A B C D E F G H I J K L M N 0 P Q R S T U V W X Y Z

CI CS JK IB QI RV eM JR KQ YB QA BQ MQ RM ZC EL

KG RM YK YQ CM BV XI AB EQ RS CQ ZC RV

GI KI EQ

KG XB EM FG

EI R- JQ

CM

VC CM YO ZE CN FM WR

ZI RV ES CV QV RO EC

BI IV II CL BP QZ PY

XB RV ET ZQ YR YK

DB HL RW ZA QV

FM ZG DI HV

ZI CL

NX

JR

JQ

YI

31

It IS there1

The letter Be

5 5

Suffixes of Qa (=Ep)

",,-~::::- ...... ::::: ...... -... ..... I Q R X L V A Z 0

2 2

Prefixes of We (=Ep )

QGKVRBIL

1 2 6

j Consider now the letter Me' It does not occur either as a prefix of Wa, or as a suffix of Qa

Hence It IS most probably a vowel, and on account of ItS lugh frequency It may be assumed to

1 2

be Op On the other hand, note that Qa occurs five tames as a prefix of Wa and tbree tunes as

5

a suffix of Qa It IS therefore a consonant, most probably Rp, for It would grve the digraph

51 12

ER (=QQa) as occurrmg three times and RE (=QWe) as occurring five times

1 2 5

9 The letter Va occurs three times as a prefix of Wa and twice as a suffix of Qa

fore a consonant, and on account of Its frequency, let It be assumed to be Tp

2 5

occurs twice as a prefix of We but not as a suffix of Qa Its frequency IS only medium, and It IS

12

probably a consonant In fact, the twice repeated digraph BWe IS once a part of the tngraph

61 2 5

GBW, and Ge, the letter of second highest frequency m Alphubet 5,l00ks excellent for Tp MIght

oIl

not the tngraph GBW be THE? It will be well to keep this poesrbility in mind

1 2 6

h The letter Go occurs only once as a prefix of Wa and doe"! not occur as a suffix of Qa It may

1 2

be a vowel, but one can not be sure The letter La occurs once as a prefix of Wa and once as 80

6 1 2

suffix of Qa It may be considered to be a consonant Raoccurs once as a prefix of Wa, and tWICe 6 1 1

80S a suffix of Qa, and IS certainly a consonant Neither the letter So nor the letter Co occurs as a

2 6 prefix of Wa or as a sufflx.. of Qa, both would seem to be vowels, but a study of the prefixes and

1 1

suffixes of these letters lends more" eight to the assumption that CO IS 'L vowel than that SO IS a e 6 5

vowel For all the prefixes of C, Vll', N,T, nnd W, ale In subsequent anulysia of .Alphabet 5 classified as consonants, as are hkewrse rts sufhves, VIZ, T, C, and B m Alphabet 2 On the other hand, 6 2 1

only one prefix, La' and one suffix, Ba, of So are later olessihed as consonants Smce vowels are

e The letters of greatest frequency In Alphabet 1 are I, M, Q, V, B, G, L, R, S, and C Ie

2 5

has already been assumed to be Ep If We and Qe=Ep , then one should he able to distinguish the

vowels from the consonants among the letters M, Q, V, B, G, L, R, S, and C by exammmg the

2 5

prefixes of We' and the suffixes of Qe The prefixes and suffixes of these letters, as shown by the

trihteral frequency dietribuuons, are these

d One now proceeds to analyze each alphabet distnbution, m an endeavor to establish

rdennflcataons of cipher aqmvulents FIrst, of course, attempts should be made to separate

the vowels from the consonants m each alphabet, using the same test as III the case of a smgle

mn..ed-alphabet cipher There seems to he no doubt about the equivalent of Ep III each alphabet

1 2 3 4 5

E=Ie. We , Ge•Ce•Qe

2-3

C G-3

C J-3

P J-3

W B-3

WF-3

W Y-3

X N-3

1-2

Q W-5

VP-3

VW-3

3-4

B R-3

GQ-4

GX-3

J R-3

N F-3

Y Z-3

5-1

G B-4

I V-3

Q Q-3

4-5

R I-3

Y Q-3

ZT-3

1-2-3

Q WB-3

V WY-2

2-3-4

C G X-2

P J E-2

WB R-2

X N F-2

3-4-5

B R I-3

G X G-2

J E L-2

Y Z T-2

Z Z G-2

4-5-1

K A G-2

X G B-2

Z G I-2

Z T C-2

R I V-3

5-1-2

I V W-2

Q R D-2

WI C-2

30

FIGt/BB 9

Condensed table oj repet~honl>

1-2-3-4-5-1-2-3

Q WB R I V WY-2

2-3-4-5-1

C G X G B-2

2-3-4-1

P J E L-2

3-4-5-1

B-R-I-V

Z-Z-G-I-2
FIGURE 10

FIGURE 11

32

ABC D E F G H I J KIL M N 0 P Q

:~ T U VIW X Y Z

1===========

I C M V

1-----1-------I-- I---I-- -I--I--

- W

I-- - - - - - 1-- - - - - I-- I-- - - I--I-- - - I-- - - - -

G

I-- - - - I-- - - - - I--- I-- - - - I-- - I-- f-- - - - - - -

I C M Q V - - - - - f-- - - - - I-- - - - - I-- - f-- f-- - - - - -

M Q V I C

MESSAGE

1 2 3 4- s

Q W B R I VWYCA I S P J L R B Z E Y QWYEU

R E R T E E E RE

L W M G W I C J C I M T Z E I M I B K N Q W B R I

E E E R 0 R 0 RE R

V W Y I G B W N B Q Q C G Q H I WJ K A GEGXN

U- A E E R E N E E E

B

33

Plain _

A

FIGURE 12

CIpher 3 _

4 _

5 _

1___ ___

2 _

C

21 Subsequent steps In solutIon -a It IS high time that the thus far deduced values, as

recorded In the reconstructaon skeleton, be mserted m the cipher text, for by t1ustime It must seem

that the analysis has certainly gone too far upon unproved hypotheses The following results

are obtained

Ii Ii Ii

c The process at once grves three definite values Mo=Bp , Vo=Gp , Io=Rp Let these deduced values be substantrated by referrmg to the frequency distnbution Smce B and G are

normally low or medium frequency letters In plam text, one should find that Mo and V0, their

hypotheticnl equrvalents In Alphabet 5, should have low frequencies As a matter of fact, they

do not appear m t1us alphabet, w1uch thusfar corroborates the assumption On the other hand,

Ii Ii

Since Io=Rp , If the values denved from symmetry of pOSItIOn are correct, I o should be of high

frequency, and reference to the distribution shows that I o IS of 1ugh frequency The position of

C IS doubtful, It belongs either under Np or Vp If the former 18 correct, then the frequency

s

of Co should be high, for It would equal Np , If the latter IS correct, then Its frequency should be

Ii

low, for It would equal Vo As a matter of fact, Co does not occur, and It must be concluded

1

that It belongs under Vp Tills In turn settles the value of Co, for It must now be placed definitely

under I p and removed from beneath Ap

d The defimte placement of C now permits the msertion of new values In Alphabet 4, and

one now has the following

Alphabet Oonsomnts Vowels

1 Q. v. B. L. R. G1 I. M. C

2 B. C. D. T W. p. I

3 J. N. D. y. F G. Z

4 y. Z. J Q C. E? R? B1

I) G. N. A. I. W. L. T Q. u

_ A B C D E F G H I J K L M N 0

p~ R S T

u~ W X Y Z

= = C9 I C9 M Q V

M Q V C9 I

.IA B C D E F G H I J K L M N 0 P Q R S T U V W X Y Z

F== - - - - - - - - - - - - - - - - - - - - - - - - 1= _C9 I C9 M Q V

f-- - - - - - - - - - - - - - - - - - - - - - - - - - - W

I--- - - - - - - - - - - - - - - - - - - - - - - - 1-

- G

I--- - - - - - - - - - - - - - - - - - - - - - - - - - C

I--- - - - - - - - - - - - - - - - - - - - - - - - 1-

- Q

Plam.

1

more often essociated with

111

consonants than with other vowels, It would seem that CO IS more

hkely to be a vowel than So At any rate CO ISassumed to be a vowel, for the present, leaving S.

unclassified

Going through the same steps WIth the remaimng alphabets, the following results are

obtained

L __

2 _

CIpher 3 _

4 _

20 Apphcation of pnnoiples of direct symmetry of posmon -a The next step IS to try

to determine a few values In each alphabet In Alphabet 1, from the foregomg analysis, the

following data are on hand

Plam . ABC D E F G H I J K L M N 0 P Q R STU V WX Y Z

CIpher C9 I C9 M Q V

Let the values of Ep already assumed III the remammg alphabets, be set down in a reconstruction

skeleton, as follows

{

t. _

CIpher 5 _

b It 18 seen that by good fortune the letter Q IS common to Alphabets 1 and 5, and the

letter C1'3 common to Alphabets 1 and 4 IfIt IS assumed that one IS dealing with a case In which

a mixed component IS shdmg agamst the normal component, one can apply the prmciples of

direct symmetry of posraon to these alphabets, as outhned In Par 18 For example, one may

msert the following values In Alphabet 5

Plam __ ___
35

b The oombmationa given are excellent throughout and no mCOnSI"ltE'nCIeS appear Note

1 2 3

the tngraph QVvB, which IS repeated in the following polygraphs (nnderlmedm the foregoing text)

12 341i1 Ii 123 41i1

Q W B R I VSQ W B I I I

R E R T REA R E

1 2 34 Ii1

Q W B I I I

REP ARE

_ A B C D E F G H I JIK

L

M

N0P

Q R S

T

U V W

XY

Z

1= 1=

=

-

S

I G B

C

M

P

Q

R V W

l-I--

- - -

- I-- - - - -

-

-

-

-

- - - - -

-

-

- I-- 1-

P

Q

RV

W

S

I

G B C

M

I--

-

-

-

-

-

-

-- - -

-

-

-

-

-

--

-

-

- l-I-

-- I--

R V

W

S

I

G B C

M

P

Q

I--

-

---

-

-

- I-

--

-

- - -

-

-

-

-

-

- - -

-

-

-

-

I G B C

M

PQ R V

W

S

I-

-

- - -

- - -

- - -

--

-

-

-

-

-

- - - -

--

- 1-

M

P

Q

R

V

W

S

I

G B C

3

c The letterB

o 18 common to both polygraphs, and

a little rmagmanon will lead to the

3

assumpnon of the value Bo=P

p

, yielding the following

128 41i1 Ii

Q W B R I VS

REP0 R TP

fL-------

2 _

CIpher 3 _

4 _

5 _

Plam_

41i1284

d Note also (in F5) the polygraphI G V P WM, which looks lIke the word ATTACK The

A T K

Ii2

frequency distnbutions are consulted to see whether the frequencies grven for Go and Po are lugh 3

enough forT

p and Ap

, respecnvely, and also whether the frequency of

WO IS good enough forCp

,

iiI

It 18 noted that they are excellent Moreover, the digraph GB" which occurs four times, looks 1

hke TH, thus makmg Bo=Hp Does the msernon of these four new valuesm our diegram of

21

alphabets brmg forth any mconsistenciesf The msertion of the value ?o=A

p and Bo=H

p grves

no mdicataons either wny, since neither letter has yet been located m any of the other alphabets

Ii

The insertion of the value Go=T

p gives

a value common to Alphabets3 and 5, f01 the value

3

Go=E

p was assumed long ago Unfortunately an moonsistenoy IS found here The letter

I

has been placed two letters to the left of G m the mixed component, and has given good results 3

in Alphabets 1 and 5, If the value Wo=C

p (obtamed above from the essumption of the word

ATTACK) 18 conect, then W, and not I, should be the second letter to the left of G WhIch shall 8

be retamed? There has been so far nothing to establish the value of G.=E

p

, tIDS value was

assumed from frequency conSIderatIOns solely Perhaps It IS wrong It certamly behaves like a vowel, and one may see what happens when one changes ItS value to Op The followmg

placements in the reconstruction skeleton result from the enalyeis, when only two or three new

values have been added as

a result of the clues afforded by the deductions

34

D I D M R U V E Z Y G

Q I G V N C T G Y

0 B P D B L

E

T

R E P

I

E

E V C G X G B K Z Z G I V X C U N T Z A

0

BW F EQ

T

E

E

E

E

E

F

Q L F C0

M T Y Z T C C B Y

Q

o P D K A G D GIG

R

E

0

I

E EA

G VPWMR

Q

I lEW I C G X G B L G Q Q

v B G R S

T

K

R

E

E ENE

T

E

H MYJJY

Q V F W Y R W NFL G X N F W M C J K X

0

R

E

0

J I D DR

U

o P J Q Q Z R H C N V W D Y

Q R D G D G

E NE

E T E

E

E

K B X DB

N P X F P U Y X N F G M P J E L SAN C D 0 E

L S E Z Z G

I BEY

U K D H C A M B J J

F K I L C

J

E

E

0

E

M MFDZT C T J R D M I Y Z

Q A C J R R S B G Z N

0

I

0

E

E

N

Q YAHQ

v E D CQ L X N C L L V V C S

Q W B I I

R

E

T E E

E

E R E A R

P I V

J R N W N B R I V P J E L TAG D N I R G

QP

R

T

E

E E N

Q A T YEW C B Y Z T E V G

QU

v P Y H L L R Z N

Q

I E N

T

E

R X I N B A I K W J

Q R D Z Y F K W F Z L G W F

JQ

E

E

E

E

E

S

Q W J YQ I B W R X

RE

E

E
36

e Many new values are produced, and these are inserted throughout the message yielding the followmg ,

1 2 3 4 5

A Q W B R I V W Y C A I S P J L R B Z E Y Q W Y E U

REPOR TE E EMY SR RE

L W M G W I C J C I M T Z E I

EWCH ES ER 0 R

A B C D E F G H I J K L M N 0 P Q R S T U V W X Y Z - - - - - - - - - - - - - - - - - - - - - - - - - - A U S T I N G B C D F J K L M 0 P Q R V W Y Z E X H - - - - - - - - - - - - - - - - - - - - - - - - - -

- P Q R V W Y Z E X H A U S T I N G B C D F J K L M 0 - - - - - - - - - - - - - - - - - - - - - - - - - - _ R V W Y Z E X H A U S T I N G B C D F J K L M 0 P Q - - - - I--- - - - - - - - - - - - - - - - - - - - - I N G B C D F J K L M 0 P Q R V W Y Z E X H A U S T - - - - - - - - - - - - - - - - - - - - - - - - - - _ L M 0 P Q R V W Y Z E X H A U S T I N G B C D F J K

TREER,

COL

37

22 Completing the solutIon -a Completion of solution IS now a very easy matter

The mixed component IS finally found to be the following sequence, based upon the word

EXHAUSTING

PlaIll_ _ _

CIpher 3 _

4 __

5 _

E X H A U S TIN G BCD F J K L MOP Q R V W Y Z

and the completely reconstructed skeleton of the CIphersquare 18 shown ill FIg 13b

FIGURB 13b

b Note that the successive equrvalents of Ap spell the word APRIL, WhICh IS the key for the

message The plain-text message IS as Iollow s

c The precedmg case IS a good example of the value of the pnneiples of direct symmetry

of posmon when applied properly to a cryptogram enciphered by the shdmg of a mixed component against the normal The cryptanalyst starts off WIth only a very hmited number of

assumpnons and builds up many new values as a result of the placement of the few ongmal

values ill the reconstructaon skeleton

23 SolutIon of subsequent messages enciphered by the same CIpher component-a

l'reltmtnary remarks-Let It be supposed that the correspondents are USlDg the same basic or

pnmary component but WIth different key words for other messages Can the knowledge of

the sequence of letters ill the reconstructed pnmary component be used to solve the subsequent

messages? It has been shown that In the case of a monoalphebenc CIpher in which a mixed

alphabet was used, the process of completing the plain component could be applied to solve

subsequent messages III which the same CIpher component was used, even though the CIpher

component was set at a different key letter A modiflcataon of the procedure used m that case

can be used ill this case, where a plurality of CIpher alphabets based upon a sliding primary

component 18 used.

REPORTED ENEMY HAS RETIRED TO NEWCHESTER ONE TROOP IS REPORTED AT HENDERSON MEETING HOUSE TWO OTHER TROOPS IN ORCHARD AT SOUTHWEST EDGE OF NEWCHESTER 2D SQ IS PREPARING TO ATTACK FROM THE SOUTH ONE TROOP OF 3D SQ IS

ENGAGING HOSTILE TROOP AT NEWCHESTER REST OF 3D SQ IS MOVING TO ATTACK

NEWCHESTER FROM THE NORTH MOVE YOUR SQ INTO WOODS EAST OF CROSSROAD 539 AND

BEPREPARED TO SUPPORT ATTACK OF 2D AND 3D SQ DO NOT ADVANCE BEYOND NEWCHESTER

MESSAGES HERE

G E G X N

G 0

B P D B L

HA D

B W F E Q

H E E

G D GIG

GOA T

VB G R S

T ROO P

MC J K X

o S

R D G D G

SOT

SAN C D

C E

K I L C J

o E

S B G Z N

C R 0

Q WB I I

REP A R

I R G Q P

E CON D

L R Z N Q

C E

G W F J Q

GEE

I W J K A

E E

C T G Y 0

I 0

N T Z A 0

M I B K N Q WB R I

o 0 PRE P 0 R

o P D K A

A

B L G Q Q

H 0 N E

G X N F W

G H

V W D Y Q

TEE

M P J E L

o A

M B J J F

o R

A C J R R

S 0 F

L V V C S

D B E P

TAG D N

o

V P Y H L

T A

K W F Z L

E

Q C G Q H

R SON

Q I G V N

ROO P

I V X C U

E D E

C C B Y Q

I S P E

I C G X G

E SOT

R WNFL

S E

Z R H C N

C E

Y X N F G

T

K D H C A

E

M I Y Z Q

o 0 E

L X N C L

E

V P J E L

T A

E V G Q U

DON

R D Z Y F

S

B WN B Q

HE DE

V E Z Y G

T T

B K Z Z G

H T

M T Y Z T

o

Q I I E W

ROM H

Q V F W Y

R D Q

o P J Q Q

A N E

P X F P U

Q M

I BEY U

E R

C T J R D

I 0

V E D C Q

TEE

W N B R I

U P 0 R

C B Y Z T

I R

I K WJ Q

E E

I B W R X

E R 0

B

C VWYIG

TEA T

D IDMRU

E W0

E V C G X G

T SOT

F Q L F C 0

R E

G VPWMR

T A C K F

H MYJJY

o

J I D 0 R U

E 0

K. B X D B N

H D

L. S E Z Z G

C T

M. M F D Z T

o

N QYAHQ

R E

P. I V J R N

E D 0

Q ATYEW

H

R. X I N B A

o D

S QWJYQ

RE E
38

b The message-Let It be supposed that the followmg message passing between the same

two correspondents as in the precedmg message has been mtercepted

MESSAGE

i'IOVBI: li/l

1 2 a • 5 6 T

C 0 FIR S T

SQUADRO

NENEMYT

ROO P DIS

MOUNTED

o N H ILL F

IVENINE

THREEWE

S T 0 F GOO

DINTENT

SHXLINE

EXTENDS

FRO MCOR

NFl E L D T

WOHUNDR

EDYARDS

SOU T H X I

ATTACKR

I C H A R D S

CAP T

OOLt1JI1II2

NPDNNMUGSHGWQENCNSBZ

1 OQEOONVHTIHXRFODOTCA

2 PRFPPOWIUJIYSGPEPUDB

3 QSGQQPXJVKJZTHQFQVEC

4 RTHRRQYKWLKAUIRGRWFD

5 SUISSRZLXMLBVJSHSXGE

6 TVJTTSAMYNMCWKTITYHF

7 UWKUUTBNZONDXLUJUZIG

8 VXLVVUCOAPOEYMVKVAJH

9 WYMWWVDPBQPFZNWLWBKI

10 XZNXXWEQCRQGAOXMXCLJ

11 YAOYYXFRDSRHBPYNYDMK

12 ZBPZZYGSETSICQZOZENL

13 ACQAAZHTFUTJDRAPAFOM

14 BDRBBAIUGVUKESBQBGPN

15 CESCCBJVHWVLFTCRCHQO

16 DFTDDCKWIXWMGUDSDIRP

17 EGUEEDLXJYXNHVETEJSQ

18 FHVFFEMYKZYOIWFUFKTR

19 GIWGGFNZLAZPJXGVGLUS

20 HJXHHGOAMBAQKYHWHMVT

21 IKYIIHPBNCBRLZIXINWU

22 JLZJJIQCODCSMAJYJOXV

23 KMAKKJRDPEDTNBKZKPYW

24 LNBLLKSEQFEUOCLALQZX

25 MOCMMLTFRGFVPDMBMRAY

39

Oot.17][" 1

FVQUPRLWVGVHIQZHVDLF

1 GWRVQSMXWHWIJRAlWEMG

2 HXSWRTNYXIXJKSBJXFNH

3 IYTXSUOZYJYKLTCKYGOI

4 JZUYTVPAZKZLMUDLZHPJ

5 KAVZUWQBALAMNVEMAIQK

6 LBWAVXRCBMBNOWFNBJRL

7 MCXBWYSDCNCOPXGOCKSM

8 NDYCXZTEDODPQYHPDLTN

9 OEZDYAUFEPEQRZIQEMUO

10 PFAEZBVGFQFRSAJRFNVP

11 QGBFACWHGRGSTBKSGOWQ

12 RHCGBDXIHSHTUCLTHPXR

13 SIDHCEYJITIUVDMUIQYS

14 TJEIDFZKJUJVWENVJRZT

15 UKFJEGALKVKWXFOWKSAU

16 VLGKFHBMLWLXYGPXLTBV

17 WMHLGICNMXMYZHQYMUCW

18 XNIMHJDONYNZAIRZNVDX

19 YOJNlKEPOZOABJSAOWEY

20 ZPKOJLFQPAPBCKTBPXFZ

21 AQLPKMGRQBQCDLUCQYGA

22 BRMQLNHSRCRDEMVDRZHB

23 CSNRMOITSDSEFNWESAIC

24 DTOSNPJUTETFGOXFTBJD

25 EUPTOQKVUFUGHPYGUCKE

j Oomb~mng the selected generatmes -After some expenmentmg With these generatnces the 23d generatnx of Column 1 and

the 1st of Column 2, which Yield the digraphs shown m FIg 17a,

are combmed The generatnces of the subsequent columns are

examined to select those which may be added to these already

selected in order to build up the plam text The results are shown

in Fig 17b 'I'his process IS a very valuable aid in the solution of

messages after the primary component has been recovered as a

result of the longer and more detaIled analysIs of the frequency

distributions of the :firstmessage intercepted Very often a short

message can be solved m no other way than the one shown,

If the primary component IS completely known

g Recovery oj the key -It may be of mterest to find the key

word for the message Assummg that enciphenng method number 1 (see Par 7j, page 6) were known to be employed, all that

IS necessary IS to set the mixed component of the Cipher alphabet

underneath the plain component so us to produce the Cipherletter

indicated a~ the equivalent of any given plain-text letter in each

of the alphabets For example, in the first alphabet It IS noted that

Cp=Sc AdJust the two components under each other so as to

bnng S of the CIphercomponent beneath Cof the plam component,

FJoVBJ: 17a thus,

1 2

C 0

S Q

NE

R 0

M 0

o N

I V

TH

S T

D I

S H

EX

FR

NF

W0

E D

S 0

A T

I C

C A

FIGURE 16

123 • 667

FNMZVYV

VPBRHXQ

QDUVQEV

UNVGHOU

PNBEXKF

RMOZPRH

LULEMTG

WG Y V I C G

VSVWKUQ

G H U KIT V

VGECMTG

HWAVRJU

IQVDGUT

QEPVPJV

ZNOLRJT

HCFRVJU

VNBKLDK

DSARGQT

LBORVJU

F Z WK

d TranscMptton ~nto period« -Let the message

be wntten m groups of seven letters, m columnar

fashion, as shown m Fig 14 The letters lD each

column belong to a single alphabet Let the letters

lD each column be converted into their plam-component equivalents by settmg the reconstructed

Cipher component against the normal alphabet at any

arbrtranly selected pomt, for example, that shown

below

SFDZR YRRKX MIWLL AQRLU RQFRT IJQKF XUWBS MDJZK

MICQC UDPTV TYRNH TRORV BQLTI QBNPR RTUHD PTIVE

RMGQN LRATQ PLUKR KGRZF JCMGP IHSMR GQRFX BCABA

OEMTL PCXJM RGQSZ VB

FIOURE 1.

1 2 3 • 6 6 7

SFDZRYR

RKXMIWL

LAQRLUR

QFRTIJQ

KFXUWBS

MDJZKMI

C Q CUD P T

VTYRNHT

RORVBQL

T I Q B N P R

RTUHDPT

IVERMGQ

NLRATQP

LUKRKGR

ZFJCMGP

IHSMRGQ

RFXBCAB

AOEMTLP

CXJMRGQ

S Z V B

c Factormg and converswn ~nto pla~n component equwalents -The presence of a repontion

of a four-letter polygraph whose mterval is 21 letters suggests a key word of seven letters There

are very few otter repetmons, and this IS to be expected in a short message with a key of such

length

Plarn__________ ABC D E F G H I J K L MN 0 P Q R STU V WX Y Z

Cipher E X H A U S TIN G BCD F J K L MOP Q R V WY Z

The columns of equivalents are now as shown m Fig 15

e Exam~natwn and selemon oj generatMces -It has been shown that lD the case of a monoalphabetic Cipher It was merely necessary to complete the normal alphabet sequence beneath

the plain-component equivalents and the plam text all reappeared on one generatnx It was

also found that in the case of a multiple-alphabet Ciphermvolvmg standard alphabets, the plaintext equrvalents of each alphabet reappeared on the same generatnx, and It was necessary only

to combine the proper generatnces m order to produce the plam text of the message In the

case at hand both processes are combmed the normal alphabet sequence IS contmued beneath

the letters of each column and then the generatnces are combmed to produce the plam text

The completely developed generatnx diagrams for the first two columns are as follows (FIg 16)
40

Plam_________ ABCDEFGHIJKLMNOPQRSTUVWXYZABCDEFGHIJKLMNOPQRSTUVWXYZ

Cipher --_ EXHAUSTINGBCDFJKLMOPQRVWYZ

It IS noted that Ap=Ac Hence, the first letter of the key word to the message ISA The 2d,

3d, 4th, 7th key letters are found In exactly the same manner, and the following 18 obtmned

VVhen C 0 FIR S T equills

S F D Z R Y R then s, suocessively equals

AZIMUTH

24 SummatIon of relatIve frequencIes as an aId to the selectIon of the correct generatnces-

a In the foregoing example, under subparagraph j, there occurs tills phrase HAfter some

expenmentmg WIth these generatnces "By tills was meant, of course, that the seleenon of

the correct nutml pair of generatnces of plain-text equivalents IS m tills processa matter of tnal

and error The test of "correctness" IS whether, when Juxtaposed, the two generatnces so

selected yield "good" digraphs, that IS, high-frequency digraphs such as occur

m normal plain

text In ills early efforts the student may have some difficultym selectmg, merely by ocular

exammanon, the most likely generatnces to try There may be III each diagram several generatnces which contam good assortments of high-frequency letters, and the number of tnals of

eombmataons of generatnces may be qurte large Perhaps

a SImple mathematical method may

be of assistancem the process

b Suppose,m FIg 16, that each letter were accompanied bya number which corresponds

to ItS relative frequency In normal Enghsh telegraphic text Then, by adding the numbers along

each horizotuai hne, the totals thus obtamed WIll serve as relative numerical measures of the

frequency values of the respective generatnees Theoretically, the generatnx WIth the greatest

value will be the correct generatnx because ItS total WIll represent the sum of the mdrvidual

values of the actual plaintext letters In actual practice, of course, the generatnx with the

greatest value may not be the correct one, but the correct one will certamly be among the three

or four generatnces WIth the largest values Thus, the number of trials may be greatly reduced,

In the attempt to put together the correct generatnces c Using the precedmg message as an example, note the respective generatnx values In FIg

18 The frequency values of the respective letters shownm the figure are based upon the normal

distnbunon for War Department telegraphic text (see Table 3, AppendIx 1, Military Cryptanalysis, Part I)

Generatrlx o123456789

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

4:1

COLUMN1

FVQUPRLWVGVHIQZHVDLF 3 203 3 8 4 2 2 2 2 3 7 0 0 3 2 4 4 3

GWRVQSMXWHWIJRAIWEMG 2 2 8 2 0 6 2 0 2 3 2 7 0 8 7 7 2 13 2 2

HXSWRTNYXIXJKSBJXFNH

3 0 6 289 820 7 0 0 0 6 1003 8 3

I Y T X SUO Z Y J Y K LTC K Y G 0 I

7 2 9063 8 0J 0 2 0 493022 8 7

JZUYTVPAZKZLMUDLZHPJ o 0 3 2 9 2 3 7 0 004 2 3 4 4 0 3 3 0

KAVZUWQBALAMNVEMAIQK o 7 2 0 3 2 0 1 7 4 7 2 8 2 13 2 7 7 0 0

LBWAVXRCBMBNOWFNBJRL

41272 0 88 1 2 1 8 828 8 1 084

MCXBWYSDCNCOPXGOCKSM

230 1 2 2648 838 302 8 8 0 62

NDYCXZTEDODPQYHPDLTN

84230011184843023344118

OEZDYAUFEPEQRZIQEMUO

8130427 3313 3 13 0 8 0 7 0132 3 8

PFAEZBVGFQFRSAJRFNVP

3 3 7130 1 2230386 7 0 8 3 8 2 3

QGBFACWHGRGSTBKSGOWQ o 2 1 3 7 3 2 3 2 8 2 6 II 1 062 8 J 0

RHCGBDXIHSHTUCLTHPXR

833214 0 7 3 6 3• 3 34• 3 3 0 8

SID H C E Y J I T I U V D M U IQ Y S

674331320797324237026

TJEIDFZKJUJVWENVJRZT

9013743 000 3 02 2138 2080 II

UKFJEGALKVKWXFOWKSAU

303013274020203820673

VLGKFHBMLWLXYGPXLTBV 2 4 2 0 3 3 124 240 223 0411 1 2

WMHLG I C NMXMYZHQYMU C W

22842 7 3 8 2 0 22 0 3 02 2 3 8 2

XNIMHJDONYNZAIRZNVDX o 8 7 2 3 0 4 8 8 2 8 0 778 0 8 2 4 0

Y 0 J N IKE P 0 Z 0 A B J SAO W E Y

2 8 0 8 7 0133 8 0 8 7 106 7 8 2132

ZPKOJLFQPAPBCKTBPXFZ o 3 0 804 3 0 8 7 8 1 8011 1 3 0 8 0

AQLPKMGRQBQCDLUCQYGA

7 04 3 0 22 8 0 1 08 443 3 02 2 7

BRMQLNHSRCRDEMVDRZHB

182048368384132248031

C S N R MOl T S D S E F N W E S A I C

3 8 8 82 8 7984 61338 2138 7 73

DTOSNPJUTETFGOXFTBJD 4 II 8 8 8 3 0 3 .13. 8 2 8 0 8 II 104

EUPTOQKVUFUGHPYGUCKE

1383 II 8 0 02 3 3 3 2 3 3 228 3 0 13

Frequency

value

57

77

66

74

49

74

73

66

91

110

82

67

82

90

83

65

50

52

86

103

51

55

88

129

102

78
42

COLUMN 2

Genemtnll:

o NPDNNMUGSHGWQENCNSBZ

83.882326322013888610

1 0 Q E 0 0 N V H T I H X R F 0 DOT C A

8018888288730888.8887

2 P R F P POW I U J I Y S G PEP U D B

3 8 3 3 8 8 2 7 8 0 7 2 6 2 8 13 3 a • 1

3 Q S G Q Q P X J V K J Z T H Q F Q V E C o 6 2 0 0 3 0 0 2 0 0 0 D 3 0 3 0 2 13 3

4 RTHRRQYKWLKAUIRGRWFD

888 8 8 0 2 0 2 • 0 7 8 7 8 282 8 •

5 SUISSRZLXMLBVJSHSXGE

6 8 7 6 6 8 0 • 02. 1 2 0 6 8 6 0 2 D

6 TVJTTSAMYNMCWKTITYHF

9 2 0 9 D 672 2 8 2 3 209 7 9 2 8 8

7 UWKUUTBNZONDXLUJUZIG

8 208 8 9 1 8 0 8 8 • 0 • 8 0 8 072

8 VXLVVUCOAPOEYMVKVAJH

20. 228 8 8 7 8 8 D 2 2 2 0 2 7 0 8

9 WYMWWVDPBQPFZNWLWBKI

222222.31088082.2107

10 X Z N X X WE Q C R Q GAO X MX C L J

o 0 8 0 0 2 13 0 8 8 0 2 7 8 0 2 0 3 • 0

11 Y A 0 Y Y X F R D S R H B P Y N Y D MK

278 220 3 8 • 6 8 8 1 8 2 8 2 • 2 0

12 Z B P Z Z Y G SET SIC Q Z 0 ZEN L

o 1 8 0 0 2 2 6 D 9 678 0 0 8 0 D 8 •

13 A C Q A A Z H T F U T J D RAP A F 0 M

730 7 7 0 3 9 3 8 9 0 • 873 788 2

14 B D R B B A lUG V U K E S B Q B G P N

1.8117732230136101238

15 C ESC C B J V H WV L FTC R C H Q 0

3 D 6 3 3 1 0 2 8 2 2 • 8 9 8 8 3 308

16 D F T D D C K WI X WMG U D S D I R P

.89 • .30 2 7 0 2 2 2 3 • 6 • 788

17 E G U E E D L X J Y X N H VET E J S Q

D 2 3 D D • • 0 0 2 0 8 3 2 D 9 D 0 6 0

18 F H V F F E MY K Z Y 0 I WF U F K T R

3 3 2 8 8 13 2 2 0 0 2 8 7 2 8 3 8 0 9 8

19 G I WG G F N Z L A Z P J X G V G L U S

2 7 2 2 2 8 8 0 • 7 0 8 002 2 2 • 8 6

20 H J X H H GOA MBA Q K Y H WH MV T

800 8 8 2 8 7 2 1 700 2 3 2 3 2 2 D

21 I K Y I I H P B NCB R L Z I X I N WU

7 0 2 7 738 188 1 8 • 0 7 0 7 828

22 J L Z J J I Q COD C SMA J Y J 0 X V

o • 0 0 0 7 0 8 8 • 8 6 2 7 0 2 0 8 0 J

23 K MA K K J R D P EDT N B K Z K P Y W

o 2 7 000 8 • 3 D • 9 8 100 0 8 2 2

24 L N B L L K SEQ FEU 0 C L A L Q Z X

.81 • • 0 6 D 0 8 D 3 8 8 • 7 • 0 0 0

25 M0 C MML T F R G F V P D MB MRAY

2 8 8 2 2 • 9 8 828 2 3 , 2 I 2 8 7 2

J'JaVBlI D

Frequency

value

90

119

84

46

88

79

94

68

73

50

60

75

85

93

73

79

77

108

76

59

59

81

56

66

85

77

I

L

43

d It will be noted that the frequency value of the 23d generatnx for the first column of

cipher letters IS the greatest, that of the first generatnx for the second column IS the greatest

In both cases these are the correct generatnces Thus the selection of the correct generatnces

in such cases has been reduced to a purely mathematical basis which 18 at tunes of much assistance

m effectmg a quick solution Moreover, an understandmg of the principles mvolved will be of

considerable value in subsequent work

25 SolutIon by the probable-word method.-a Oecasionally one may encounter a cryptogram which 18 so short that It contams no recurrences even of digraphs, and thus grves no mdicanons of the number of alphabets involved If the shdmg mixed component IS known, one may

apply the method illustrated in Par 15, assummg the presence of a probable word, checkmg It

against the text and the sliding components to estabhsh a key, If the correspondents are using

key words

b For example, suppose that the presence of the word ENEMY 18 assumed in the message

in Par 23b above One proceeds to check It agamst an unknown key word, sliding the already

reconstructed mixed component agamst the normal and startmg WIth the first letter of the

cryptogram, in tlus manner

When ENEMY equals

SFDZR then Ap successrvely equals

XENFW

The sequence XENFW spells no mtelhgible word Therefore, the location of the assumed word

ENEMY 18 shifted one letter forward in the CIpher text, and the test 18 made again, Just as was

explained in Par 15 When the group AQRLU IS tned, the key letters ZIMUT are obtamed,

wluch, taken as a part of a word, suggests the word AZIMUTH The method must YIeldsolution

when the correct assumptions are made

c The danger to cryptographic secunty resultmg from the melusion of cryptographed

addresses and SIgnatures in cryptographic messages becomes quite obVIOUS in the light of

solunon by the probable-word method To illustrate, reference 18 made to the message employed

m Pars 19-22 It will be noted in Par 22b that the message earned a signature (Treer, Col)

and that the latter was enciphered Suppose that this were an authonzed pracuce, and that

every message could be assumed to conclude WIth a cryptographed signature The signature

"TREER COL" would at once afford a very good baSIS for the quick solunon of subsequent messages emanatmg from the same headquarters as did the first message, because presumably tlus

same signature would appear in other messages It 18 for this reason that addresses and signatures must not be cryptographed, If they must be included they should be cryptographed in a

totally different system or by a wholly dIfferent method, perhaps by means of a special address

and SIgnature code It would be best, however, to omit all addresses and SIgnatures, and to

let the call signs of the headquarters concerned also convey these parts of the message, leaving

the dehvery to the addressee a matter for local action

26 SolutIon when the plam component IS a mixed sequence, the CIpher component, the

normal-a ThIS falls under Case B (2) outlined m Par 6 It IS not the usual method of

employmg a smgle mixed component, but may be encountered oecasionally in CIpher devices

b The prehmmary steps, as regards factonng to determme the length of the penod, are

the same as usual The message IS then transcribed into ItS penods Frequency distnbutions

are then made, as usual, and these are attacked by the pnnciples of frequency and recurrence,

An attempt IS made to apply the pnnciples of direct symmetry of position, but this attempt

will be futile, for the reason that the plain component IS m thia case an 'Unknown mixed sequence
44

(See Par 18d) .Anyattempt to find symmetry In the secondary alphabets based upon the normal

sequence can therefore disclose no symmetry because the symmetry which eXIsts IS based upon a

wholly different sequence

c However, If the pnnciples of direct symmetry of poaition ale of no avail In this case,

there are certam other pnnciples of symmetry which may be employed to great advantage

To evplam them an actual example will be used Let It be assumed that It 18 known to the

cryptanalyst that the enemy IS usmg the general system under discussion, mz, a mixed sequence,

variable from day to day, IS used as plain component, the normal sequence IS used as cipher

component, and a repeatmg key, variable IIIIII message to message, IS used in the ordinary

manner

The followmg message has been mtercepted

1 2 3 4, S 6

A Q E 0 V K LRMLZ J V G T G NDLVK E V N T Y ERMUE

B V R Z M 0 YAAMP n K E I J S F M Y 0 YHMME G Q A M B

C UQAXR HUFBU KQYMU N E L V T K Q I L E K Z B U E

0 U LIB K N 0 A X B XU 0 G L LADVK P 0 A Y 0 DKKYK

E LAD H Y B V N F V U E E M E F F M T E G V W B Y T V 0 Z L

F S P B H B X V A Z C U 0 Y U E LKMMA E U 0 0 K N C F S H

G HSAHY T M G U J H Q X P P OK 0 U E XU Q V B F V W B X

H N X ALB T COL M I V A A A N S Z I L o V W V P Y A G Z L

J SHMME G Q 0 H 0 Y H I V P NCR R E X K 0 Q Z G K N C G

K NQGUY J I W Y Y TMAHW XRLBL o A 0 L G N Q GUY

L J U U G B JHRVX E R F L E G W GUO XED T P o K E I Z

M VXNWA F A A N E M K G H B S S N L 0 K J C B Z T G G L 0

N PKMBX HGERY T M W L Z N Q C Y Y T M W I P DKATE

P FLNUJ NOT V X J R Z T L o P A H C o F Z Y Y DEY C L

Q G P G T Y T E C X B HQEBR KVWMU N I N G J I Q 0 L P

R J KAT E GUWBR HUQWM V R Q B W Y R F B F KMWMB

S T M U L Z LAAHY J G 0 V K LKRRE X K N A 0 NOS B X

T X C G Z A HDGTL VKMBW I S A U E F 0 N W P N L Z I J

V S R Q Z L AVNHL GVWVK FIG H P G E C Z U K Q A P

d A study of the recurrences and factonng their mtervals discloses that five alphabets are

mvolved Umhteral frequency distributaons are made and are shown in FIg I9a

ALPHABET I

:::

:::::

-~~ ::::::::-

~~~~~~~~~-~~::::- ~~~~ ~~ ABCDEFGHIJKLMNOPQRSTUVWXYZ

ALPHABET 2

45

ALPHABET 3 :::::

:::: ::::::

~§

~::::::~~~~~ ~~~~:::::: ~~~~:::::: 2~~~

ABCDEFGHIJKLMNOPQRSTUVWXYZ

ALPHABET 4

ALPHABET 5

- :::::

~~ ~~ ~-

~~:::::: ~~~~ ~~~~ ~~ -~-~~~~

ABCDEFGHIJKLMNOPQRSTUVWXYZ

FIGlIBJ: 19a

e SInce the cipher component m this case IS the normal alphabet, 'ttfollows that the five

fr6lfU6ncy d'tst1'1,lYut'tons are based upon a sequence wh'tch '/,8 knoum, and therefore, thefive frequency

d'tStrilYutwns shouldmanifest a d'trect symmetry of d'tst1'1,butwn of crests and troughs By VIrtue of

this symmetry and by sluftmg the five distnbutrons relative to one another to proper supenmposmons, the several distnbutaons may be combined mto a single umlrteral distnbutaon Note

how tlus sluftlng has been done In the case of the five illustrative distnbutions

ALPHABET 1 ;:::

:::::

~~:::::: ~~- ~~~ ~~~~ ~~-~~ -- ~~~~~~~~~-~~:::::- ~~:::::::::: ~~

ABCDEFGHIJKLMNOPQRSTUVWXYZ

ALPHABET 2

:::: :::: ::::::::::::::: ~~

-~ ~~~~~~~~~~~ ,~~~~ ~~­

XYZABCDEFGHIJKLMNOPQRSTUVW

ALPHABET 3 :::::

:::: ::::::

~~

_:::: ~~~~~::::~~~~~ ~~~~:::::: ~~­

TUVWXYZABC DEFGHIJKLMNOPQRS

ALPHABET 4

- - ~~~ ~~

::::::_~-~~~~~~~::::~::::::- -~~~ ~~­

OPQRSTUVWXYZABCDEFGHIJKLMN

ALPHABET 5

::::: - ~- ~~ ~~

-~-~~~~~~:::::: ~-~- ~~~~ ~~ RSTUVWXYZABCDEFGHIJKLMNOPQ

J'lGlIBB 196

1112018-88--4
46

The umhteral frequency distnbution for this converted text follows Note that the frequency

of each letter 18 the sum of the five frequencies In the correspondmg columns of FIg 19b

:::::: ::::

§§

~~ ~~

- ~~ ~~ ~~

~~~ ~~ ~~

~~~ -~:::: ~~ ~~

~~~~~~~ ~~~ ~~

~~~~~~~ ~~~:::: ~~

~~~~~~~~~~~~~ ~~~~ ~~

~~~~~~~~~~~~~~~~::::~~~~~ ~~~

ABC DEFGHIJKLMNOPQRSTUVWXYZ

7 II 8 U 7 • M M D 8 00 U " 2 3 W " a una

Flot1U20

47

h The reconstruction of the plain component 18 now a very simple matter It 18 found to

be as follows

HYDRAULICBEFGJKMNOPQSTVWXZ

Note also, In Fig 19b, the keyword for the message, (HEAVY), the letters bemg in the columns

headed by the letter H

The solutron of subsequent messages With different keys can now be reached directly, by

a SImplemodificataon of the prmcrples explained in Par 18 'I'his modification consists rn usmg

for the completion sequence the mixed plain component (now known) instead of the normal alphabet, after the Cipher letters have been converted into their plam-eomponent equrvalents Let

the student confirm this by experiment

:J The probable-word method of aolution discussed under Paragraph 20 IS also applicable

here, In case of very short cryptograms Tills method presupposes of course, possession of the

mixed component and the procedure IS essentially the same as that In Par 20 In the example

discussed In the present paragraph, the letter A on the plain component was successively set

agamst the key letters HE!VY, but this IS not the only possible procedure

k The student should go over carefully the pnneiple of "conversion into monoalpbabetic

terms" explained m subpaiagraph j' above until he thoroughly understands It Later on he will

encounter cases in which this principle IS of very great assistance In the cryptanalysis of more

complex problems (Another example will be found under Par 45)

l The prmciple illustrated m subparagraph e, that IS, slnftmg two or more monoalphabctic

frequency distnbunons relatively '30 as to bnng them into proper alignment for amalgamation

Into a SIngle monoalphabetic distnbution, IS called match~ng It IS a very Important cryptanalytic pnnoiple Note that Its practical apphcation consists In shdmg one monoalphabetic

distnbution agaInst the other so as to obtain the best comotdence between the enhre sequence

of crests and troughs of one distnbution and the ennre sequence of crests and troughs of the other

distnbunon When the best POInt of coincidence has been found, the two sequences may be

amalgamated and theorehcally the SIngle resultant distnbunon WIll also be monoalphabetic In

character The successful apphcation of the pnnciple of matching depends upon several factors

FIrst, the cryptographic situatron must be such that matchmg IS a correct cryptographic step

For example, the distnbutaons In figure 19b are properly subject to matching because the Cipher

component In the basic sequences concerned III this problem 18 the normal sequence, while the

plain component IS a mixed sequence But It would be futile to tly to match the distnbutions

m figure 9, for m that case the CIpher component IS a mixed sequence, the plain component IS

the normal sequence Hence, no amount of sluitmg or matching can bnng the drstnbunons of

9 The problem having been reduced to monoalphabeue terms, a tnhteral frequency distnbunon can now be made and solutron readily attamed by SImple pnnciples It yields the

followmg

JAPAN CONSULTED GERMANY TODAY ON REPORTS THAT THE COMMUNIST INTERNATIONAL

WAS BEHIND THE AMAZING SEIZURE OF GENERALISSIMO CHIANG KAI SHEK IN CHINA

TOKYO ACTED UNDER THE ANTICOMMUNIST ACCORD RECENTLY SIGNED BY JAPAN AND GERMANY THE PRESS SAID THERE WAS INDISPUTABLE PROOF THAT THE COMINTERN INSTIGATED THE SEIZURE OF GENERAL CHIANG AND SOME OF HIS GENERALS MILITARY OBSERVERS SAID THE COUP WOULD HAVE BEEN IMPOSSIBLE UNLESS GENERAL CHANG HSUEN

LIANG HOTHEADED FORMER WAR LORD OF MANCHURIA HAD FORMED AN ALLIANCE WITH THE

COMMUNIST LEADERS HE WAS SUPPOSED TO BE FIGHTING SUCH AN ALLIANCE THESE

OBSERVERS DECLARED OPENED UP A RED ROUTE FROM MOSCOW TO NORTH AND CENTRAL

CHINA

I

EUTGN

GTHYK

K C I G N

DNRKT

TYKLU

NFMEQ

FYDNG

YDNLU

GNU 0 P

NTNGH

DNLUI

TJNXX

DNHFN

DHFOU

ITKXY

KPDYK

NGZNG

N 0 GUS

KTHB

A

B

C

D

E

F

G

H.

J.

K

L

M.

N

P.

Q

R

S

T

V

J The supenmposruon of the respective distnbutions enables one to convert the cipher

letters of the five alphabets into one alphabet Suppose It 18 decided to convert Alphabets

2, 3, 4, and 5 mto Alphabet 1 It IS merely necessary to substitute for the respective letters In

the four alphabets those which stand above them In Alphabet 1 For example, In Fig 19b, XI

In Alphabet 2 IS directly under AI in Alphabet 1, hence, If the supenmpositaon IS correct then

2, I

XI= AI Therefore, In the cryptogram It IS merely necessary to replace every Xc In the second

position by AI Agam TI In Alphabet 3=A, ill Alphabet 1, therefore, In the cryptogram one

replaces every T, m the tlurd posiuon by Ac The entire process, hereinafter designated as

conv6rswn ~nto monoalphabetu terms, gives the followmg convertedmessage

1 2 8 4 II

QHVHT LUTXI JYNFP NGSHT EYUFH

VUGYX YDHYY DNLUS SITKX YKTYN

UTHJA HXMND KTFYD NHSHC KTPXN

UOPNT NGHJK XXKSU LDKHT PRHKX

LDKTH BYURE UHLYN FITFN GYDNH

SSITK XYHLL UGFGN LNTYJ EXKPT

HVHTH TPNGS HTEBY DNVGN XXXHK

NAHXK TFKXV IYHMJ NVGUU OYDHY

SKTYN GTKTX YKPHY NFYDN XNKCI

NTNGH JLDKH TPHTF XUSNU ODKXP

JXBSK JKYHG EUMXN GZNGX XHKFY

VAUIJ FDHZN MNNTK SVUXX KMJNI

PNTNG HJLDH TPDXI NTJKH TPDUY

FOUGS NGAHG JUGFU OSHTL DIGKH

GSNFH THJJK HTLNA KYDYD NLUSS

JNHFN GXDNA HXXIV VUXNF YUMNO

TPBXI LDHTH JJKHT LNYDN XNUMX

XFNLJ HGNFU VNTNF IVHGN FGUIY

SUXLU AYUTU GYDHT FLNTY GHJLD
48

figure 9 mto proper supenmposinon for correct amalgamation (If the occurrencesin the vanous

distnbunons in figure 9 had been distributed aceordmg to the sequence of letters in the mixed

component, then matching would be possible, but in order to be able to distnbute these occurrences according to the mixed component, the latter has to be known-and that IS Just what IS

unknown until the problem has been solved) A second factor involved m successful matching

18 the number of elements m the two distnbunons formmg the subject of the test If both

of them have very few tallies, there 18 hardly sufficient mformation to permit of matching with

any degree of assurance that the work IS not in vain If one of them has many tallies, the other

only a few, the chances for success are better than before, because the posraona of the blanks m

the two distnbunons can be used as a guide for their proper supenmposinon

m There are certain mathematical and statistical procedures wluch can be brought to bear

upon the matter of eryptanalytic matching These will be presented in a later text However,

until the student has studied these mathematical and statistical methods of matchmg distnbuttons, he will have to rely upon mere ocular exammanon as a guide to proper supenmposmon

ObVIOUsly, the more data he has in each distnbunon, the easier 18 the correct supenmposmon

ascertained by any method

SECTION VI

REPEATING.KEY SYSTEMS WITH MIXED CIPHER ALPHABETS, II

Paragraph Further cases to be considered ------ ------ 27

IdentIcal pnmary mixed components proceeding In the same direction • • ••_ 28

Cryptograph.J.ng and decryptograph.J.ng by means of Identical pnmary mrxed components 29

Principles of solution • •• •••---••--.--- 30

27. Further cases to be considered -a Thus far Cases B (1) and (2), mentioned m Paragraph 6 have been treated There remains Case B (3), and this case has been further subdrvided

as follows

CASE B (3) Both components are mixed sequences

(a) Components are Identical mixed sequences

(1) Sequences proceed m the same direction (The secondary alphabet are nnxed

alphabets )

(2) Sequences proceed m OPPOSIte direcnons (The secondary alphabets are

reciprocal mixed alphabets)

(b) Components are different mixed sequences (The secondary alphabets are mixed

alphabets )

b The first of the foregomz subcases will now be examined

28 IdentIcal prImary mixed components proceedmg in the same direcnon -a It IS often

the case that the mixed components are derived from an easily remembered word or phrase,

so that they can be reproduced at any time from memory Thus, for example, given the key

word QUESTIONABLY, the following mixed sequence IS derived

QUESTIONABLYCDFGHJKMPRVWXZ

b By usmg this sequence as both plain and CIpher component, that IS, by shdmg this

sequence agamst Itself, a senes of 26 secondary mixed alphabets may be produced In enciphermg a message, shdmg stnps may be employed WIth a key word to designate the particular and

successrve posrnons 10 which the stnps are to be set, the same as was the case III preVIOUS examples

of the use of shdmg components The method of designating the pOSItIOns, however, reqmres

a word or two of comment at this point In the examples thus far shown, the key letter, as

located on the CIpher component, was always set opposite A, as located on the plain component,

possibly an erroneous impression has been created, mz, that this IS mvanably the rule This

18 decidedly not true, as has already been explained in paragraph 7c If It has seemed to be the

case that 81< always equals Ap , It 18 only beca use the text has dealt thus far principally with cases in

which the plum component IS the normal sequence and Its mntal letter, which usually constitutes the index for juxtaposmg CIpher components, IS A It must be emphasized, however,

that vanous convennons may be adopted 10 this respect, but the most common of them IS to

employ the mrtial letter of the plam component as the index letter That IS, the index letter,

8 11will be the untral letter of the mixed sequence, in this case, Q Furthermore, to prevent the

possibilrty of ambiguity It will be stated again that the pair of sncrphermg equations employed

m the ensumg diacussron will be the first olthe 12 set forth under Par 7J, mz, eJ3=8JI, 8P/I=8./3

In tlus case the subscnpt 1/1" means the plain component, the subscnpt 1/2", the cipher

component, so that the enciphering equation IS the followmg eJo=eJp, 8';p=8J.

(49)
50

Secondary alphabet (1)

Plam ABC D E F G H I J K L MN 0 P Q R STU V WX Y Z

Cipher H J P R L V W X D Z Q K U G F E A S Y C BTl 0 MN

c By settmg the two shdmg components against each other m the two posrtions shown

below, the cipher alphabets labeled (1) and (2) grven by two key letters, A and B, are seen to be

different

QUE S T ION A B L Y C D F G H J K M P R V WX Z

U EST ION A B L Y C D F G H J K M P R V WX Z Q

EST ION A B L Y C D F G H J K M P R V WX Z Q U

S T ION A B L Y C D F G H J K M P R V WX Z QUE

T ION A B L Y C D F G H J K M P R V WX Z QUE S

ION A B L Y C D F G H J K M P R V WX Z QUE S T

o NAB L Y C D F G H J K M P R V WX Z QUE S T I

NAB L Y C D F G H J K M P R V WX Z QUE S T I 0

A B Lye D F G H J K MP R V WX Z QUE S T ION

B L Y C D F G H J K M P R V WX Z QUE S T ION A

L Y C D F G H J K M P R V WX Z QUE S T ION A B

Y C D F G H J K M P R V WX Z QUE S T ION A B L

C D F G H J K M P R V WX Z QUE S T ION A B L Y

D F G H J K M P R V Wx Z QUE S T ION A B Lye

F G H J K M P R V Wx Z QUE S T ION A B L Y C D

G H J K M P R V Wx Z QUE S T ION A B L Y C D F

H J K M P R V Wx Z QUE S T ION A B Lye D F G

J K M P R V Wx Z QUE S T ION A B L Y C D F G H

K M P R V Wx Z QUE S T ION A B L Y C D F G H J

M P R V Wx Z QUE S T ION A B L Y C D F G H J K

P R V Wx Z QUE S T ION A B Lye D F G H J K M

R V Wx Z QUE S T ION A B L Y C D F G H J K M P

V WX Z QUE S T ION A B L Y C D F G H J K M P R

WX Z QUE S T ION A B L Y C D F G H J K M P R V

X Z QUE S T ION A B Lye D F G H J K M P R V W

Z QUE S T ION A B L Y C D F G H J K M P R V Wx

51

FIGURB 21

29. Cryptographmg and decryptographmg by identical primary mixed components -There

IS nothing of special interest to be noted in connection WIth the use either of rdentical mixed

components or of an equivalent quadncular table such as that shown m FIg 21, m enciphering or

deciphenng a message The baSIC pnnciples are the same as in the case of the sliding of one

mixed component against the normal, the displacements of the two components being controlled

by changeable key words of varymg lengths The components may be changed at Will and so on

All this has been demonstrated adequately enough m Elementary M~htary Oryptography, and

Advanced Md~tary Oryptography

30. Principles of eolunon.v-c Basically the pnnciples of solution m the case of a cryptogram enciphered by two Identical mixed shding components are the same as m the preceding

case Primary recourse IS had to the pnncrples of frequency and repetinon of smgle letters,

digraphs, tngraphs, and polygraphs Once an entering wedge has been forced into the problem,

the subsequent steps may consist merely in contmumg along the same lines as before, building

up the solution bit by bit

b Doubtless the question has already arisen in the student's mmd as to whether any

pnnciples of symmetery of pOSItIOn can be used to assist in the solution and in the reconstruction

of the CIpher alphabets m cases of the kind under considerataon ThIS phase of the subject Will

be taken up m the next section and will be treated m a somewhat detailed manner, because the

theory and principles involved are of very WIde application m cryptanalytics.

el .j.

QUESTIONABLYCDFGHJKMPRVWXZQUESTIONABLYCDFGHJKMPRVWXZ

QUESTIONABLYCDFGHJKMPRVWXZ

t

ek

el .j.

QUESTIONABLYCDFGHJKMPRVWXZQUESTIONABLYCDFGHJKMPRVWXZ

QUESTIONABLYCDFGHJKMPRVWXZ

t

ek

KEY LETTER=A

Plain component _

CIpher component _

KEY LETTER=B

Plain component.. _

CIpher component _

Secondary alphabet (2)

Plain ABC D E F G H I J K L MN 0 P Q R STU V WX Y Z

Cipher J K R V Y WX Z F Q U ME H G S B T C D L ION P A

d Very frequently a quadncular or square table IS employed by the correspondents, mstead

of shdmg stnps, but the results are the same The Ciphersquare based upon the word QUESTIONABLY IS shown m FIg 21 It will be noted that It does nothrng more than set forth the successive

pOSItIOns of the two primary sliding components, the top line of the square IS the plain component,

the successive honzontal hnes below It, the CIpher component in ItS vanous juxtapositrons The

usual method of employing such a square (1 e , corresponding to the enciphering equatrons

ek/c=el/p, ep/p=ec/c) IS to take as the CIpher equivalent of a plam-text letter that letter which

hes at the mtersectron of the vertical column headed by the plum-text letter and the horizontal

row begun by the key letter For example, the Cipher equrvalent of Ep With keyletter T IS the

letter Oc, or Ep (Tk)=Oe The method given in paragraph b, for determmmg the Cipher equrvalents by means of the two shdmg stnps yields the same results as does the Cipher square
SECTION VII

THEORY OF INDIRECT SYMMETRY OF POSITION IN SECONDARY ALPHABETS

Paragraph

Reeonstruetron of pnmary components from secondary alphabets____________________________________ 31

31 Reconstrucnon of prImary components from secondary alphabets -a Note the two

secondary alphabets (1) and (2) grven in paragraph 28c Externally they show no resemblance

or symmetry despite the fact that they were produced from the same pnmary components

Nevertheless, when the matter IS studied with care, a symmetry of positron IS discoverable

Because It IS a hidden or latent phenomenon, It may be termed latent symmetry oj pomt~on

However, m previous texts the phenomenon has been designated as an ~nd~rect symmetryoj p08U~on

and this termmology has grown mto usage, so that a change IS perhaps now madvisable

Indirect symmetry of pOSItIOn IS a very mterestmg and exceedmgly useful phenomenon in

cryptanalytlcs

b Consider the following secondary alphabet (the one labeled (2) in paragraph 28c)

(2) {Plam_ ------ ABC D E F G H I J K L M N 0 P Q R STU V W X Y Z

CIpher J K R V Y W X Z F Q U M E H G S B T C D L ION P A

c Assummg It to be known that this IS a secondary alphabet produced by two pnmary

rdenncal mixed components, It IS desired to reconstruct the latter Construct a cham of alternatmg plain-text and CIpher-text equrvalents, begmmng at any pomt and contmumg until the

cham has been completed Thus, for example, begmnmg With Ap=Jc, Jp=Qc, Qp=Bc, , and

droppmg out the letters common to successrve purrs, there results the sequence AJ Q B By

completmg the cham the following sequence of letters IS established

AJQBKULMEYPSCRTDVIFWOGXNHZ

d 'I'his sequence consists of 26 letters When slw aga~n8t '/,tselj U 'lJJ11l produce e.cactly the

same secondary alpJtabets as do the p1'1,mary components based upon the word QUESTIONABLY

To demonstrate that this IS the case, compare the secondary alphabets given by the two settmgs

of the externally different components shown below

Plain component., QUESTIONABLYCDFGHJKMPRVWXZQUESTIONABLYCDFGHJ~PRVWXZ

CIpher component_____ QUESTIONABLYCDFGHJKMPRVWXZ

Secondary alphabet (1)

Plam ABC D E F G H I J K L M N 0 P Q R STU V W X Y Z

CIpher J K R V Y W X Z F Q U M E H G S B T C D L ION P A

Plam oomponent., AJQBKULMEYPSCRTDVIFWOGXNHZAJQBKULMEYPSCRTDVIFWOGXNHZ

CIpher component..c., AJQBKULMEYPSCRTDVIFWOGXNHZ

Secondary alphabet (2)

Plam___________ ABC D E F G H I J K L M N 0 P Q R STU V W X Y Z

CIpher J K R V Y W X Z F Q U M E H G S B T C D L ION P A

(52)

53

e Smce the sequence A J Q B K grves exactly the same equrvalentsin the secondary

alphabets as the sequence QUE S T grves, the former sequence IS cryptographically

equivalent to the latter sequence For tlns reason the A J Q B K sequence IS termed

an equ~valent p1'1,mary component I If the real or ongmal primary component IS a key-word mixed

sequence, It IS hidden or latentwithin the equivalent primary sequence, but It can be made patent

by decimataon of the equivalent pnmary component The procedure IS as follows Fmd three

letters in the equivalent primary component such as ale hkely to have formed an unbroken

sequence m the ongmal pnmary component, and see If the interval between the first and second

IS the same as that between the second and third Such a case IS presented by the letters W, X,

and Z m the equivalent primary component above Note the sequence W0 G X NHZ,

the distance or interval between the letters W, X, and Z IS two letters Contmumg the cham by

adding letters two mtervals removed, the latent original primary component IS made patent

Thus

123 C a 8 7 8 9 ro II U H M U n H

WXZQUESTIONABLYCDFGHJKMPRV

j ItIS possible to perform the steps given m c and e in a combmed smgle operation when the

ongmal primary component IS a key-word mixed sequence Startmg WIth any pair of letters (in

the CIpher component of the secondary alphabet) hkely to be sequent in the key-word mixed

sequence, such as JKc in the secondary alphabet labeled (2), the following cham of digraphs may

be set up Thus, J I K,m the plain component stand over QI U, respectively, in the CIpher component, QI UI m the plam component stand over BI LI respectively, in the CIphercomponent, and

so on Connectmg the pairs in a senes, the followmg results are obtained

~~~~~~~~~~u~~~~~re~ffi~~~~~~~

TI DF VW IO FG WX ON GH XZ NA HJ ZQ AB JK

These may now be united by means of their common letters

JK MP PR RV etc =J K M P R V WX Z QUE S T ION A B L Y C D F G H

The ongmal pnmary component IS thus completely reconstructed

g Not all of the 26 secondary alphabets of the senes Yielded by two shdmg primary components may be used to develop a complete equivalent pnmary component If exammataon be made,

It will be found that only 13 of these secondary alphabets will YIeld complete equivalent pnmary

components when the method of reconstruction shown m subparagraph c above 18 followed For

example the following secondary alphabet, which 18 also derived, from the pnmary components

based upon the word QUESTIONABLY will not Yield a complete cham of 26 plam text-cipherplam text equivalents

Plam________ ABC D E F G H I J K L M N 0 P Q R STU V W X Y Z

Cipher C D H J 0 K M P B R V F W Y L X T Z N A I QUE G S

I Such an equivalent component IS merely a sequence which has been or can be developed or derived from

the ongmal sequence or baSIC pnmary component by applying a dec~mahon process to the latter, conversely,

the original or baSIC component can be derrved from an equivalent component by applying the same sort of

process to the equivalent component By decimation 18 meant the soleetion of elements from a sequence accordmg to some fixed mterval For example, the sequence A ElM 18 denved, by deermanon, from the

normal alphabet by selectmg every fourth letter
54

Equivalent primary component

1 2 3 • II I 7 8 II 10 11 12 13 11 2 3

A C H P X E 0 L F K V Q T A C H (The A C H sequence begins agam )

h It IS seen that only 13 letters of the cham have been estabhshed before the sequence begms

to repeat Itself It IS evident that exactly one-half of the cham has been established The other

half may be established by begmnmg with II letter not in the first half Thus

1 2 3 • II I 7 8 II 10 11 12 13 I 1 2 3

B D J R Z S N Y G M W U I B D J (The B D J sequence begms agam )

It IS now necessary to distnbute the letters of each half-sequence within 26 spaces, to

correspond WIth their placements In a complete alphabet 'I'his can only be done by allowmg a

constant odd number of spaces between the letters of one of the half-sequences Distnbutions

are therefore made upon the baSIS of 3, 5, 7, 9, spaces Select that distnbution which

most nearly coincides WIth the distnbution to be expected In a key-word component Thus, for

example, WIth the first half-sequence the distnbution selected IS the one made by leaving three

spaces between the letters I t IS as follows

123 • II I 7 8 D ro 11 U U U a a n u s

A-L-C-F-H-K-P-V-X-Q-E-T-O-

:J Now mterpolate, by the same constant interval (three m this case), the letters of the other

half-sequence Notmg that the group F - H appears in the foregoing distnbunon, It IS apparent

that G of the second half-sequence should be inserted between F and H The letter which immediately follows Gm the second half-sequence, mz, M, IS next mserted in the posmon three spaces to

the nght of G, and so on, until the mterpolation has been completed ThIs YIelds the ongmal

pnmary component, which ISas follows

123 • II I 7 8 D ro 11 U n u a a n u m Z s

ABLYCDFGHJKMPRVWXZQUESTION

k Another method of hsndlmg cases such as the foregomg IS mdicated m subparagraph J

By extending the pnnciples set forth m th-it subparagraph, one may reconstruct the following

cham of 13 pears from the secondary alphabet given III subparagraph 9

1 2 3 • II I 7 8 D 10 11 12 13 I 1

CD -+ HJ -+ PR -+ XZ -+ ES -+ ON -+ LY -+ FG -+ KM -+ VW -+ QU -+ TI -+ AB -+ CD. • •

Now find, III the foregoing cham, two pears hkely to be sequent, for example HJ and KM and count

the interval between them III the cham It 1"1 7 (countmg by pairs) If this decimation mterval

ISnow applied to the cham of pairs, the following IS established

123 • II I 7 8 D ro 11 U U U a a n u m

HJKMPRVWXZQUESTIONABLYCDFG

l The reason why a complete cham of 26lettels cannot be constructed from the secondary

alphabet given under subparagmph 9 IS that It represents a case m which two pnmary components of 26 letters were slid an even number of intervals apart (Tlus will be explained m

further detail III subparagraph r below) There are 111 all 12 such cases, none of which will

admit of the construction of a complete cham of 26 letters In additaon, there IS one case wherem, despite the fact that the pnmary components are an odd number of mtervals apart, the

secondary alphabet cannot be made to YIeld a complete cham of 26 letters for an equivalent

pnmary component This IS the case m which the displacement IS 13 mtervals Note the

secondary alphabet based upon the primary components below (which are the same as those

shown m subparagraph d)

55

PRIMARY COMPONENTS

QUESTIONABLYCDFGHJKMPRVWXZ

DFGHJKMPRVWXZQUESTIONABLYC

SECONDARY ALPHABET

Plam__________ ABC D E F G H I J K L M N 0 P Q R STU V W X Y Z

Cipher R V Z Q G U E S K T I WOP M N D A H J F B L Y X C

m If an attempt IS made to construct a cham of letters from this secondary alphabet alone,

no progress can be made because the alphabet IS completely reciprocal However, the cryptanalyst need not at all be baffled by this case The attack will follow along the hnes shown below

ill subparagraphs n and 0

n If the ongmal pnmary component IS a key-word mixed sequence, the cryptanalyst may

reconstruct It by attemptmg to "dovetail" the 13 reciprocal pairs (AR,BV, CZ,DQ,EG,FU, HS,

IK, JT , LW, MO, NP, and XY) mOO one sequence The members of these parrs are all 13 mtervals

apart Thus • 1 23. II 0 7 8 II ro 11 U n

A R

B V

C Z

D Q

E G

F U

H S

I K

J T

L W

M 0

N • • • • P

X Y

FIGtJllB

WrIte out the series of numbers from 1 to 26 and insert as many pRIrS mto pOSItIOn as possible

bemg gmded by consrderatrona of probable partial sequences m the key-word mixed sequence,

Thus

o 1 2 8 • II I 7 8 II ro 11 U n u u a

ABCD RVZQ

It begins to look as though the key-word commences WIth the letter Q, in which case It should

be followed by U ThIs means that the next pair to be inserted IS FU Thu"!

o 1 2 3 • " 6 7 8 D ro 11 12 13 U 1& 10 17

ABCDF RVZQU

The sequence ABC D F means that E ISm the key Perhaps the sequence IS ABC D F G H

Upon tnsl, usmg the parrs EG and HS, the followmg placements are obtamed

o 1 23. " 6 7 8 D ro 11 U U U u a n u m

ABCDFGH RVZQUES

ThIs suggests the word QUEST or QUESTION The pair JT ISadded

o 1 2 8 • " I 7 • D ro 11 n u u a n u m

ABCDFGHJ ••••• RVZQUEST
FIOUBlI:2S

57

Or, usmg hnes 2 and 3

WTYKZODPUAGVSLJXICMQNFREBH

PTFXAKECVOHQL and M S D W N J U Y RIG Z B

TIns IS an equrvalent pnmary CIpher component By finding the values of the suooessrve

letters of tIns cham m terms of the plain component of secondary alphabet number 1 (the zero

hne), the following IS obtained

TZPGLIQRHYOUVJCNEWKDASXMFB

ASPTFGHUVJZEBWKNRLXOCMIYQD

The sequence ASP T IS an equivalent pnmary plain component The ongmal keyword mixed components may be recovered from each of the equivalent pnmary components

That for the pnmary plam component IS based upon the key PUBLISHERS MAGAZINE, that for

the pnmary CIphercomponent IS based upon the key QUESTIONABLY

(2) Usmg only lmes 1 and 2, the followmg cham IS constructed

TZPGLIQRHYOUVJCNEWKDASXMFB

9____ ABC D E F G H I J K L M N 0 P Q R STU V W X Y Z

L__ T V ABU L I Q X Y C W S N D P FEZ G R H J K M 0

2____ Z J S T V I Q R M 0 N K X E A G B W P L H Y C D F U

The ongmal key-word mixed pnmary component (based on the word QUESTIONABLy) can

be recovered from either of the two foregoing equivalent pnmary components But If lines 1

and 3 are used, only half-chams can be constructed

TIns IS because 1 and 3 are both odd-mterval secondary alphabets, whereas 2 IS an evenmterval secondary It may be added that odd-interval secondaries are characterized by havmg

two cases m which a plain-text letter IS enciphered by Itself, that IS, e p IS identical with eo

Tins phrase "identical With" will be represented by the symbol =, the phrase "not Identical

With" will be represented by the symbol t= (Note that m secondary alphabet number 1 above,

Fp=Fo and Up=Uo, m secondary alphabet number 3 above, Mp=Mo and Op=Oo) TIns charactensnc will enable the cryptanalyst to select at once the proper two secondaries to work With m

case several are available, one should show two cases where ep=eOl the other should show

none

q (1) When the pnmary components are different mixed sequences, their reconstrucnon

from secondary CIpher alphabets follows along the same lmes as set forth above, under b to 3,

mclusrve, With the exception that the selection of letters for building up the cham of eqwvalents

for the pnmary CIphercomponent IS restncted to those below the zero lme in the reconstruction

skeleton Havmg reconstructed the pnmary CIpher component, the plain component can be

readily reconstructed 'I'his will become clear If the student will study the following example

The sequence GH J suggests GH J K, which places an I after T Enough of the process has

been shown to make the steps clear

o Another method of circumventmg the difflculties mtroduced by the 14th secondary

alphabet (displacement interval, 13) IS to use It in conjuncuon WIth another secondary alphabet

winch IS produced by an even-interval displacement For example, suppose the followmg two

secondary alphabets are available 1

9-------- ABC D E F G H I J K L M N 0 P Q R STU V W X Y Z

L______ R V Z Q G U E S K T I WOP M N D A H J F B L Y X C

L __ XZESKTIORNAQBWVLHYMPJCDFUG

FIOUIIE 23

56

The first of these secondaries IS the 13-mterval secondary, the second IS one of the evenmterval secondanes, from which only half-cham sequences can be constructed But If the construction be based upon the two sequences, 1 and 2 in the foregomg wagram, the followmz IS

obtamed b

RXUTNLDHMVZEIAYFJPWQSOBCGK

TIns IS a complete equivalent primary component The ongmal key-word mixed component

can be recovered from It by decunatron based upon the 9th interval

RVWXZQUESTIONABLYCDFGHJKMP

P (1) When the primary components are identical mixed sequences proceeding in oppo8'Lte

directions, all the secondary alphabets will be reciprocal alphabets Reconstructaon of the

pnmary component can be aocomphshed by the procedure mdrcatod under subparagraph 0

above Note the followmg three reciprocal secondary alphabets

1 2 8 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 2S 26

~_ABCDEFGHIJKLMNOPQRSTUVWXYZ

L__ P M H G Q F D C W Y L K B R V A E N Z X U 0 I T J S

2____ W V M K S J H G Q F D R C X Z Y I LEU T BAN P 0

3____ T S S Z L X W V N R P E M I 0 K C J BAY H G F U D

(2) USIng lines 1 and 2, the following cham can be constructed (equivalent pnmary component)

PWQSOBCGKRXUTNLDHMVZEIATFJ

I The method of wntmg down the secondaries show n m figure 23 will hereafter be followed In all eases when

alphabet reconstruction skeletons are necessary The top lme will be understood to be the plain component It

IS common to all the secondary alphabets, and IS set off from the CIpher components by the heavy black hne

ThiS top hne of letters WIll be designated by the digit 9, and Will be referred to as "the zero hne" In the diagram

The sueeeserve hnes of letters, which occupy the space below the zero line and which contam the vanous Cipher

components of the several secondary alphabets, Willbe numbered serially These numbers may then be used as

reference numbers for designating the honzontal lmes in the diagram The numbers standing above the letters

may be used as reference numbers for the vertical column'! m the diagram Hence, any letter in the reconstruction skeleton may be designated by coordmates, giving the horizontal or X coordmate first Thus D (2-11)

means the letter D standing lD line 2, Column 11 '

FIOUBE24"
FIOUKB 26

58

(3) Another method of accomphslung the process indicated above can be illustrated graphically by the following two chams, based upon the two secondary alphabets set forth m subparagraph q (1)

Number ofcharactersIn orlglnll1 prlI11ll1'Y component

32 30 28 27 26 25 24 22 21 20 18 16

16 15 14 27 13 25 12 11 21 10 9 8

32 10 28 9 26 25 8 22 7 20 6 16

8 15 7 27 13 25 6 11 21 5 9 4

32 6 28 27 26 5 24 22 21 4 ]8 16

16 5 14 9 13 25 4 11 7 10 3 8

32 30 4 27 26 25 24 22 3 20 ]8 16

4 15 7 27 13 25 6 11 21 5 9 2

32 10 28 9 26 25 8 22 7 20 2 16

16 3 14 27 13 5 12 11 21 2 9 8

32 30 28 27 26 25 24 2 21 20 18 16

8 5 7 9 13 25 2 11 7 5 3 4

32 30 28 27 2 25 24 22 21 20 18 16

16 15 2 27 13 25 12 11 3 10 9 8

32 2 28 9 26 5 8 22 7 4 6

2 15 7 27 13 25 6 11 21 5 9

32 30 28 27 26 25 24 22 21 20

16 5 14 9 13 25 4 11 7 10

32 30 28 27 26 25 24 22 21

8 3 7 27 13 5 6 11

32 10 4 9 26 25 8

16 15 14 27 13 25 12

32 30 28 27 26 25

4 5 7 9 13

32 6 28 27

16 15 14

32 10

8 15

32

16

14 6 10 16 10 18 16 8 10 6 4 6

Declmabon

Intarvll1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

Total number

ofcomplete

I18quences

59

yIeld 3 chams of 8 letters each It also follows that in the case of an ongmul primary component m wlnch the total number of characters II!> a pnme number, all decimanon mtervals will

yIeld complete equrvalent primary components The followmg table has been drawn up in the

light of these observations, for original pnmary sequences from 16 to 32 elements (All primenumber sequences have been omitted) In this table, the column at the extreme left grves the

vanous decimation mteivals, omittmg in each caee the first interval, wInch merely gives the

ongmal pnmary sequence, and the last interval, which merely gives the ongmal sequence

reversed The top line of the table gives the vanous lengths of ongmal primary sequences from

32 down to 16 (The student should bear in mind that sequences contammg characters m addition to the letters of the alphabet may be encountered, he can add to this table when he 18

mterested m sequences of more than 32 characters) The numbers withm the table then show,

for each combmation of decimataon interval and length of, ongmal sequence, the lengths of the

chams of characters that can be constructed (The student may note the symmetry In each

column) The bottom hne shows the total number of complete equivalent pnmary components

wluch can be denved for each drlferent length of ongmal component

T (2--4) D (0-4), -.

B (2-17) -. Q (0-17), -.

F (2-25) -. Y (ft-25), -.

M(2-9) -+ I (0-9), -.

X (2-13) -+ M (0-13), -+

S (2-3) -. C (0-3), -+

Coil Col 2

A (0-1) T (1-1),

D (0-4) B (1--4),

Q (0-17) F (1-17),

Y (0-25) M (1-25),

I (0-9) X (1-9),

M (0-13) S (1-13),

etc etc

123 , 6 678 9 W II U H u D

ft-------- ABC D E F G H I J K L M N 0 P Q R STU V W X Y Z

l..______ T V ABU L I Q X r c wS N D P FEZ G R H J K M 0

~--_ Z J S TV I Q R M0 N K X E A G B WP L H Y C D F U

(4) By jommg the letters in Column 1, the following cham 18 obtained A D Q Y I If, etc

If thia be examined, It will be found to be an equrvalent pnmary of the sequence based upon

PUBLISHERS MAGAZINE By joimng the letters in Column 2, the followmg cham IS obtained

T B F M X S This ISan equrvalent pumary of the sequence based upon QUESTIONABLY

r A final word concernmg the reconstruction of pnmary components m general may be

added It has been seen that in the case of a 26-element component shdmg agamst Itself (both

components proceeding in the same direction), It 18 only the secondary alphabets resultmg from

odd-interval displacements of the pnmary components which permit of reconstructmg a single

26-1etter cham of equivalents This 18 true except for the 13th mterval displacement, which

even though an odd number, still acts hke an even number displacement m that no complete

cham of equrvalents can be established from the secondary alphabet ThIS exception gives the

clue to the basic reason for this phenomenon It ISthat the number 26 has two factors, 2 and 13, which enter into the picture With the exeepnon of dIsplacement-mterval 1, any dUlplacement

'Lnterl'al wh'l.ch '1.8 a 8'Ub-mult'Lple oj, 01 has ajactor 'Ln commonW'Lth the number of letters 'l.n the prtmary

sequence wtll yteld a secondary alphabet from wh'Lch no complete cha'l.n oj 26 equ'Lvalents can be

denved jor th6 construdwn oj a complete el[l.l'I.valent pnmary component ThIS general rule IS

apphcable only to components which progress m the same direction, If they progress in opposite

directaons, all the secondary alphabets are reciprocal alphabets and they behave euctly like

the reciprocal secondanes resultmg from the IJ-mterval displacement of two 26-1etter Identical

components progressing in the same direction

s The foregomg remarks give me to the followmg obeervations based upon the general

rule pointed out above Whether or not a complete equivalent pnmary component ISdenvable

by decimataon from an ongmal primary component (and If not, the lengths and numbers of chams

of letters, or mcomplete components, that can be constructed in attempts to denve such eqUIValent components) will depend upon the number of letters m the ongmal pnmary component

and the specific decunation interval selected For example, in a 26-letter original primary component, decimation interval 5 will yield a complete equivalent primary component of 26 letters,

whereas decimation intervals 4 or 8 will yield 2 chams of 13 letters each In a 24-letter component, decimataon mterval5 Will also Yield a complete equivalent pnmary component (of 24 letters),

but decnnanon mterval 4 WIll Yield 6 chams of 4 letters each, and decnnanon interval 8 will
FJI"I'IlF 71

1112018-88--11

o ....,g V Q W E J M K G H DO G 5 U GOP 0 T H X 55 R Z X

RR YBWEWVMDWJ -- -

MM YUXOPPYOXZ - ----:::;:-

Y GKBWTLFDUZ -=--- -

X GHXEROQPSE - --

N X E R 0 Z J CC G C K W D V B L 5 E - ---

L J C Q R Q F V M L H AA K L B P C JOT X E PP U K W P E F X ENEM 5 R Q E W M L N A BB H 5 P 0 P N M D L M.... QQ .J; C U G D W P E U H

NN.JjOZOWMXCGQ

K Q C B Z E X Q T il Z 0 C D H LM Z T U Z 00 J J U G D W U V M

J B B J I U F J H D

I G 0 X 5 N Z H A 5 E - -

61

THE CRYPTOGRAM

(Repetrtiona undcrhned)

1 2 3 4 & 8 7 8 9 10 1234&878910 1234&878910

A WFUPCFOCJY P R C V 0 P N B L C W EE UDZFMTGQJ

B G B Z D P F B 0 U 0 Q LQZAAAMDCH FF L F U Y D T Z V !:LQ..

C GRFTZMQMU R BZZCKQOIKE... GG ZGWNKXJTRN

- ---

D KZUGDYFTRW 5 -.QFB5CVX C !!.JL HH -XTXCQ...lMVLW - --

E GJXNLWYOUX T ~TZ5DMXWCM II B G B W W 0 Q R G N - -- F IKWEUZOKZ U RKUHEQEDGX JJ HHVLAQQVAV

G PRXDWLZICW V FKVHPJJKJY KK J Q WOO T T N V Q

H GKQHOLODVM W YQD~XLLL LL BKXD50ZR5N -=-- -- -- - -

32 Applymg the principles to a speclfl.c example -a The precedmg seonon, with the

many details covered, now forms a suffiCIent base for proceeding with an exposition of how the

princrples of indirect symmetry of position can be applied very early m the solution of a polyalphabetic substItutIOn cipher in which shdmg pnmary components were employed to produce

the secondary cipher alphabets for the enciphermg of the cryptogram

b The case descnbed below will serve not only to explain the method of applying these

pnncrples but will at the same time show how therr apphcanon greatly facilitates the solution

of a single, rather difficult, polyalphabetae substrtution CIpher It IS realized, of course, that the

cryptogram could be solved by the usual methods of frequency and long, patient expernnentataon

However, the method to be described was actually applied and very materially reduced the

amount of time and labor that would otherwise have been reqmred for solunon

33 The cryptogra.m employed in the exposltlon -a The problem that will be used in this

exposmon mvolves an actual cryptogram submitted for solution m connectlon With a CIpher

device having two concentnc disks upon wlnch the same random mixed alphabet appears, both

alphabets progressmg m the same direction Tms was obtained from a study of the descnptlve

circular accompanymg the cryptogram By the usual process of factonng, It was determmed

that the cryptogram mvolved 10 alphabets The message as arranged accordmg to Its penod

IS shown in FIgure 27, m which all repetataons of two or more letters are mdicated

b The tnhteral frequency distnbutions are given m Figure 28 It will be seen that on

account of the breVIty of the message, considering the number of alphabets mvolved, the frequency du,tnbutlons do not YIeld many clues By a very careful study of the repetataons,

tentatrve mdrvidual determinations of values of CIpher letters, as illustrated m Figures 29,30,

31, and 32, were made These are given m sequence and ill detail ill order to show that there IS

notlung artrliclal or arbItrary m the prehmmary stages of analysis here set forth

(60)

SECTION VIII

APPLICATION OF PRINCIPLES OF INDIRECT SYMMETRY OF POSITION

PBrlllll'Bph

ApplYlllg the pnnclples to So specific example --------------- 32

The cryptogram employed In the exposltlon_____________ _ -------------- 33

Fundamental theory ---------- 34

AppllcatlOn of prmcrples ------------------------------------------------ 35

General remarks________ _ _ --------------- 36
62 63

TRILITERAL FREQUENCY DISTRIBUTIONS V

I A B C D E F G H I J K L M N 0 P Q R S T U V W X Y Z

A B C D E F G H I J K L M N 0 P Q R S T U V W X Y Z AA PF GY ZX ZM CQ NW SZ HL DF RF EO DO WL DL TM

LQ SV SM WJ NX OT EQ EO EM

EB FF XK YB ES XK ZC VZ WQ ZC ZR DC HC HR MK -F YQ QT PJ WV HQ IQ HM

HZ FC OR NH VQ ZL JF MK NT QG PJ GP PF ON WO

XK WJ ZO QJ JZ NU YT HJ OM

WG WK HB GP ON EV

QK MO GW OP

ES GW

EV VI

LH

EK A B C D E F G H I J K L M N 0 P Q R S T U V W X Y Z

MC AM CO EM WZ ZQ PB RZ DO PZ DZ CX LY EQ DF NH

ES PB PJ 00 WL PM RQ DM PF OT DB DQ KJ

QV CX TF DX WQ PY KO WM DP

II EX CO WZ SZ EE

FT AQ

A B C D E F G H I J K L M N 0 P Q R S T U V W X Y Z WX

GZ QB WU ZW GX GX IW KB GX LZ GF GX ZZ YX GQ KU

BJ JQ CB BB HV JU GQ HZ YD PX HP YX BZ VII

YW RV LU RU JW SQ GU RX A B C D E F G H I J K L M N 0 P Q R S T U V W X Y Z

00 FV FO QD YT ZA JK MN JK FC WE MM MG FM VC WO QO

GK GB NL QJ XT AD LO XT TN MW PO LI

CU BO VL LD NO QI OP JL OJ

BX PV JT OR MC MT

UW VD PT QV FE TV

III

WR OR

E F G H I J K L M N 0 P Q R S T U V W X Y Z VIII

A B C 0

CZ QP RT BI CW SO KH FP CO KE IN BD A B C 0 E F G H I J K L M N 0 P Q R S T U V W X Y Z

FS CH CR ZG KH GN RD QA HS OJ OV XN TQ ZC FH MG BC QA LA BU QS QG FR ZH XC

KW KZ RE KH HL QO OS ZC XH YC PU OK ZS JJ XL VL TV YU ZS QX ML

LP VW SG KP SE TS XG EG BS ZK QV ZU QA

GW FY BE HE 00 FU YX OX

JG TC ML OH

CG KO MY JR

UO IX

ZA B C 0 E F G H I J K L M N 0 P Q R S T U V W X Y Z

IV MV IW KH JO CY OZ MH EF GJ TW AE 00 DM TZ DJ

N 0 P Q R S T U V W X Y Z NE LW DX CQ KY IF LL TN JE OX NQ TE

A B C 0 E F G H I J K L M VV DH RN TX DM PE DZ RM OZ

ZA ZK ZP WP UD QO JP VA XL VP UC QQ XN FZ QE UD BE WM CQ VQ VW LE TZ

XD XW QW UD UE WK PP DC BC BT OF RN EH

XS XR UD VP WO BC ZO KO

XR UD DW XP WE BW X

WW ZW A B C D E F G H I J K L M N 0 P Q R S T U V W Z Y Z

HQ SB KC LS QL LG VG RY UG HZ AK RG UI JG KP

AG NC GR YR CR GH HZ AJ CG GF JY XJ

SG CB LG SY VB CL HB UO

SG UY VU GJ LB UK

XH XH

, SG

FIGUBIl: 28
64

INITIAL VALUES FROM ASSUMPTIONS

NN H 0 Z 0 W M X C G Q

SS R Z X

H E

00 J J U G D W2-R V M

THE

PP U K WP E F X ENE

E T

QQ Q C U G D WP E U H

THE

RR YBWEWVMDYJ

LL B K X'D S 0 Z R S li

E E T

MM YUXOPPYOXZ

1 2 d 4 6 6 7 8 9 10

EE B K D Z F MT G Q J

E

FF L F U Y D T Z V H-Q

T E

GG G WN K X J T R li

JJ H H V L A Q Q V A V

WI

KK J Q WOO T T N V Q

HH YTXCDPMVLW

E E

II B G B WW0 Q R G N

FIGURB 30

Q L Q Z A A A M D C H

12346678910

P R C V 0 P N B L C W

R B Z Z C K Q 0 IKE

H

S Q F B S C V X C H-Q

H

T T Z S D MX WC M

E

U RKUHEQEDGX

E T

V F K V HPJ J K J Y

E E

W Y Q D P C J X L L L

THE

X G H X E R 0 Q P S E

E E T H

Y g K B WT L F D U Z

E E

Z 0 C D H WMZ T U Z

AA K L B P C JOT X E

T THEBB H S P 0 P N MD L M

N

CC g C K WD V B L S E

E E T H

DD g S U G D P 0 T H X

E NTH E

M S R Q E WML N A

W H

N G S X E R 0 Z J S E

ENE T H

o g V Q WE J MK G H

E E

K Q C B Z E X Q T X Z

65

ADDITIONAL VALUES FROM ASSUMPTIONS (I)

2

Refer to lme DD III FIgure 29, So assumed to be Np

9

Refer to hne MIII figure 29, Ac assumed to be Wp

9 10 1 2 3 4 6

Then III hnes C-D, AVKZ UGD IS assumed to be WITH THE

L JCQRQFVMLH

1 2 8 4 6 6 7 8 9 10

A WFUPCFOCJY

T T H

B G B Z D P F B 0 U 0

E

C G R F T Z MQ MA V

E WI

D KZUGDYFTRrl

T H THE

E g J X N L WYOU X

E E -

F I K WE E-Q Z 0 K Z

EG PRXDWLZICW E -=

H G K Q H 0 L 0 D V M

EE -

I g 0 X S N Z H A S E

E E T H

00 J J U G D W V M J B B J I E-Q F J H D

THE

PP U K WP E F X ENE

E T

QQ Q C U G D WP E U H

THE

RR YBWEWVMDYJ

SS R Z X

E

LL B K X D S 0 Z R S li

E E T

MM YUXOPPYOXZ

KK JQWOOTTNVQ

JJ H H V L A Q Q V A V

HH YTXCDPMVLW

E E

II B G B WW0 Q R G N

128411678910

EE B K D Z F M T G Q J

E

FF L F U Y D T Z V H-Q

T E

GG G WN K X J T R li

FIGURB 29

Q L Q Z A A A MD C H

S Q F B S C V X C H-Q

H

T T Z S D MX WC M

E

U RKUHEQEDGX

E T

V F K V HPJ J K J Y

E E

W Y Q D P C J X L L L

THE

X G H X E R 0 Q P S E

E E T H

Y g...K B W T L F D U Z

E E

Z 0 C D H WM Z T U Z

R B Z Z C K Q 0 IKE

1 2 3 4 6 6 7 8 9 10

P R C V 0 P N B L Q.J'!

CC g C K WD V B L S E

E E T H

DD g S U G D P 0 T H X

ETHE

AA K L B P C JOT X E

THE - H

BB H S P 0 P-li M D L M

1284116789 10

A WF U P C F 0 C J Y

T T H

B G B Z D P F B 0 U 0

E

C GRFTZMQMAV

E

D K Z U G Q Y F T R

THE

E GJXNLWYOUX E E --

F I K WE E-Q Z 0 K Z

E--

G P R X D WL Z I C W

E -

H GKQHOLODVM

E E -=

I g 0 X S N Z H A S E

E E T H

J B B J I E-Q F J H D

1 2 3 6

Gc=Ep , Ko=Ep , Xo=Ep , and Do=Ep , from frequency consideranons.

346 456 901

UGD=THE, PCJ=THE, and SEG=THE, from study of repetatrons

M S R Q E WML N A

H

N g S X E R 0 Z J S E

E E T H

o g V Q WE J MK G H

E E

L J C Q R Q F V ML H

K Q C B Z E X Q T X Z
66

ADDITIONAL VALUES FROM ASSUMPTIONS (II)

3 4 5 6

Refer to FIgure 30, hnesNand X,where repetitron X E R 0 occur", assume EACH

E - - - 1 2 3 4 5 6 7 8 9 w

EE B K D Z F MT G Q J

E

FF L F U Y D T Z V tl-Q

UTE I N

GG G WN K X J T R N

G

HH YTXCDPMVLW

E E

II B G B WW R G N

H

JJ H H V L A Q Q V A V

WI

KK J Q WOO T T N V Q

I N

LL B K X D S 0 Z R S N

E E H T

MM Y U X 0 P P Y 0 X Z

- -----

I N

NN H 0 Z 0 WMX C G Q I--G N

00 J J U G D W2-B V M

THE

PP U K WP E F X ENE

E T 0

QQ C U G D WP E U H

THE

RR YBWEWVMDYJ

A -

SS R Z X

H E

FIGURE 32

1 2 3 4 5 6 7 8 9 10

P R C V 0 P N B L C W

I N G

Q L Q Z A A A M D C H

R B Z Z C K Q 0 IKE

H U

S C F B S C V X C H-Q

-U H GIN

T T Z S D M X WC M

G E

U RKUHEQEDGX

E T

V F K V HPJ J K J Y

ENE H

W Y Q D P C J X L L L

THE

X G H X E R 0 Q P S E

E E A C H T H

Y G K B WT L F D U Z EE --- Z 0 C D H WM Z T U Z

AA K L B P C JOT X E

T THE U --- H

BB H S P 0 P N MD L M

N I N G

CC g C K WD V B L S E

E E T H

DD G S U G D POT H X EH THE U I

90 1

HQ2-assume ING from repetinon and frequency

67

ADDITIONAL VALUES FROM ASSUMPTIONS (III)

4 5 6

OPN-assume ING from repetitaon and frequency

L J COR Q F V M L H

o

M S R Q E WML N A

A - WH

N G S X E R 0 Z J S E ENE A C H T H

o g V Q WE J M K G H

E E

1 2 3 4 5 6 7 8 9 10

A WF U P C F 0 C J Y

BUT THO UGH

B G B Z D P F B 0 U 0

E N 0

C G R F T Z M Q MA V

E W I

D K Z U G D Y F TRW

T H THE

E G J X N L WYOU X E E ----

F I K WE E-Q Z 0 K Z

EAN

G P R ~-2 WL Z I C W

E -

H g K Q H 0 L 0 D V M

E E UI g 0 X S N Z H A S E

E E T H

J B B J I E-Q F J H D

N I

K Q C B Z E X Q T X Z NN H 0 Z 0 WMX C G Q -=- G

00 J JUG D W2-B V M

THE

PP U K WP E F X ENE

E T 0

QQ C U G D WP E U H

THE

RR YBWEWVMDYJ

A

SS R Z X

HE

LL B K X D S 0 Z R S N

E E H T

MM YU X 0 P P Y 0 X Z

1 2 3 4 5 6 7 8 9 10

EE B K D Z F MT G Q J

E

FF L F U Y D T Z V H-Q

UTE

GG G WN K X J T R N

HH YTXCDPMVLW

E E

II B G B WW R G N

H

JJ H H V L A Q Q V A V

WI

KK J Q WOO T T N V Q

FIGURE 31

Q L Q Z A A A MD C H

1 2 3 4 5 6 7 8 9 10

P R C V 0 P N B L C W

R B Z Z C K Q 0 IKE

H U

S F B S C V X C tl-Q

U H G

T T Z S D MX WC M

E

U R K U H E QED G X

E T

V F K V HPJ J K J Y

E E H

W Y Q D P C J X L L L

THE

X G H X E R 0 Q P S E

E E A C H T H

Y g K B WT L F D U Z

E E

Z 0 C D H WM Z T U Z

AA K L B P C JOT X E

T THE U --- H

BB H S P 0 P N MD L M

N

CC g C K WD V B L S E

E E T H

DD G S U G D POT H X

EN THE U

12345678910

Refer to FIgure 30, hne A, W F U P C F 0 C J Y, assume to be BUT THOUGH

--TTH-----

12345678910

K Q C B Z E X Q T XZ

L J C Q R Q F V M L H

o

M S R Q ~W ML N A

A WH

N g S X E R 0 Z J S E

ENE A C H T H

o g V Q WE J M K G H

E E

A WF U P C FOe J Y

BUT THO UGH

B G B Z D P F B 0 U 0

E 0

C G R F T Z M Q MA V

E W I

D K Z U G D Y F TRW

T H THE

E G J X N L WYOU X - ---=- E E

F I K WE E-Q Z 0 K Z

EA

G P R X D WL Z I C W

E -

H G K Q H 0 L 0 DVM EE U-=

I g 0 X S N Z H A S E

E E T H

J B B J I E-Q F J H D
68

c }i rom the imtial and subsequent tentatrv e rdennficatrons shown In FIgures 29, 30, 31,

and 32, the values obtained were arranged m the form of the secondary alphabets m a reconstrucnon skeleton, shown in FIgure 33

I 2 3 4 6 7 8 0 10 11 12 13 14 15 16 17 18 10 20 21 22 23 24 215 211

.!.I~I!.. c D E F G1H _:J:J~ L M N 0 P Q R S -:.I~ V w X YIZ

1 w G Z K - - - - - - - - - - - - - - - - - - - - - - - - - - - 2 <;. K Z S F

e- - - - - - - - - - - - - - - - - - - - - - 1-

3 X U

e-- --- - - - --- - - - - - - - - - - - - - - - - - - 4 E T K G 0 P - - - - - - - - - - - - - - - - - - - - - - - - - - 5 R D C P - - - - - - - - - - - - - - - - - - - - - - - - - - - 6 V J N 0 F L- - - - - - - - - - - --- - - - - - - - - - - - - - ,-

7 0 ,...- - - - - - - - - - - - - - - - - - - - - - - - - - - 8 C - - - - - - - - - - - - - - - - - - - - - - - - - - - 9 J H S A - - - - - - - - - - - - - - - - - - - - - - - - - - - 10 D T E V Q y

FIGl1BJ: 33

34 Fundamental theory -a In paragraph 31, methods of reconstructmg pnmary components from secondary alphabets were given In detaal It IS necessary that those methods be

fully understood before the followmg steps be studied It was there shown that the pnmary

component can be one of a senes of equivalent pnmary sequences, all of which will grve exactly

similar results so far as the secondary alphabets and the cryptographic te-,:t are concerned

It IS not necessary that the Identical or ongmal pnmary component employed In the cryptographmg be reconstructed, any equivalent pnmary sequence WIll serve The whole question IS

one of estabhshmg a sequence of letters the interval between which IS either idenncal WIth that

m the ongmal primary component or else IS an exact constant multiple of the interval separatmg

the letters in the ongmal primary component For example, suppose K P X N Q forms a

sequence In the original pnmary component Here the Interval between K and P, and P and X,

X and N, N and QIS one, in an equivalent pnmary component, say the sequence K P X

N. Q, the interval between K and P IS three, that between P and X also three, and so on,

and the two sequences will yield the same secondary alphabets So long as the interval between

Kand P, P and X, X and N, Nand Q, ,IS a constant one, the sequence will be cryptographically

equivalent to the original pnmary sequence and will yield the same secondary alphabets as do

those of the ongmal primary sequence However, In the case of a 26-letter component, It IS

necessary that tlus mterv al be an odd number other than 13, as these are the only cases which

will yIeld one unbroken sequence of 26 letters Suppose a secondary alphabet to be as follows

(1){PlaIn___________ ABC D E F G H I J K L M N 0 P Q R STU V W X Y Z

Cipher_________ X K N P

69

It can be said that the pnmary component contains the following sequences

XN KP NQ PX

These, when umted by means of their common letters, yield K P X N Q

Suppose also the followmg secondary alphabet IS at hand

(2) {Plmn.__________ ABC D E F G H I J K L M N 0 P Q R STU V W X Y Z

Cipher_________ P X K N

Here the sequences PN, XQ, KX, and NZ can be obtained, which when united yield the two sequences KXQ and PNZ

By a comparison of the sequences K P X N Q, K X Q, and P N Z, one can estabhsh the

following

K P X N Q

K X Q

P • N Z

It followsthat one can now add the letter Z to the sequence, making It K P X N Q Z

b The reconstructIon of a primary component from one of the secondary alphabets by the

process grven In paragraph 31 requrres a complete or nearly complete secondary alphabet

ThIS IS at hand only after a cryptogram has been completely solved But If one could employ

several very scant or skeletonized secondary alphabets SImultaneously WIth the enalysis of the

cryptogram, one could then possibly build up a primary component from fewer data and thus

solve the cryptogram much more rapidly than would otherwise be possible

c Suppose only the cipher components of the two secondary alphabets (1) and (2) given

above be placed mto juxtaposraon Thus

128 4 & 6 7 8 0 W II n a u u u a 20 n 22 23 24 215 •

(1) X K N P

(2) P X K N

The sequences PX, XN, and KP are grven by juxtaposition These, when united, yield KPXN

as part of the primary sequence It follows, therefore, that onecan employthe c'tpher components

of seeondaru alphabets assources oJ'tndependent data to assist, m building up the pnmary sequences

The usefulness of this pomt will become clearer subsequently

35 Apphcation of prmclples -a Refer now to the reconstmcnon skeleton shown In

Figure 33 Hereafter, In order to aVOId all ambiguity and for ease In reference, the posrtion of

a letter In FIgure 33 will be indicated as stated In footnote 1, page 56 Thus, N (6-7) refers to

the letter NIn hne 6 and In column 7 of Figure 33

b (1) Now, consider the following pairs of letters

E (0-5) J (6-5)

G (0-7) N (6-7)

{

H (0-8) 0 (6-8) } _ o (0-15) F (6-15) HO, OF-HOF

(One IS able to use the hne marked zero In Figure 33 since this IS a mixed sequence sliding agmnst

'tIsetj)
71

V H 0 F

E J G N

6 (1) AgaIn USIng HOF and EJ, but In a different supenmposinon

H 0 F

E J

(2) Refer now to H (9-9), J (9-8) DIrectly under thebe letters IS found V (10-9), E (10-8)

Therefore, the V can be added immediutely before H 0 F, making the sequence V H 0 F

J (1) Now take V H 0 F and Juxtapose It WIth E J, thus

V H 0 F

E J

(2) Refer now to FIgure 33, and find the followmg

V (10-9) E (10-8)

H (9-9) J (q-8)

o (4-9) G (4-8)

I (0-9) H (0-8)

(3) From the value 0 G It follows that G can be set next to J mE J Thus

V H 0 F

E J G

(4) But G N already 18 known to belong to the same set of displacement-interval values

as E J Therefore, It IS now possible to combme E J, J G, and G N Into one sequence, E J G N,

yieldmg

There IS one 00

The frequency

H (0-8)

'I (1-8)

E (0-5)

G (1-5)

(2) Refer now to FIgure 33

9 (1) Refer now to FIgure 33

V (0-22) E (0-5)

'7 (1-22) G (1-5)

'7 (2-22) K (2-5)

'7 (3-22) X (3-5)

? (5-22) D (5-5)

'7 (6-22) J (6-5)

(2) The only values which can be Inserted are

o (1-22) G (1-5)

H (6-22) J (6-5)

(3) ThIS means that Vp=Oe In alphabet 1 and that Vp=He In alphabet 6

in the frequency distnbution for alphabet I, and no He In that for alphabet 6

distribution IS, therefore, corroborative insofar as these values are concerned

(h) (1) Further, tskmg E J G N and V H 0 F, supenmpose them thus

E J G N

VHOF

•

When Hp=Ee, then Op=J e

(3) Refer now to alphabet 10, Fieure 33, where It IS seen that Hp=Ee The denved value,

Op=J e, can 'Lmmed'Lately be inserted 'Ln the same alphabet and substituted In the cryptogram

(4) The student may possibly get a clearer Idea of the pnnciples Involved If he will regard

the matter as though he were dealing witb ar.thmetacal proportion For Instance, given any

three terms In the proportion 2 8=4 16, the 4th term can easily be found Furthermore, given

the pair of values on the left-hand SIde of the equation, one may find numerous pairs of

values which may be Inserted In the nzht-hand SIde, or VIce versa For Instance, 2 8=4 16

IS the same as 2 8=5 20, or 9 36=4 16, and so on An illustrataon of each of these prmcrples

will now be given, reference bemg made to FIgure 33 As an example of the first principle, note

that E (0-5) H (0-8)=J (6-5) 0 (6-8) Now find E (10-8) H (0-8)=9 (10-15) 0 (0-15)

It IS clear that J may be inserted as the 3d term In this proportion, thus givmg the 10

Important new value, Op=Je, which IS exactly what was obtained directly above, by means of

the parnal shdmg components As an example of the second pnnciple, note the followmg pairs

E (0-5) H (0-8)

K (2-5) Z (2-8)

D (5-5) C (5-8)

J (6-5) 0 (6-8)

(2) The immediate results of tlus set of values "Ill now be given Having HOF as a sequence,

WIth EJ as belonging to the same displacement Interval, suppose HOF and EJ are placed Into

juxtaposrtion as portions of shdmg components Thus

Plam, H 0 F

Clpher____ E J

H (0-8) G (0-7)

o (6-8) N (6-7)

'I'his checks lip the value In alphabet 6, Gp=Ne

d (1) AgaIn supenmpose HOF and GN

(2) Note thia corroboration

These addinonal pairs are also noted

70

H 0 F

G N

K (1-20) Z (1-7)

T to-20) G (0-7)

Therefore, E H=K Z=D C=J O=-T G, and T may be Inserted in pOSItIOn (4-5)

c (1) Agam, GN belongs to the same set of displacement-mterval values as do EJ and HOF

Hence, by supenmposinon

Plam, ___ H 0 F

Cipher.c., G N

(2) Refernng to alphabet 4,when Hp=Ge, then Op=Ne Therefore, theletterN can be mserted 4

In position (4-15) In FIgure 33, and the value Ne=Op can be substituted In the cryptogram

(3) Furthermore, note the corroborution found from this particular supenmposinom

o (6-8) G (4-8)

F (6-15) N (4-15)

which has Just been Inserted In FIgure 33, as stated above
72

and referrmg to Figure 33

A V H 0 F

.EJGNZ

P (4-20)

E (4- 1)

G (4- 8)

o (4- 9)

H (0- 8) Z (2- 8)

E (0- 5) K (2- 5)

N (0-14) S (2-14)

U (0-21) F (2-21)

E J G N Z C sequence the letters Hand Z are separated

T (0-20)

A (0- 1)

H (0- 8)

I (0- 9)

1 2 3 • Ii II 7 8

Z

K

• • • • S •

• • F •

H

E

• • • N

• • • U

l (1) From Figure 33

(2) Smce in the I AVH 0 F

by 8 mtervals one can wnte

.EJGNZ

• • • T K • S •

73

k (1) Consider all the values based upon the displacement interval correspondJ.ng to JG

J (6-5) G (1-5)- J (9- 8) G (4- 8)

N (6-7) Z (1-7) H (9- 9) 0 (4- 9)

S (9-20) P (4-20)- S (2-14) P (5-14)

Z (2- 8) C (5- 8)

K (2- 5) D (5- 5)

(2) Smce J and G are sequent m the E J G N Z sequence, It can be said that all the letters

of the foregomg p8J.rS are also sequent Hence Z C, S P, and K Dare available as new data

These give E J G N Z C and T K D S P

(3) Now consider

1 2 3 • Ii II

Now m the T K D S P sequence the interval between T and P 18 T P

Hence the interval between A and E IS6 also It followstherefore that the sequences A V H 0 F

and E J G N Z C should be united, thus

123 4 Ii II

•• AVHOF.EJGNZC

(4) Corroboration 18 found in the mterval between H and G, which 18 &!so SIX The letter I

can be placed mto position, from the relation I ijJ--9) 0 (4-9), thus

1 2 3 4 Ii II

I .AVHOF EJGNZC

whereupon the value S can be inserted

G (1-5)

Z (1-7)

T (0-20)

K (1-20)

U (3-20)

EJGNZ

T K

E (0-5)

G (0-7)

E (0-5)

G (1-5)

K (3-5)

(3) In the set there are also

Then m the supenmposraon

(2) Then

(3) From the wagram of supenmposrnon the value G (1-5) F (1-8) can be Inserted, which

gIves Hp=Fe In alphabet 1

'" (1) AgaIn, V H 0 F and E J G N are Juxtaposed

• VHOF

EJGN

(2) Refer to FIgure 33 and find the following

H (0-8) G (4-8)

A (0-1) E (4-1)

ThIS means that It ISpossible to add A, thus

A V H 0 F

E J G N

(3) Now refer to the followmg

E (0-5) K (2-5)

N (0-14) S (2-14)

E J G N

E J G N

It 18 possible to add Z under G, makmg the sequence E J G N Z

(4) Then taking

H (0-8) N (0-14)

o (6-8) '1 (6-14)

It will be seen that O=Z from supenmposmon, and hence m alphabet 6 Np=Ze, an Important

new value, but occurrmg only once in the cryptogram Has an error been made? The work

so far seems too corroborative m mterlocklng details to thmk so

3 (1) The possibihnes of the supenmposition and sliding of the AVHOF and the EJGNZ

sequences have by no means been exhausted as yet, but a httle different tral1 this time may

be advtsable
74

1 2 3 4 5 8 7 8 II 10 11 12 13 14 15 18 17 18 III 21 28

UIMYAVHOFLEJGNZCTQKDXSPBRW

1 234 II 8 7 8 9 ro u u a « w re u u n •

From U I A V H 0 F E J G N Z C T K D S P R

one can wnte

U I T

BKDZFMTGQJ

S ELF W ILL G 0

LFUYDTZVHQ

OUT B E COM I N

ZGWNKXJTRN

GACOLDANDL

YTXCDPMVLW

IFELESSMAS

BGBWWOQRGN

SANDTHESOL

HHVLAQQVAV

ARSYSTEMWI

J Q WOO T T N V Q

LLCIRCLEUN

BKXDSOZRSN

SEENGHOSTL

YUXOPPYOXZ

IKEINSPACE

HOZOWMXCGQ.

A W A I TIN G 0 N

JJUGJWQRVM

LYTHERESUR

UKWPEFXENF

R E C T ION 0 F A

CCUGDWPEUH

NOTHERCOSM

YBWEWVMDYJ

ICC A T A S T R 0

RZX

PH E

FIGw}, 34

R C V 0 P N B L C W

P 0 SIN G THE S

LQZAAAMDCH

OLARSYSTEM

B Z Z C K Q 0 I K F

SHALLTURNA

C F B S C V X C H Q

NUNCHANGIN

ZTZSDMXWCM

GFACEINPER

RKUHEQEDGX

PET U I T Y TOT

FKVHPJJKJY

HESUNEACHW

Y Q D P C J X L L L

ILLTHENHAV

GHXEROQPSE

EREACHEDTH

GKBWTLFDUZ

E END 0 FIT S E

OCDHWMZTUZ

VOL UTI 0 N S E

K L B P C JOT X E

TIN THE U N C H

HSPOPNMDLM

ANGINGSTAR

GCKWDVBLSE

EOFDEATHTH

GSUGDPOTHX

ENTHESUNIT

o The primary component appears to be a random-mixed sequence, no key word IS to be

found, at least none reappears on expenmentation With various hypotheses as to enciphering

equations Nevertheless, the random construe non of the pnmary component did not comphcate or retard the solution

75

Havmg the pnmary component fully constructed, decipherment of the cryptogram can be

completed WIth speed and precision The text IS as follows

WF U P C F 0 C J Y

BUTTHOUGHW

G B Z D P F B 0 U 0

ECANNOTASY

GRFTZMQMAV

ETREVIEWWI

KZUGDYFTRW

THTHEMINDS

GJXNLWYOUX

EYEOURPAST

I T WE P Q Z 0 K Z

WECANTOANE

P R X C WL Z I C W

XTENTFORES

GKQHOLODVM

EEOURFUTUR

GOXSNZHASE

EWECANWITH

B B J I P Q F J H D

SCI E N T I F I C

QCBZEXQTXZ

CON F IDE N C E

JCQRQFVMLH

LOOKFORWAR

SRQEWMLNAE

DTOATIMEWH

GSXEROZJSE

ENEACHOFTH

GVQWEJMKGH

E BOD I ESC 0 M

I I

,

X

D R

1 2 8 4 II

T (0-20)

E (0- 5)

U (3-20)

X (3- 5)

E

and

and

making the sequence

1 284 II 8 7 8 9 ro u u u « w re u u n

UI. AVHOF EJGNZCT KDXSP R

(3) Another denvation

(3) Hence one can make the sequence

128411878

I AVHOF ~JGNZC oK

Then I A V H 0 F E J G N Z C T K D S P

and y I A V HOE E J G HZ C T K D § P

128411878 123411878

m (1) Subsequent denvataons can be indicated very briefly as follows

E (0-5) C (0-3)

D (5-5) R (5-3)

1 284 8 8 7 8 9 ro II U « w re u u n •

FromUI AVHOF EJGNZCT KD SP

one can wnte E C

1 2 S 4 8

making the sequence

1 234 II 878 9 ro u a « re u u n •

UI AVHOF EJGNZCT KD SP R

(2) Another denvanon

E (0-5) G (1-5)

B (0-2) W(1-2)

From EJG

one can wnte E G

and then B W

There IS only one place where B Wcan fit, mz, at the end

1 234 8 8 7 8 9 ro una M re u a u n

UI AVHOF EJGNZCT KDXSP~R!l

n Only four letters remain to be placed into the sequence, mz, L, M, Q, and Y Their

positions are easily found by apphcation of the pnmary component to the message The complete sequence IS as follows
SI!.C'l'ION IX

REPEATING-KEY SYSTEMS WITH MIXED CIPHER ALPHABETS, III

Paragraph Solution of messages enciphered by known pnmary components _ _ 37

Solution of repeatmg-key crphers In which the identical mrxed components procced In OPPOSIte directIOns______ 38

Solution of repeatmg-koy eiphers m whu.h the primarv components are different mixed sequences_______________ 39

Solutron of subsequent messages after the pnmary components have bcen recovered.________________________________ 40

37 Solution of subsequent messages enciphered by the same pnmary components -a In

the discussion of the methods of solving repeating-key ciphers usmg secondary alphabets derived

from the shdmg of a mixed component against the normal component (Section V), It was shown

how subsequent messages enciphered by the same parr of pnmary components but WIthdifferent

keys could be solved by apphcation of prmcrples mvolvmg the completion of the plain-component

sequence (paragraphs 23, 24) The present paragraph deals WIth the apphcation of these same

pnnciples to the case where the primary components are identical mixed sequences

b Suppose that the following primary component has been reconstructed from the analysia

of a lengthy cryptogram

QUE S T ION A B Lye D F G H J K M P R V W X Z

A new message exchanged between the same correspondents IS intercepted and IS suspected

of having been enciphered by the same primary components but WIth a different key The

message IS as follows

N F W W P NOM K I W P I D SeA A E T Q V Z S E

Y 0 J seA A A F G R V N H D W D seA E G N F P

FOE M T H X L J W P NOM K I Q D B J I V N H L

T F N C S B G C R P

c Factonng discloses that the period IS 7 letters The text IS transcnbed accordingly, and

IS as follows

NFWWPNO

MK I WP I D

SeA A E T Q

VZSEYOJ

SCAAAFG

RVNHDWD

seA E G N F

PFOEMTH

X L J WP N 0

MKIQDBJ

I V N H L T F

N C S B G C R

P FIGtlBB 37

(78)

79

d The letters belongmg to the same alphabet are then employed as the mitral letters of

eompletion sequences, m the manner shown III paragraph 23e, usmg the already reconstructed

pnmary component The completion diagrams for the first five letters of the first three alphabets

are as follows

ALPBABBT I ALPJIABBT 2 ALPBABBT 3

N M S V S F K C Z C W I A S A

APT W T G M D Q D X 0 B T B

B R I X I H P F U F Z N L I L

L V 0 Z 0 J R G E G Q A Y 0 Y

YWNQN K V H S H U BeN C

C X A U A MWJTJ E L DAD

D Z B E B P X K I K S Y F B F

F Q L S L R Z MOM T C G L G

GUY T Y V Q P N P I D H Y H

*H Eel C W U R A R o F J C J

J S DOD X E V B V NGKDK

K T F N F Z S W L W AHMFM

MIG A G Q T X Y X B J P G P

P 0 H B H U I Z C Z LKRHR

R N J L J E 0 Q D Q Y M V S V

VAKYK S N U F U C P W K W

W B M C M T A E G E DRXMX

X L PDP I B S H S F V Z P Z

ZYRFR o L T J T G W Q R Q

Q C V G V N Y I K I HXUVU

UDWHW *A COM 0 J Z EWE

E F X J X B D N P N KQSXX

S G Z K Z LFARA M U T Z T

T H Q M Q Y G B V B PEl Q I

I J U P U C H L W L R SOU 0

o K ERE DJYXY *V T N E N

FIOtlBB 38

e Exammmg the successrve generanves to select the ones showmg the best assortment of

high-frequency letters, those marked III Figure 38 by astensks are chosen These are then assembled III columnar fashion and Yield the followmg plam text

123 4 5 6 7

HAV

E C T

CON

I ME

CON

FIGtlBB 39
80

FIOt1BJ: 40

The message may now be completed With ease It IS as follows

FIOt1U ..

1 2 8 , II 8

OSUMUH

QPFQKG

EQBMUP

WMMMWI

QYUVTU

WAHVBH

MKJXTJ

IQPKTJ

OSUMUJ

PAFUEY

NKCMEA

WTDXTJ

GSHQJZ

XEAEUF

PCLTNC

ZHCTOZ

WDFSZE

c Factonng discloses that the penod IS 6 and the message IS accordingly transcnbed mto 6 columns, Fig 42

The letters of these columns are then converted mto theIr

plam component equrvalents by juxtaposing the two pnmary components at any pomt of coincidence, for example Qp=Zo The converted letters are shown m FIg 43

The letters of the mdrvidual columns are then used as the

rmtial letters of completion sequences, using the

QUESTIONABLY pnmary sequence The final step IS the

selection and assembling of the selected generatnces

The results for the first ten letters of the first three columns

are shown below

MVXOX B Z I Y Z NLWZH o X I E 0 o 0 E P Z

FXSRX E J B S H BON A U R A P Z I NRAMV.

X 0 X A I J Y X W F K N DOW J E R C U R A L V L c

A Q U W J W X Y I DGRKD QBDRM Q E C Y V

Q W

81

I That IS, messages Intercepted after the primary components have been reconstructed and enciphered by

keys cWJerent from those used In the messages upon whIch the reconstruction of the pnmary components was

accompbshed

sequences, the solution of subsequent messages liS a relatively easy matter In both cases, however, the student must remember that before the method illustrated in paragraph 37 can be

applied It IS necessary to convert the cipher letters into their plain-component equivalents

before completmg the plam-eomponent sequence From there on, the process of selecting and

assemblmg the proper generatnces IS the same as usual

b Perhaps an example may be advisable Suppose the enemy has been found to be using

pnmary components based upon the keyword QUESTIONABLY, the plain component runmng

from left to nght, the cipher component in the reverse directaon The followmg new message

has arnved from the intercept station

128'118

MVXOXB

ZIYZNL

WZHOXI

E 0 0 0 E P

ZFXSRX

EJBSHB

ONAURA

PZINRA

MVXOXA

IJYXWF

KNDOWJ

ERCURA

LVBZAQ

UWJWXY

IDGRKD

QBDRMQ

ECYVQW

JOURNEY

S A INC E I

PFOEMTH

NTHEDIR

XLJWPNO

ECTIONO

MKIQDBJ

FHORSES

IVNHLTF

HOEFALL

NCSBGCR

S

P

FIOt1BBC

JOURNEY

HAVEDIR

NFWWPNO

ECTEDSE

MKIWPID

CONDREG

SCAAETQ

IMENTTO

VZSEYOJ

CON D U C T

SCAAAFG

THOR ORE

RVNHDWD

CONNAIS

SCAEGNF

f The corresponding key-letters are sought, usmg enciphenng equations 9t/o=el/p, 9pfp=

ec/o, and are found to be JOU, which suggests the keyword JOURNEY Testmg the key-letters

RNEY for alphabets 4,5,6, and 7, the following results are obtained

123 , II 8 7

JOURNEY

NFWWPNO

HAVEDIR

SCAAETQ

ECTEDSE

38 SolutIon of repeatIng-key Ciphers 10 whIch the Identical mixed components proceed 10

opposite dIreotlons.-The secondary alphabets 10 this case (paragraph 6, Case B (3) (a) (II)

are reciprocal The steps 10 solution are essentially the same as 10 the preceding case (paragraph 28), the pnneiples of indirect symmetry of posinon can also be applied With the necessary

modiflcations mtroduced by VIrtue of the reciprocity exIStIng within the respective secondary

alphabets (paragraph 3lp)

39 SolutIon of repeatIng-key Ciphers In whIch the pnmary components are different Dllxed

sequences.-Thlsls Case B (3) (b) of paragraph 6 The steps m solutaon are essentially the same

as 10 paragraphs 28 and 31, except that 10 applying the principles of indirect symmetry of POSItion It 18 necessary to take cogmzance of the fact that the pnmary components are drlferent

mixed sequences (paragraph 3lq)

40 SolutIon of subsequent messages after the primary components have been recovered.-

a In the case 10 which the pnmary components are Identical mixed sequences proceeding 10

opposite direcnons, as well as 10 that 10 which the pnmary components are different mixed
FIGURS 46

Columnar assemblmg of selected generatrices gives what IS shown In FIg 45

,

The entire

NUMBER

E LAY I N

IJYXWF

GPO SIT

KNDOWJ

ION AND

ERCURA

W ILL P R

LVBZAQ

o T E C T L

UWJWXY

EFTFLA

IDGRKD

NKOFBR

QBDRMQ

IGADEX

ECYVQW

FloURS 46

e If the pnmary components are different nuxed sequences, the procedure IS identical wrth

that ust indicated The Important point to note IS that one must not fall to convert the letters

mto :heu plam-component equivalents before the eompletion-sequence method IS applied

83

d The key letters are sought, and found to be NUM, which suggests NUMBER

message may now be read with ease It IS as follows

NUMBER

FIR S T C

MVXOXB

AVALRY

Z I Y Z N L

LESSTH

WZHOXI

IRDSQU

E 0 0 0 E P

ADRONW

ZFXSRX

ILL 0 C C

EJBSHB

UPYAND

ONAURA

DEFEND

PZINRA

FIRSTD

MVXOXA

COLVIlIf 3

UFBMUHJPUF

EGLPEJKREG

SHYRSKMVSH

TJCVTMPWTJ

I K D W I P R X I K

OMFXORVZOM

NPGZNVWQNP

ARHQAWXUAR

BVJUBXZEBV

LWKELZQSLW

YXMSYQUTYX

C Z P T CUE I C Z

D Q RID E SOD Q

FUVOFSTNFU

G E W N G T I AGE

HSXAHIOBHS

J T Z B JON L J T

K I Q L K NAY K I

MOUYMABCMO

P N E C P B L D P N

*R A S D R L Y F R A

VBTFVYCGVB

WLIGWCDHWL

XYOHXDFJXY

ZCNJZFGKZC

QDAKQGHNQD

82

COLl1XIf 2

FIGUBB 44

SPQMYAKQSA

TRUPCBMUTB

*I V E R D L P E I L

o W S V F Y R SOY

NXTWGCVTNC

AZIXHDWIAD

B Q 0 Z J F X 0 B F

LUNQKGZNLG

YEAUMHQAYH

C S B E P J U B C J

DTLSRKELDK

FIYTVMSYFM

G 0 C I WP T C G P

HNDOXRIDHR

JAFNZVOFJV

KBGAQWNGKW

MLHBUXAHMX

P Y J L E Z B J P Z

RCKYSQLKRQ

VDMCTUYMVU

WF P DIE C P W E

XGRFOSDRXS

ZHVGNTFVZT

Q J WH A I G W0 I

UKXJBOHXUO

EMZKLNJZEN

COLl1l11f 1

1 234 S 6

FIR

AVA

L E S

I R D

ADR

ILL

U P Y

DE F

FIR

E L A

o Q E WQ WMI 0 P

NUSXUXPONR

AETZEZRNAV

B S I Q S Q v A B W

LTOUTUWBLX

YIN E I E X L Y Z

C 0 A S 0 S Z Y C Q

D N B TNT Q C D U

*F A L I A IUD F E

G B Y 0 B 0 E F G S

HLCNLNSGHT

J Y DAY A T H J I

K C F B C B I J K 0

MDGLDLOKMN

PFHYFYNMPA

R G J C G CAP R B

VHKDHDBRVL

WJMFJFLVWY

XKPGKGYWXC

ZMRHMHCXZD

Q P V J P J D Z Q F

URWKRKFQUG

EVXMVMGUEH

SWZPWPHESJ

TXQRSRJSTK

I Z U v Z V K TIM
SECTION X

REPEATING·KEY SYSTEMS WITH MIXED CIPHER ALPHABETS, IV

Paragraph General remarks - - - -- -- -- __ _ __ __ _ _ __ __ _ _ _ _ 41

Denvmg the secondary alphabets, the pnmary components, and the key, given a cryptogram WIth Its plain text -- -___ 42

Denvmg the secondary alphabets, the primary components, and the keywords for messages, given two or more cryptograms m different keys and suspected to eontam Identical plain text ._____ 43

The case of repeatmg-key systems_ _ __ _ __ _ _ .____ 44

The case of Identical messages enciphered by keywords of different lengths __ _ • 45

Concluding remarks 46

41 General remarks -The preceding three sections have been devoted to an elucidation

of the general principles and procedure in the solution of typical cases of repeatmg-key Ciphers

Tills secnon will be devoted to a considerataon of the vananons m cryptanalytic procedure ansmg

from special Circumstances It may be well to add that by the designanon "special Circumstances" It IS not meant to imply that the latter are necessarily unu8'Ual circumstances The

8tudent shouldalwaysbeon thealert to 8e~ze upon any opportUnthe8 thatmay appeartn wh'tch hemay

apply the methods to be descMbed In practical work such opportumties are by no means rare and

are seldom overlooked by competent cryptanalysts

42 Denvmg the secondary alphabets, the pnmary components, and the key, given a

cryptogram Wlth Its plam text -a It may happen that a cryptogram and Its equivalent plain

text are at hand, as the result of capture, pilferage, compromise, etc 'I'his, as a general rule,

affords a very easy attack upon the whole system

b Takmg first the case where the plain component IS the normal alphabet, the CIpher component It mixed sequence, the first thmg to do IS to write out the CIpher text With ItS letter-forletter decipherment From this, by a slight modification of the pnnciples of "factoring", one dIScovers the length (\f the key It IS obVIOUS that when a word of three or four letters IS enciphered

by the same Cipher text, the interval between the two occurrences IS almost certainly a multiple

of the length of the key By noting a few recurrences of plain text and CIpher letters, one can

quickly determine the length of the key (assuming of course that the message IS long enough to

afford sufficient data) Having determmed the length of the key, the message IS rewntten accordmg to Its periods, wrth the plam text hkewise m penods under the CIpher letters From this

arrangement one can now reconstruct complete or partial secondary alphabets If the secondary

alphabets are complete, they willshow direct symmetry of pOSItIOn, rl they are but fragmentary

m several alphabets, then the pnmary component can be reconstructed by the apphcation of the

pnneiples of direct symmetry of posraon

c Ifthe plam component IS a mixed sequence, and the CIpher component the normal (direct or

reversed sequence), the secondary alphabets will show no direct svmmetry unless they are arranged ill the form of deciphering alphabets (that IS, Ao Z, above the zero line, With their

equivalents below) The student should be on the lookout for such cases

d (1) If the plain and CIpher primary components are identical mixed sequences proceeding

m the same direction, the secondary alphabets willshow indirect symmetry of POSItIon, and they

cenbe used for the speedy reconstruction of the pnmary components (Paragraph 31a to 0)

(84)

85

(2) If the plam and the cipher pnmary components are Identical mixed sequences proceeding

m opposite directions, the secondary alphabets will be completely reciprocal secondary alphabets

and the pnmary component may be reconstructed by applying the principles outlined m paragraph 31p

(3) If the plain and the cipher pnmary components are different mixed sequences, the

secondary alphabets will show indirect symmetry of positaon and the primary components may

be reconstructed by applying the pnnciples outlmed m paragraph 31q

e In all the foregoing cases, after the primary components have been reconstructed, the

keys can be readily recovered

43. DenVlng the secondary alphabets, the pnmary components, and the keywords for

messages, given two or more cryptograms m dl1ferent keys and suspected to contam Identical

plam text-a The simplest case of this kmd IS that mvolvmg two mono alphabetic substitution

Ciphers with mixed alphabets derived from the same pair of shdmg components An understandmg of this case IS necessary to that of the case mvolvmg repeating-key Ciphers

b (1) Amessage 18 transmitted from stataon Ato stationB B then sends Asome operatmg

signals which indicate that B cannot decipher the message, and soon thereafter A sends a second

message, Identical m length with the first 'I'lus leads to the Suspicion that the plam text of both

messagee 18 the same The mtercepted messages are superimposed Thus

f I v I;; t UA1)/2. 0"S#' u STr t. / A,P" (I" t 'I'h Pi . "", '1 ~"',.) -r A -.,.A , '" .. w;" r I r C

1 NXGRV MPUOF ZQVCP VWERX QDZVX WXZQE TBDSP VVXJK RFZWH ZUWLU IYVZQ FXOAR

2 EMLHJ FGVUB PRJNG JKWHM RAPJM KMPRW ZTAXG JJMCD HBPKY PVKIV QOJPR BMUSH

(2) Inraanng a cham of CIpher-text equivalents from message 1 to message 2, the followmg

complete sequence IS obtained

123 t I 6 7 8 9 ro II U U U n H n u

NEWKDASXMFBTZPGLIQRHYOUVJC

(3) Expenmentation along already-mdioated lmes soon discloses the fact that the foregoing

component IS an equivalent pnmary component of the ongmal primary based upon the keyword

QUESTIONABLY, decimated on the 21st interval Let the student decipher the cryptogram

(4) The foregoing example IS somewhat amficaal m that the plam text was consciously

selected WIth a VIew to making It contam every letter of the alphabet The purpose in doing

tlus was to permit the construction of a complete cham of equivalents from only two short

messages, m order to give a simple illustration of the pnnciplesinvolved Ifthe plain-text message

does not contain every letter of the alphabet, then only partial chams of equivalents can be constructed These may be united, If Circumstances will permit, by recourse to the vanous pnnciples elucidated m paragraph 31

(5) The student should carefully study the foregomg example ill order to obtam a thorough

comprehension of the reason why It was possible to reconstruct the pnmary component from the

two CIpher messages without having any plam text to begin with at all Since the plain text of

both messages 18 the same, the relative displacement of the pnmary components m the case of

message 1 dIffers from the relative displacement of the same pnmary components in the case of

message 2 by a jiud mterval Therefore, the distance between N and E (the first letters of the

two messages), on the pnmary component, regardless of what plam-text letter these two

Cipher letters represent, 18 the same as the distance between E and W(the 18th letters), Wand K

(the 17th letters), and so on Thus, this fixed interval permits of estabhshmg a complete cham

of letters separated by constant mtervals and tins cham becomes an equivalent pnmary com·

ponent.
86

~~~«~~~~~roM~MMMMMM~OO

1 BDJEZALVIDTROQSUHAFK

2 LCBLQNETOCVSNZRBJNOQ

~~~~~z~~~~n~~M~OO~~~~

1 DYSABPCQTUNGKFAZEFIZ

2 FLRHLMTAIQZWMDQNSDWN

1 2 3 4

B P C Q

L M T A

o Q S U

NZRB

0 A B C DIE F G H I J K L M N 0 P Q R S T U V W X Y Z

- - - - - - - - - - - - - - - - - - - - - - - - - - 1 L F S J 0 M Y N I Z C Q

- - - - - - - - - - - - - - - - - - - - - - - - - - -

2 N C D G B M Z Q L

- - - - - - - - - - - - I--- - - - - - - - - - - - - - 3 Q U T 0 W B E Z C R V F S

- - - - - - - - - - - - - - - - - - - - - - - - - - -

4 H L W Q A S B T N

87

(2) By the usual methods, construct the pnmary or an equivalent pnmary component

Takmg hnes 0 and 1, the followmg sequences are noted

BL, DF, ES, HJ, IO, KM, LY, ON, TI, XZ, YC, ZQ,

which, when muted by means of common letters and study of other sequences, YIeld the complete

ongmal pnmary component based upon the keyword QUESTIONABLY

QUESTIONABLYCDFGHJKMPRVWXZ

(3) The fact that the pair of hnes With which the process was commenced YIeld the original

pnmary sequence IS purely aecidental, It might have Just as well YIelded an equivalent pnmary

sequence,

(3) The factor 4IS the only one common to every one of these mtervals and It may be taken

as beyond question that the length of the penod IS 4

d Let the messages now be supenmposed according to their penods

123412341234123412341234

1 YHYE XUBU KAPV LLTA BUVV DYSA

2 CGSL ZQUB MNCT YBVH LQFT FLRH

1 TUNG KFAZ EFIZ BDJE ZALV IDTR

2 I Q Z W M D Q N S D WN L C B L Q NET 0 C V S

1 H A F K

2 J N 0 Q

e (1) Now distnbute the supenmposed letters into a reconstruction skeleton of "secondary

alphabets "

Thus

Repetition Interval Factors RepetitIOn Interval Factors

1st EL to 2d EL.. ----- 40 2,4,5,8,10 1st TVto 2d TV_ -- 36 2, 3, 4, 6, 9, 12

1st UQ to 2d UQ _ --- - 12 2,3,4,6,12 1st AH to 2d AM. ____ 8 2,4,8

2d UQ to 3d UQ 12 2,3,4,6,12 1st BL to 2d BL -- 8 2,4,8

1st UB to 2d UB 48 2, 3, 4, 6, 8, 12 2d BL to 3d BL ---- 16 2,4,8

1st KM to 2d KM_ . 24 2, 3, 4, 6, 8, 12 1st SR to 2d SR. -- 32 2,4,8

1st AN to 2d AN _ -- 36 2,3,4,6,9, 12 1st FD to 2d FD. -- 4 2, 4

2d AN to 3d AN - 12 2,3,4,6,12 1st ZN to 2d ZN - 4 2,4

1st VT to 2d VT .. ----_.. 8 2,4,8 1st DC to 2d DC.. _ 8 2,4,8

2d VT to 3d VT__ - 28 2,4,7

and factors applicable to cases of idenucal supenmposed pairs IS as follows (factors above 12

are omitted)

F L R H L

NET 0 C

D Y S A B

A L V I D

ABU V V

B D J E Z

H L Q F T

L C B L Q

MESSAGE 1

Y H Y E X UBUKA P V L L T

P C Q T U N G K F A Z E F I Z

T R 0 Q S UHAFK

MESSAGE 2

C G S L Z QUBMN C T Y B V

M T A I Q Z W M D Q N S D W N

V S N Z R B J N 0 Q

1 2 a 4 6 6 7 8 9 W " U U M M U

1 YHYEXUBUKAPVLLTABUVV

2 CGSLZQUBMNCTYBVHLQFT

(2) The first step IS to try to determine the length of the penod The usual method of

factonng cannot be employed because there are no long repetitions and not enough repetanons

even of digraphs to grve any eonvmcmg mdicataons However, a subterfuge Will be employed,

based upon the theory of faetonng

c (1) Let the two messages be supenmposed

44 The case of repeatmg-key systems -a WIth the foregoing basic pnncrples m mmd

the student ISready to note the procedure m the case of two repeatmg-key CIphers having identical

plain texts FIrst, the case m which both messages have keywords of identical length but different

compositions WIll be studied

b (1) GIven the followmg two cryptograms suspected to eontam the same plain text

4 «

E E

(2) Now let a search be made of cases of idenncal supenmpositron For example, Land L

6 18

U U U

are separated by 40 letters, Q, Q, and Q are separated by 12 letters Let these mtervals between

rdentacal supenmposrnons be factored, Just as though they were ordinary repennons That

factor which ISthe most frequent should correspond With the length of the penod for the following

reason If the period IS the same and the plam text IS the same m both messages, then the condinon of identity of supenmposiuon can only be the result of idennty of encipherments by

identical CIpher alphabets ThIS ISonly another way of saymg that the same relative posraon in

the keymg cycle has been reached m both cases of identity Therefore, the distance between

idenncal supenmposrnons must be either equal to or else a multiple of the length of the penod

Hence, factonng the mtervals must YIeld the length of the period The complete hst of mtervals
88

f (1) Having the pnmary component, the soluuon of the messagesIS now a relatively simple

matter An apphcanon of the method elucidated m paragraph 37 IS made, mvolvmg the completion of the plain-component sequence for each alphabet and selecting those generatnces which

contain the best assortments of high-frequency letters Thus, using Message 1

J'mBT ALPJI.LBBT SBCOND ALPHABBT TmRD ALPHABBT FOURTH ALPHABBT

YXKLB HUALU YBPTV EUVAV

C Z MY L J E BYE C L R I W SEW B W

D Q P C Y K S L C S DYVOX T S X L X

FUR D C MTYDT F C WN Z I T Z Y Z

G E V F D PIC F I GDXAQ o I Q C Q

H S WG F ROD G 0 HFZBU NOUDU

JTXHG VNFHN JGQLE *A N E F E

K I Z J H WAGJA KHUYS BAS G S

M0 Q K J XBHKB MJ E C T LBTHT

PNUMK ZLJML P K S D I Y L I J I

RAEPM QYKPY RMTFO C Y 0 K 0

V B S R P UCMRC V PIG N DCNMN

WLTVR EDPVD WROHA FDAPA

X Y I WV SFRWF XVNJB GFBRB

Z COX W TGVXG ZWAKL HGLVL

QDNZX IHWZH QXBMY JHYWY

UFAQZ OJXQJ U Z L P C K J C X C

E G B U Q NKZUK E Q Y R D MKDZD

S H LEU AMQEM S U C V F PM F Q F

T J Y S E B PUS P TEDWG RPGUG

I K C T S *L RET R I S F X H VRHEH

o MD I T Y V S I V o T G Z J WV J S J

N P F 0 I C WTOW NIH Q K XWKTK

*A R G N 0 DXINX A 0 J U M ZXMIM

BVHAN F Z 0 A Z BNKEP Q Z POP

L WJ B A GQNBQ *L A MS R UQRNR

FJGUBB 48

(2) The selected generatnces (those marked by astensks In Fig 48) are assembled m

columnar manner

ALLA

RRAN

GEME

NT S F

ORRE

FJo1lU 4lI

89

(3) The key letters are sought and give the keyword SOUP The plain text for the second

message IS now known, and by reference to the cipher text and the pnmary components, the

keyword for tills message IS found to be TIME The complete texts are as follows

SOUP TIME

ALLA ALLA

YHYE CGSL

RRAN RRAN

XUBU ZQUB

GEME GEME

KAPV MNCT

NTSF NTSF

LLTA YBVH

ORRE ORRE

BUVV LQFT

LIEF LIEF

DYSA FLRH

OFYO OFYO

BPCQ LMTA

UROR UROR

TUNG IQZW

GANI GANI

KFAZ MDQN

ZATI ZATI

EFIZ SDWN

ONHA ONHA

BDJE LCBL

VEBE VEBE

ZALV QNET

ENSU ENSU

IDTR OCVS

SPEN SPEN

OQSU NZRB

DEDX DEDX

HAFK JNOQ

FJO'llBli IiO

46 The case of IdentIcal messages enCiphered by keywords of different lengths -a In the

foregoing case the keywords for the two messages, although dUferent, were identacal In length

When tills 18 not true and the keywords are of drlferent lengths, the procedure need be only

shghtly modified
b Given the following two cryptograms suspected of contammg the same plain-text enCiphered by the same pnmary components but With different keywords of dIfferent lengths, solve

the messages

MESSAGE No 1

V MY Z G E A U N T P K FAY J I Z M B UMYKB V F I V V

S E 0 A F SKXKR Y W C A C Z 0 R D 0 Z R D E F B L K F E

S M K S F A F E K V Q U R C M Y Z V 0 X V A B T A YYUOA

Y T D K F E N W N T DB Q K U L A J L Z IOU M A BOA F S

K X Q P U Y M J P W Q T D B T o 5 I Y S M I Y K U R 0 G M W

C T M Z Z VMVAJ

MESSAGE No 2

Z GAM W 10M 0 A COD H A C L R L P M 0 Q 0 J E M 0 Q U

D H X B Y UQMGA U V G L Q DB S P U o A B I R PWXYM

o G G F T M R H V F G W K N I V A U P F A B R V I L A Q E M

Z D J X Y ME D D Y B 0 S V M P N L G X X D Y D 0 P X B Y U

QMNKY FLU Y Y G V P V R D N C Z E K J Q 0 R WJXRV

G D K D S X C E E C

A B C D E F G H I J K L M N 0 P Q R S T U V W X Y Z - - - - - - - - - II-I I J P D Q G C E K 0 R Z - - - - - - - - - - - - - - - - - - - - 1-

2-2 H V N G U W E D M L X - - - - - - - - - - - - - - - - - - - - - - 1-

3-3 E M X G I D J N R A 0 - - - - --- - - - - - - - - - - - - - - - - - - -

4-4 X 0 C D K A F Y Q V M - - - - - - - - --- - - - - - - - - - - - - - - - 1-5 B T W L R E N Y Q U A - - - - - - - - - - - - - - - - - -

2-6 M 0 I C D U V F R - - - - - - - - - - - - - - - - - - - - - - - 3-1 0 G R L P S D Z - - - - - - - - - - - - - - - - - - - - -

4-2 L P H U V E D M F - - - - - - - - - - - --- - - - - - - - - - - 1-3 Q J V W K 0 X Y M A - - - - - - - - - - - --- - - - - - - - 2-4 B J X P 0 A F Y D - - - - - - - - - - - - - - - - - - - - - - IS 3-5 N R Y B C G Q 1-

4-6 M L 0 S U V W X

FIGUU61

91

e There are more than sufficient data here to parmrt of the reconstrucnon of a complete

equrvalent pnmary component, for example, the following

1 2 3 4 5 6 7 8 9 ro n n M " w

ITKNPZHMWBQEULFCSJAXRGDVOY

f The subsequent steps in the actual decipherment of the text of either of the two messages

are of considerable mtetest Thus far the cryptanalyst has only the Cipher component of the

pnmary sliding components The plam component may be identical With the Cipher component and may progress in the same direction, or in the reverse direcuon, or, the two components may be different If different, the plam component may be the normal sequence,

direct or reversed Tests must be made to ascertain which of these vanous possibilitaes 18 true

g (1) It will first be assumed that the pnmary plain component IS the normal direct

sequence Applymg the procedure outhned m Par 23 to the message With the shorter key

(Message No 1, to give the most data per secondary alphabet), an attempt IS made to solve

the message It IS unnecessary here to go further mto detaJ1 m tlus procedure, suffice It to

indicate that the attempt IS unsuccessful and It follows that the plain component IS not the

normal direct sequence A normal reversed sequence IS then assumed for the plain component

and the proper procedure applied AgaIn the attempt IS found useless Next, It IS assumed

that the plain component IS identacal With the Cipher component, and the procedure outlmed In

Par 37 18 tried 'I'his also 18 unsuccessful Another attempt, assummg the plam component

runs m the reverse direction, 18 likewise unsuccessful There remains one last hypothesis, VIZ,

that the two pnmary components are different mixed sequences

d A reconstruction skeleton of "secondary alphabets" IS now made by distnbutmg the

letters In respective lmes corresponding to the 12 different supenmposed paars of numbers For

example, all pairs corresponding to the supenmposinon of posinon 1 of Message 1 With position 1

of Message 2 are distributed In lines 0 and 1 of the skeleton Thus, the very first supenmposed

paar 18 \ ;, the letter Z IS inserted In lme 1 under the letter V The next g parr 18 the 13th supernnposmon, With{~, the letter DIS Inserted m hne 1 under the letter F, and so on The skeleton

IS then as follows

90

No 1

No 2

No 1

No 2

No 1

No 2

No 1

No 2

No 1

No 2

No 1

No 2

No 1

No 2

c The messages are long enough to show a few short repetltlons wluch pemut factoring

The latter discloses that Message 1 has a period of 4 and Message 2, a penod of 6 letters The

messages are superimposed,Withnumbers marking the position of each letter In the correspondmg

penod, as shown below

123 4 1 234 123 4 1 234 1 234 1 234

VMYZGEAUNT PKFAYJ I 2MBUMYK

ZGAMWIOMOACODHACLRLPMOQO

1 234 6 6 1 2 3 4 6 6 1 2 S 4 6 6 1 2 S 4 5 6

1 234 1 2 S 4 1 234 1 234 1 234 1 234

BVFIVVSEOAFSKXKRYWCACZOR

JEMOQUDHXBYUQMGAUVGLQDBS

123 4 561 2 345 6 1 2 3 4 6 6 1 2 3 4 6 6

1 234 1 234 1 2 3 4 1 234 1 234 1 234

DOZRDEFBLKFESMKSFAFEKVQU

PUOABIRPWXYMOGGFTMRHVFGW

1 234 661 234 561 234 561 234 5 6

1 234 1 234 1 234 1 234 1 234 1 234

RCMYZVOXVABTAYYUOAYTDKFE

KNIVAUPFABRVILAQEMZDJXYM

123 4 6 6 1 234 561 2 3 4 6 6 1 234 6 6

123 4 1 234 1 234 1 234 1 2 3 4 1 234

NWNTDBQKULAJLZIOUMABOAFS

EDDYBOSVMPNLGXXDYDOPXBYU

1 2 3 4 5 6 1 2 S 4 5 6 1 2 3 4 5 6 1 234 5 6

1 2 S 4 1 234 1 2 3 4 I 234 1 234 1 2 S 4

KXQPUYMJPWQTDBTOSIYSMIYK

QMNKYFLUYYGVPVRDNCZEKJQO

1 234 561 234 661 2 3 466 1 2 3 4 5 6

1 234 1 2 S 4 1 234 1 234

UROGMWCTMZZVMVAJ

RWJXRVGDKDSXCEEC

1 2 3 4 6 6 1 2 3 4 6 6 1 234
(2) Here IS Message No 1 transcribed ill penods of four letters Umhteral frequency

distnbutions for the four secondary alphabets are shown below ill FIg 52, labeled la, 2a, 3a,

and 4a These distnbutaons are based upon the normal sequence A to Z But since the reconstructed Cipher component IS at hand these distnbutaons can be rearranged according to the

sequence of the Cipher component, as shown ill distnbutaons labeled 16, 26, 36, and 46 ill FIg 52

The latter dutrtlnttwns may 6e com6~n6d 6y 8h'ift~ng dutrt6utwn8 26, S6, and 46 to proper 8uper-

."mp08u~on8 'UJ'l,th re8pect to 16 80 asto yuld a 8'tngle monoalphabmc d~Bt,."buhon for the enhreme88age

In other 'Words, the polyalphabetw me88age can be converted ~nto monoalphabetw term8, thUB very

consu1erably 8~mplif~ng the 8oluhon

HMPCBL RSW ODUGAFQKIYNETV

::::: :::::::

~-~~ -~ ~- ~~ K N P Z H MWB Q E U L F C 5 J A X R G D V 0 Y I T

-- ~~ ~~--~ ~~~-~~ PZHMWBQEULFCSJAXRGDVOYITKN

- i!1!

~§:- :::: §§~~

~~~~2:::._:::::~",,~~~_

Q E U L F C S J A X R G D V 0 Y

- ~~-~ ~~~-~ -~~~~~ ITKNPZHMWBQEULFCSJAXRGDVOY

- - ~~- ~-~~-~ ~~ 26 EULFCSJAXRGDVOYITKNPZHMWBQ

36

16

46

V D V T G I S W N S K 0 F M V L I R Z Z U D V 0 B UUDVU

ENEMY HAS C A P T U R E D H ILL ONE T W o 0 N E 0

F MOM U U K W I S Y V L F C R D S D L N S DIU Z L J U M

U R T R 0 o P S H A V E DUG I NAN D CAN H 0 L D FOR

S DIU F MUMKU W W R P Z G Z U D C VMMVA F V W 0 M

A N H 0 U R 0 R P 0 S SIB L Y LON G ERREQ U EST R

V V D J U M N V T V DOW 0 U K S L L R o R U D S Z 0 M U U

E I N F 0 R C E M E N T S T 0 PAD D I T ION A L T ROO

K W W I U F Z L P V W V D 0 Y R S C V U MeV 0 U BDJMV

P S S H 0 U L D B E SEN T V I AGE 0 R GET 0 WNFRE

LVMRN XMUSL

D E RIC K R 0 AD

- 1b-4b ~::::: :::: ::::

combined I T K N P Z H M W B

98

(6) The keywords for both messages can now be found, If desirable, by findmg the equivalent

of Ap ill each of the secondary alphabets of the ongmal polyalphabettc messages The keyword

for No 118 STAR, that for No 2 Ie; OCEANS

152018-38--7

l!IGllBB 53

(5) Having the plain text, the denvation of the CIpher component (an eqwvalent) IS an

easy matter It ISmerely necessary to base the reconstruction upon any of the secondary alphabets, SInce the plam tC'1l:t-Clpher relauonship IS now known directly, and the pnmary cipher

component 1'1 at hand The pnmo.ry plam component ISfound to be as follows

(4) Thc letters belonging to alphabets 2,3, and 4 of FIg 52 may now be transcnbed ill terms

of alphabet 1 That IS, the two E's of alphabet 2 become L's , the L of alphabet 2 becomes a K,

the C becomes a P, and so on Likewise, the two K's of alphabet 3 become I'c;, the N becomes

a T, and so on The entire message IS then 0. monoalphabet and can readily be solved It 1&as

follows

(3) Note ill FIg 53 how the four drstnbutaons are shifted for supenmpositaon and how the

eombmed distnbunon presents the eharactenaties of a typical monoalphabetae distnbutaon

~-~~-~ ~~ ~~- ~~ ITKNPZHMWBQEULFCSJAXRGDVOY

- ~-~ ~~~ - ~~- ~-~ ITKNPZHMWBQEULFCSJAXRGDVOY

3b

2b

~~~ -- ~~ %~--~ ~~, 4b I T K N P Z H MWB Q E U L F C S J A X R G D V 0 Y

FIGllU52

~~~ ~~§ - - ~~~~~ 2a ABC D E F G H I J K L M N 0 P Q R STU V W X Y Z

92

~-~ - ~-~~~ ~- ~~ 3a ABC D E F G H I J K L M N 0 P Q R STU V W X Y Z

~~-- -. ~~~~~ -~~~~- 1b I T K N P Z H M W B Q E U L F C S J A X R G D V 0 Y

--~~ ~- - ~~~~~~ -~ ~~ ~­ 1a ABC D E F G H I J K L M N 0 P Q R STU V W X Y Z

MESSAGE No 1

VMYZ VABT

GEAU AYYU

NTPK o A Y T

FAYJ DKFE

IZMB NWNT

UMYK DBQK

B V F I U L A J

V V S E L Z I 0

OAF S UMAB

KXKR OAF S

YWCA KXQP

C Z 0 R UYMJ

DOZR P W Q T

D E F B D B T 0

L K F E SlY S

SMKS M I Y K

FAFE U R 0 G

KVQU M W C T

RCMY M Z Z V

Z V 0 X MVAJ
l1lK--PlaIn Compo- II8IIt Mbed (26)

904-Mlzed Cipher Alphabets (Primary CompoDents are not both Normal) (16, 17)

11113-PlaID (,ompo- nent NormaJ (18-211)

loo~Reversed Stand

ard Cipher Alphabets

(lSI l~ 15)

ll02-ldentlcal Com

ponents (28-30)

llOIi-Penodlc (2)

om-Normal Cipher Alphabets (Primary CompoDents are both

Normal) (13-14)

4O~Poly llphabetlc SYlItemq (I, 2)

Analytical Key for Military CryptanalysIS, Part II *

(Numbers In parentheses reler to Paragraph Numbers In this text)

1--------1-------1 taken Ii06-Aperlodlc up later) (to be I

---...,...---

Itaken 7OD-Pr0grll881ve op later) (to be I ISystems 7l6-Bepeatulg (3) Key1

I

8111With IDterrelat 81~Wlth Jndepend

ed C pher Alphabets ent Cipher Alphabets

(6-12) (Sc)

12111-8equences pro- 12~8equences proceed In same direction ceed 10 opposite dIrec

(31-37 ~) tlOns (38, 40-46)

·For m:p\aDlltlon of the IIB8 of tillS chart see Par IiO of Mlllt&ry CryptanalysIS Part I

(95)

lI11-F1ulble PerlCl- 1I12~FIDd Periodic- dlclty (26) (to be taken Ity (26) up later)

1101-DllJerent Com

poneots (30)

1001-Dtreot 'ltand

erd Cipher AI'pliabets

(ldll-ll 144-1. 16)

94

(7) The student may, If he WIShes, try to find out whether the prunary components reconstructed above are the ongmal components or are equrvalent components, by ex&TDJDJDg all the

possible decimations of the two components for evidences of denvanon from keywords

h As already stated m Par 26l, there are certain statIStical and mathematical tests that

can be employed m the process of "matching" distributions to ascertam proper superimpositions

for monoalphebetierty In the case Just considered there were sufficient data m the distributaons

to permit the process to be applied successfully by eye, Without necessitating statastacal tests

'l. ThIS case IS an excellent illustration of the apphcanon of the process of convert'l.ng a

polyalphabet'l.c mpher ",nto monoalphabetw terms Because It IS a very valuable and Important

cryptanalytic "trick," the student should study It most carefully m order to gam a good understanding of the pnnciple upon which It IS based and Its sigmficance m cryptanalysis The

conversion m the case under discussion was possible because the sequence of letters formmg the

CIpher component had been reconstructed and was known, and therefore the umhteral distnbunons for the respective secondary Cipher alphabets could theoretically be shifted to correct

superrmpositaons for monoalphabencity It also happened that there were sufficient data m

the distnbutaons to give proper indicataons for their relative displacements Therefore, the

theoretical possibihty in this case became an actuahty Without these two necessary conditions

the supenmposmon and conversion cannot be aeeomphshed The student should always be

on the lookout for situations in which this IS possible

48 Concludmg remarks -a The observant student w111 have noted that a large part of

thIS text 18 devoted to the elucidanon and apphcation of a very few basic pnnciples These

pnnciples are, however, extremely Important and their proper usage III the hands of a skilled

cryptanalyst makes them practically mdispensable tools of IDs art The student should therefore

dnlllumself III the apphcation of these tools by having someone make up problem after problem

for hun to practice upon, until he acquires facility 1Il their use and feels competent to apply

them 1Il practice whenever the least opportunity presents Itself 'I'his w111 save him much time

and effort m the solution of bona fide messages

b Continuing the analytieal key introduced 1Il Military Cryptanalysis Part I, the outline

for the studies covered by Part II follows herewith
Components

'fABLE II

A H L U R G P S 0 V Y B X D E I M K Q T W Z C F J N - - - - - - T A E N K Z I L H 0 R U Q w X B r-! F D J M P S V Y C

- W A J G V E H D K N Q M S T X B Z F I L 0 R U Y 1£

G N R A X M V Y U B E H D J K 0 S .s. W Z C F I L p

II - .....-

J Q U D A P Y B X E H K G M N R V T Z C F I L 0 S -

~I B F 0 L A J M I P S V R X Y C G E K N .s T W Z D -!

L S W F C R A D Z G J M I 0 P T X V B E H K N Q u y

-I E H K N R I P T C Z 0 X A W D G J F L M Q u s 11 y B

M T X G D S B E A H K N J P Q u y w C F I L 0 R V Z

- - - - - - - - - - - - - - - - M Q Z W L U X T A D G C I J N R P V Y B E H K 0 I~

C J N W T I R U Q X A D Z F G K 0 14 S V Y B E H L P

- - - 1-

Z G K T Q F 0 R N U X A W- e D H L J P S V y B E I I~ -

D K 0 X U J S V R Y B E A G H L P N T W Z C F I 14 1..9.., - E I R 0 D M P L S V Y U A B F J H N Q T W Z C G -.!

w D H Q N C L 0 K R U X T Z A E I G M P S V Y B F J

I-- - - - --- - - - - - - - - - - - - --- - - - - S Z D M J Y H K G N Q T P V W A E C I L 0 R U X B F

- I--

0 V Z I F U D G C J M P L R S W A Y E H K N Q T X a - - - - - - - - - - - - - - - - - - - - - - - - - X B K H W F I E L 0 R N T U Y C A G J M P S v z IEK R V E B Q z C Y F I L H N 0 S W U A D G J M P T X

H 0 S B Y N W Z v C F I E K L P T R X A D G J M .s. U

I--

E L P Y V K T W S Z C F B H I M Q 0 U X A D G J N

- N D G [i B I M V S H Q T P W Z C y E F J L R U X A K

-

Y F J S P E N Q M T W Z V B C G K I 0 R U X A D H L

- X J... C G P M B K N J Q T W S Y Z D H F L 0 R U A E I

R Y C L I X G J F M P S 0 U v Z D B H K N Q T W A E

-I FlaTI - - - - - IN U Y H E T C L 0 K Q R V Z X D G J M P S W A

(1) ABC D E F G H I J K L M N 0 P Q R STU V W X Y Z

(2) F B P Y R C Q ZIG S E H T D J U M K V A L W N 0 X

91

PLAIN '1EXT

ABC D E F G H I J K L M N 0 P Q R STU V W X Y Z

A

B

C

D

E

F

G

H

I

J

K

L

~M

r"'l

~N

o

p

Q

R

S

T

U

V

W

x

y

z

1

I

I

L

APPENDIX 1

(96)

THE 12 TYPES OF CIPHER SQUARES

(See Paragraph 7)

TABLE I-B I

Components

(1) ABC D E F G H I J K L M N 0 P Q R STU V W X y Z

(2) F B P Y R C Q ZIG S E H T D J U M K V A L W N 0 X

Enclphenng equations Ok/2=81/1, 8P /I=8c /2 (8./1 IS A)

J ThIS table IS labeled "Table I-B" because It 18 the same as Table I-A on page 7, except that the honsontal

hnes of the latter have been shifted so 88 to begm the succeesive alphabets with the successive letters of the normal

sequence
TABI,E III

98

PLAIN TEXT

ABC D E F G H I J K L M N 0 P Q R STU V W X Y Z

U B F 0 L A J M I P S V R X Y C G E K N Q T W Z D H

- 1-

--------- Q T W S Y Z D H F L 0 R U X A E I V C G P M B K N J

--------- R U X T Z A E I G M P S V Y B F W D H Q N C L 0 K 1---------- S V Y U A B F J H N Q T W Z C G K X E I ROD M P L 1-

f---------- T W Z V B C G K I 0 R U X A D H L Y F J S PEN Q M --------- U X A W C D H L J P S V Y B E I M Z G K T Q FOR N --------- V Y .!. X D E I M K Q T W Z C F J N

A H L U R G P S 0 - 1---------- W Z C Y E F J N L R U X A D G K 0 B I M V S H Q T P - 1--------- X A D Z F G K 0 M S V Y B E H L C J N W T I R U Q -!- --------- D K 0 X U J S V R Y B E A G H L P N T W Z C F I M I~ r---------- Z C F B H I M Q 0 U X A D G J N R E L P Y V K T W S - ---------- A D G C I J N R P V Y B E H K 0 S F M Q Z W LUX T --------- B E H D J K 0 S Q W Z C F I L P T G N R A X M V Y U - - '---------

C F I E K L P T R X A D G J M U H 0 S B Y N W Z V

--------- D G J F L M Q U S Y B E H K N R V I PTe Z 0 X A W - --------- E H K G M N R V T Z C F I L 0 S

f J Q U D A P Y B X

--------- F I L H N 0 S W U A D G J M.!- T K R V E B Q Z C Y - ---------- G J M I 0 P T X V B E H K N Q U Y L S W FeR ADZ 1-

~--------H K N J P Q U Y W C F I L 0 R V Z

M T X G D S B E A - 1----------I L 0 K Q R V Z X D G J M P S W A

N U Y H ETC F B i---------- J M P L R S W A Y E H K N Q T X B o V Z I F U D G C 1-

--------- K N Q M S T X B Z F I L 0 R U Y C

P W A J G V E H D

--------- L 0 R N T U Y C A G J M P S V Z D

Q X B K H W FIE ~---------M P S 0 U V Z D B H K N Q T W A E

Rye L I X G J F 1-

r-f--------- N Q T P V W A E C I L 0 R U X B F

S Z D M J Y H K G 1--------- 0 R U Q W X B F D J M P S V Y C G

T A E N K Z I L H

PLAIN TEXT

ABC D E F G H I J K L M N 0 P Q R STU V W X Y Z

TABLE IV

Components

(1) ABC D E F G H I J K L M N 0 P Q R STU V W X Y Z

OOFBPYRCQZIGSEHTDJUMKVALWNOX

Enciphenng equations 0,,/,=8,/2, Op/2=8C/1 (8'/JIb F)

99

A

B

C

D

E

F

G

H

I

J

K

>oL

r;;l M

:=: N

o

P

Q

R

S

T

U

V

W

X

Y

Z

F B P Y R C Q Z I G ~I.!. H T D J U M K V A L W N 0 X

X F B P Y R C Q Z I G S E H T D J U M K V A L W N 0 - - - - - - - - - - - - - - - - - - - - - - - - - - 0 B P Y R C Q Z I G S E H T D J U M K V A L W N

N 0 X F B P Y R C Q Z I G S E H T D J U M K V A L W --- - - I-- - - - - - - W N 0 X F B P Y R C Q Z I G S E H T D J U M K V A L - - - - - - - - - - - - - - - - - - - - - - - - - - L W N 0 X F B P Y R C Q Z I G S E H T D J U M K V A - - - - - - - - - - - - - - - - - - - - - - - - - - A L W N 0 X F B P Y R C Q Z I G S E H T D J U M K V - - - - - - - - - - - - - --- - - - - - - - - - - - V .!- W N 0 X F B P Y R C Q Z I G S E H T D J U M K - - - K V A L W N 0 X F B P Y R C Q Z I G S E H T D J U M - - - - - - - - - - - - --- - M K V A L W N 0 X F B P Y R C Q Z I G S E H T D J U i-- - - f--

M K V A L W N 0 X F B P Y R C Q Z I G S E H T D J - J U M K V A L W N 0 X F B P y R C Q Z I G S E H T D - - - - - - - - - - - - J U M K V A L W N 0 X F B P Y R C Q Z I G S E H ..'!'-

1- D J U M K V A L W N 0 X F B P y R C Q Z I G S E H

H T D J U M K V A L W N 0 X F B P Y R C Q Z I G S E - - - - - - -

!- H T D J U M K V A L W N 0 X F B P y R C Q Z I G S - - S E H T D J U M K V A L W N 0 X F B P Y R C Q Z I G - S E H T D J U M K V A L W N 0 X F B P Y R C Q Z I

- I G S E H T D J U M K V A L W N 0 X F B P Y R C Q Z

I G S E H T D J U M K V A L W N 0 X F B P y R C Q

J- Z I G S E H T D J U M K V A L W N 0 X F B P y R C

Q Z I G S E H T D J U H K V A L W N 0 X F B P y R '--

C Q Z I G S E H T D J U M K V A L W N 0 X F B P Y

-!.. R C ..9.- Z I G S E H T D J U M K V A L W N 0 X F B P

P Y R C Q Z I G S E H T D J U M K V A L W N 0 X F B - - - - - - - - - - - - - --- - - VTAIL- - - - B P Y R C Q Z I G S E H T D J U M K W N 0 X F

Components

(1) ABC D E F G H I J K L M N 0 P Q R STU V W X Y Z

(2) F B P Y R C Q ZIG S E H T D J U M K V A L W N 0 X

Enciphenng equanons 8,,/1=8,/2, 8p/l=e~/2 (9'/2 IS F)

A

B

C

D

E

F

G

H

I

J

K

L

>oM

~N

o

P

Q

R

S

T

U

V

W

X

Y

Z
100

TABLE V

Components

(1) ABC D E F G H I J K L M N 0 P Q R STU V W X y Z

~) F B P Y R C Q ZIG S E H T 0 J U MK V A L WN 0 X

Enciphenng equations 81</2=8 p/1 , 8 1/1= 8 0/2 (8 1/ 1 IS A)

PLAIN TEXT

ABC 0 E F G H I J K L M N 0 P Q R STU V W X Y Z

101

TABLE VI

Components

(1) ABC D E F G H I J K L M N 0 P Q R STU V W X Y Z

~) F B P Y R C Q ZIG S E H T 0 J U MK V A L WN 0 X

Enciphenng equations 81<12=80/ 1 , 81/1=Op/2 (8 111 IS A)

A T P G J U L I M F C Z 0 X W S 0 Q K .!!-I~ B Y V R N

I--

Ji- A W N Q B S P T M J G K E 0 Z V X R 0 L I F C y U

- - - - - - - - - - - - - - - - - -

L E A R U F W T X Q N K 0 I H 0 Z B V S P M J G C y

U N J A 0 0 F C G Z W T X R Q M I K E B Y V S P L H ,..-- - - - - - - - - - - - - - - - - - - - - - - - - R K G X A L C Z D W T Q U 0 N J F H B Y V S P M I E

G Z V M P A R 0 S L I F J 0 C Y U w Q N K H E B X T ,..-

P I E V Y J A X B U R 0 5 M L H D F Z W T Q N K G C ,..-- - - - - - - - - - - - - - - - - - - - - - - - - L H Y B M 0 A E X U R V P 0 K G I C Z W T Q N J F

0 H D U X I Z W A T Q N R L K G C E Y V S P M J F B

- - - - - - - - - - - - - - - - - - - -

!-, 0 K B E P G D H A X U y S R N J L F C Z W T M I

Y R N E H S J G K 0 A X B V U Q M 0 I F C Z W T P L

I--

B U Q H K V M J N G 0 A E Y X T P R L I F C Z W S 0

f-- - - - - - - - - - - - - - - -- X Q M 0 G R I F J C Z W A U T P L N H E B Y V S 0 K

'--

D W 5 J M X 0 L P I F C G A Z V R T N K H E B Y U ,~ f--

E X T K N Y P M Q J G 0 H B A W S U 0 L I F C Z V R I--- - - - - - - - - - - - - - - - - - - - - - - - I B X 0 R C T Q u N K H L F E A W Y S P M J G D Z V

- - - - - - - - - - - - - - - - - - - - - - - - - M F B 5 V G X U Y R 0 L P J I E A C W T Q N K H D Z

- - - - - - - - - - - - - - - - - - - - - - - - - K D Z Q T E V S W P M J N H G C Y A U R 0 L I F B X

I--

S. J F W Z K B Y C V S P T N M I E G A X U R 0 L H D

- T M I Z C N E B F Y V S w Q P L H J D A X U R 0 K G - - - - - - - - - - - - - - - P L C F Q H E I B y V Z T S 0 K M G 0 A X U R N J

I--

Z S 0 F I T K H L E B Y C W V R N P J G D A X u Q M ,-- - - - - - - - - - C V R I L W N K 0 H E W F U T U Q S M J G D V X T P

I-- - - - - - - - - F Y U L 0 Z Q N R K H A I C B X T V P M J G D A W 5

I- - - - - - - - - - - - - J C Y P S D U R V 0 L I M G F B X Z T Q N K H E A W - - - - - - - - - - N G C T W H Y V Z S P M Q K J F B D X U R 0 L I E A

A

B

C

o

E

F

G

H

I

J

K

>oL

f:I:lM

~N

o

P

Q

R

S

T

U

V

W

X

Y

Z

A V K M U J D T H E S G I Z Q C R Y P B F X 0 N W L

1-----

B F X 0 N W L A V K M U J

1-

---- 0 T H E S G I Z Q C R Y I~ CRY P B F X 0 N W L A V K M U J 0 T H E S G I Z ---- .s

D THE S G I Z Q C R Y P B F X 0 N W L A V K M U J ----

E S G I Z Q c R y P -

---- B F X 0 N W L A V K M U J D T -!! J.--!.~!!- w L A V K M U J 0 T H E S G I Z Q C R Y P

G I Z Q C R Y P B F X 0 N W L A V K M U J D T H E S

1-----

H E S G I Z Q c R y P B F X 0 N -

'----- - W L A V K M U J D -!.

I Z Q C R Y P B F X 0 N W L A V K M U J D T H E S G ---- J D T H E S G I Z Q C R Y P F -

---- B X 0 N W L A V K M U

K M U J D T H E S G I Z Q C R Y -

---- P B F X 0 N W L A J....

L A V K M U J D T H E S G I Z Q C R Y P B F X 0 N W

1-----

M U J 0 T H E S G I Z Q C R Y P '-

~---

B F X 0 N W L A V .!.

N WL A V K M U J 0 T H E S G I Z Q C R Y P B F X 1-----

o N W L A V K M U J D T H E S G I Z Q C R Y P B F X '-----

P B F X 0 N W L A V K M U J 0 T H E S G I Z C R Y ----

~I----

Q CRY P B F X 0 N W L A V K M U J D T H E S G I .~ R Y P B F X 0 N W L A V K M U J 0 T H E S G I Z Q C

i-----

S G I Z Q C R Y P B F X 0 N W L A V K M U J D T H 1----- '-!

THE 5 G I Z Q C R Y P B F X 0 N W L A V K M U J D ----

U J D T H E 5 G I Z Q C R Y -

---- P B F X 0 N W L A V K M

V K M U J D T H E 5 G I Z Q C R Y P B F X 0 N W L A ----

W L A V K M U J D T H E 5 G I Z Q C R Y P B F X 0 N

1-----

X 0 N W L A V K M U J 0 T H E S

'-

1----- G I Z Q C R Y P B l'..

Y P B F X 0 N W L A V K M U J 0 T H E 5 G I Z Q C R ---- Z Q C R Y P B F X 0 N W L A

IV K M U J 0 T H E S G I

PLAIN TEXT

ABC D E F G H I J K L M N 0 P Q R STU V W X y Z

A

B

C

D

E

F

G

H

I

J

K

L

~M

~N

Q

p

Q

R

S

T

U

V

W

X

y

Z
A

B

C

D

E

F

G

H

I

J

K

>oL

~M

N

o

P

Q

R

S

T

U

V

W

X

Y

Z

102

TABLE VII

Components

(1)- ABC D E F G H I J K L M N 0 P Q R STU V W X Y Z

(2)- F B P Y R C Q ZIG S E H T D J U M K V A L W N 0 X

Em iphenng equations Okf2=Op/II 8112= 0 0 / 1 (8 1/ 2 Ib F)

PLAIN TEXT

ABC D E F G H I J K L M N 0 P Q R STU V W X Y Z

G H I J K L M N 0 P Q R S T U V W X Y Z A B C D E F I-- - - - - - - - - - - - - - - - - - A B C D E F G H I J K L M N 0 P Q R S T U V W X Y,- V W X Y Z A B C D E F G H I J K L M N 0 p Q R S T U I--- --- - - - - - - - - - - - - - - - - - - - - - N 0 P Q R S T U V W X Y Z A B C D E F G H I J K L

Q R S T U V W X Y Z A B C D E F G H I J K L M N 0

A B C D E F G H I J K L M N 0 P Q R S T U V W X Y Z - - - - - - - - - - - - - - - - - - - - - - - - - - S T U V W X Y Z A B C D E F G H I J K L M N 0 p Q

P Q R S T U V W X Y Z A B C D E F G H I J K L M N

T U V W X Y Z A B C D E F G H I J K L M N 0 P Q R

L M N 0 P Q R S T U V W X Y Z A B C D E F G H I J K I--

T U V W X Y Z A B C D E F G H J. J K L M N 0 P Q R S

F G H I J K L M N 0 p Q R S T U V W X Y Z A B C D E

J K L M N 0 p Q R S T U V W X Y Z A B C D E F G H I - - E F G H I J K L M N 0 P Q R S T U V W X Y Z A B i~ C D E F G H I J K L M N 0 p Q R S T U V W X Y Z A B - - - - - - - - - - - - - - - - - - - - - - - - - Z A B C D E F G H I J K L M N 0 p Q R S T U V W X

U V W X Y Z A B C D E F G H I J K L M N 0 P Q R S T I-- - - - - - - - - - - - - - - - - - - - - - - - - - X Y Z A B C D E F G H I J K L M N 0 P Q R S T U V

~~ R S T U V W X Y Z A B C D E F G H I J K L M N 0 P

N 0 P Q R S T U V W X Y Z A B C D E F G H I J K L M

K L M N 0 P Q R S T U V W X Y Z A B C D E F G H I J - - - - - - - - - - - - - - - --- - - - - H I J K L M N 0 P Q R S T U V W X Y Z A B C D E F G I-- - - - - - - - - - - - - - - - F G H I J K L M N 0 p Q R S T U V W X Y Z A B C D

B C D E F G H I J K L M N 0 P Q R S T U V W X Y Z A

IX Y Z A B C D E F G H I J K L M N 0 P Q R S T U V W --- --- - - - - - - - --- - - - - - - - - - - - - T U V W X Y Z A B C D E F G H I J K L M N 0 P Q R S

A

B

C

D

E

F

G

H

I

J

K

>oL

r.:l

~M

N

o

P

Q

R

S

T

U

V

W

X

Y

Z

103

TABLE VIII

Componets

(1) ABC D E F G H I J K L M N 0 P Q R STU V W X Y Z

~) F B P Y R C Q ZIG S E H T D J U M K V A L W N 0 X

Enciphenng equations Okf2-=-OO/l' 9 1/2=9p/l (91/ 2 Ie;; F)

PLAIN TEXT

ABC D E F G H I J K L M N 0 P Q R STU V W X Y Z

U V W X Y Z A B C D E F G H I J K L M N 0 P Q R S T

I-- - - - - --- - - - - - - - - - - - - B C D E F G H I J K L M N 0 P Q R S T U V W X Y Z A

I--

F G H I J K L M N 0 P Q R S T U V W X Y Z A B C D E

I--

0 P Q R S T U V W X Y Z A B C D E F G H I J K L M N

I--

S T U V W X Y Z A B C D E F G H I J I.!. L M N 0 P Q R

I--

A B C D E F G H I J K L M N 0 P Q R S T U V W X Y Z

I--

K L M N 0 P Q R S T U V W X Y Z A B C D E F G H I

M N 0 P Q R S T U V W X Y Z A B C D E F G H I J K L

I--

I J K L M N 0 P Q R S T U V W X Y Z A B C D E F G H I--

P Q R S T U V W X Y Z A B C D E F G H I J K L M N 0

I--

S T U V W X Y Z A B C D E F G H I J K L M N 0 p Q R

I--

V W X Y Z A B C D E F G H I J K L M N 0 P Q R S T U

R S T U V W X Y Z A B C D E F G H I J K L M N 0 P Q I--

Y Z A B C D E F G H I J K L M N 0 p Q R S T U V W

Y Z A B C D E F G H I J K L M N 0 P Q R S T U V W X

C D E F G H I J K L M N 0 P Q R S T U V W X Y Z A B I--

G H I J K L M N 0 P Q R S T U V W X Y Z A B C D E F ~-

E F G H I J K L M N 0 P Q R S T U V W X Y Z A B C D 1--

K L M N 0 P Q R S T U V W X Y Z A B C D E F G H I J

N 0 P Q R S T U V W X Y Z A B C D E F G H I J K L M

Q R S T U V W X Y Z A B C D E F G H I J K L M N 0 P

IT U V W X Y Z A B C D E F G H I J K L M N 0 P Q R

X Y Z A B C D E F G H I J K L M N 0 P Q R S T U i:i

Z A B C D E F G H I J K L M N 0 p Q R S T U V W X

I-- I-!

D E F G H I J K L M N o P Q R S T U V W X Y Z A B I~ H I J K L M N 0 P Q R STT U V W X Y Z A B C D E F G
TABLE IX 2

104:

A B C D E F G H I J K L M N 0 P Q R S T U V W X Y Z - - - - - - - - - - - 1-

L P Q J H B S T G U V W K 0 X Y Z C E D M A N F R I~ I--

W Y Z U T P E D S M A N V X F R I Q H J K L 0 B C G

- - - - - - - - -

N R I M D Y H J E K L 0 A F B C G Z T U V W X P Q S

- - -

0 C G K J R T U H V W X L B P Q S I D M A N F Y Z E

I-- - - - - - - - - - - - -

-!. Q S V U C D M T A N F W P Y Z E G J K L 0 B R I H

F Z E A M Q J K D L 0 B N Y R I H S U V W X P C G T

r-- - - - - - - - - - - - - - - - - - - - - -

B I H L K Z U V J W X P 0 R C G T E M A N F Y Q S D

--- - - - -

P G T W V I M A U N F Y X C Q S D H K L 0 B R Z E J

I--- - - - - - - - - - - - - - - - - -

Y S D N A G K L M 0 B R F Q Z E J T V W X P C I H U

f-- - - - - - - - - - -

R E J 0 L S V W K X P C B Z I H U D A N F Y Q G T I-!

---C H U X W E A N V F Y Q P I G T M J L 0 B R Z S D K

f-- - - - - - - - - - - - - - - - - -

S T M F N H L 0 A B R Z Y G S D K U W X C I E J V - Z D K B 0 T W X L P C I R S E J V M N F Y Q G H U A

l- I-- I--

I J V P X D N F W Y Q G C E H U A K 0 B R Z S T MI~

---G U A Y F J 0 B N R Z S Q H T M L V X P C I E D K Il!-

lS M L R B U X P 0 C I E Z T D K W A F Y Q G H J V N

--- I-- - - - - IE K W C P M F Y X Q G H I D J V N L B R Z S T U A 0

I-- - - - - - - - - - - - - - - - - - -

H V N Q Y K B R F Z S T G J U A 0 W P C I E D M L X

--- - - - - - - - -

T A 0 Z R V P C B I E D S U M L X N Y Q G H J K W F

--- - - - - - -

D L X I C A Y Q P G H J E M K W F 0 R Z S T U V N B

I-- - - - - - - - - - - - - - -

J W F G Q L R Z Y S T U H K V N B X C I E D M A 0 P

I-- I--

U N B S Z W C I R E D M T V A 0 P F Q G H J K L X Y

I- - - - - - - - 1-

M 0 P E I N Q G C H J K D A L X Y B Z S T U V W F I~

---K X Y H G 0 Z S Q T U V J L W F R P I E D M A N B I~ I--

V F R T S X I E Z D M ATi) W N B C Y G H J K LIO P Q

PLAIN TEXT

ABC D E F G H I J K L M N 0 P Q R STU V W X Y Z

TABLE X3

105

Components

(1) ABC D E F G H I J K L M N 0 P Q R STU V W X Y Z

OOFBPYRCQZIGSEHTDJUMKVALWNOX

A

B

C

D

E

F

G

H

I

J

K

L

><M

I";I;"l

~N

o

P

Q

R

S

T

U

V

W

X

Y

Z

J Footnote 2 to Table IX, page 104, also apphes to this table, except that the key letters at the left Will

follow the order of the direct mrxed component

ABC D E F G H I J K L !!I~ ° P Q R S T U V !!-I~ Y Z -----

V F R T S

IX I E Z D M A U W N B C Y G H J K L 0 P Q -----

K X Y H G 0 Z S Q T U V J L W F R P I E D M A N B C

1------

MOP E I N Q G C H J K D A L X Y B Z S T U V W F R -----

U NBS Z W C I R E D M T V A o P F Q G H J K L X Y

1------

~!!-~~~ L R Z Y S T U H K V N B X C I E D M A 0 P

D L X I C A Y Q P G H J E M K W F 0 R Z S T U V N B -----

TAO Z R V P C B I E D S U M L X N Y Q G H J K W F -----

H V N Q Y K B R F Z S T G J U A 0 W P C I E D M L X -----

E K W C P M F Y X Q G H I D J V N L B R Z S T U A

1----- I~ S M L R B U X P o C I E Z T D K W A F Y Q G H J V N f------

G U A Y F J 0 B N R Z S Q H T M L V X P C I E D K W -----

I J V P X D N F W Y Q G C E H U A K 0 B R Z S T M L '------

Z D K B 0 T W X L P C I R S E J V M N F Y Q G H U A ------

Q T M F N H L 0 A B R Z Y G S D K U W X P C I E J V -----

C H U X W E A N V F Y Q P I G T M

1-

1----- J L 0 B R Z S D K

R E J 0 L S V W K X P C B Z I H U D A N F Y Q G T M

1-----

Y S DNA G K L M 0 B R F Q Z E J

1-

----- T V W X P C I H U

P G T W V I M A U N F Y X C Q S D H K L 0 B R Z E J

1------

B I H L K Z U V J W X p o R C G T E M A N F -.!-~ S D f------

F Z E A M Q J K D L o B N Y R I H S U V W X P C G T -------

X Q S V U C D M T A N F W P Y Z E G J K L 0 B R I H -----

o C G K J R T U H V W X L B P Q S -- ------ I D M A N F Y Z E

N RIM D Y H J E K L 0 A F B C G Z T U V W X P Q S

WYZIUT P E D S M AjN ------ V X F R I Q H J K L o B C I~ L P Q J H B S T (flu V w K 0 X YIZ C E D M A N F R I

Components

(1) ABC D E F G H I J K L M N 0 P Q R STU V W X Y Z

~) F B P Y R C Q ZIG S E H T D J U M K V A L W N 0 X

Eneiphcnng equations 8k/1=8p /2, 81/1=00 /2 (8 1/ 1 IS A)

N

o

P

Q

R

S

T

U

V

W

X

Y

Z

PLAIN TEXT

ABC D E F G H I J K L M N 0 P Q R STU V W X Y Z

A

B

C

D

E

F

G

H

I

J

K

~L

I"'l

~M

J An mterestmg fact about this ca~e II> that If the plam component IS made Identical with thc CIpher component (both bemg the sequence FBPY ), and If the eneiphermg equations are the same as for Tablc l-B,

then the resultant cipher square IS rdentreal with Table IX, except that the key letters at the lcft arc m the

order of the re, ersed mixed component, FXON In other words, the secondary CIpheralphabets produced

by the Interaction of two Identical mixed components are the same all these given by the mtersctron of a

mixed component and the normal component,
107

TABLE XII

(1) ABC D E F G H I J K L 14 N 0 P Q R STU V W X Y Z

(2) F B P Y R C Q ZIG S E H T 0 J U M K V A L W N 0 X

PLAIN 1EXr

ABC D E F G H I J K L 14 N 0 P Q R STU V W X Y Z

Components

!- X ~!!J~ L A V K M U J D T H E S G I Z Q C ~I.!- P B

F X 0 N W L A V K M U J D T H E S G I Z C R Y P

--- H E S G I Z Q C R y P B F X 0 N W L A V K M U J D T

--- H E S G I Z Q C R J.. P B F X 0 N W L A V K M U J D T --- T !!l~ S G I Z Q C .!. y P B F X 0 N W L A V K M U J D

--- D T H E S G I Z Q ..9.- R Y P B F X 0 N W L A V K M U J --- D T H E S G I Z

S- C R Y P B F X 0 N W L A V K M U J

--- U J D T H E S G I Q CRY P B F X 0 N W L A V K M

--- M U J D T H E S G Z J!..£..!. Y p B F X 0 N W L A V K

I Z Q C R y P B F X 0 N W L A V K M U J D T H E ,~ ---

G I Z Q c R y P B F X 0 N W L A V K M U J D T H E

--- A V K M U J D T H S G I Z Q C R y P B F X 0 N W L

--- A V K M U J D T

li- E S G I Z !i c R y P B F X 0 N W L

---

-!. H E S G 1- Z Q c R Y P B F X 0 N W L A V K M U J D

--- -

E. T H E S G I Z Q c R y P B F X 0 N W L A V K M U J

--- -

d.- D THE S G I Z Q c R y P B F X 0 N W L A V K M U

---- F X o N W L A V K 14 U J D T H E S G I Z Q c R y P B -- X 0 N W L A V K U J D T H E S G I Z Q C R u P B F 1--"--

.!-.!. U J D T H E G I Z Q C R Y P B F X 0 N W L A s. --- V K M U J D T H E S G I Z Q c R y P B F X 0 N W L A

--- I-

.!. V K M U J D T H E S G I Z Q c R Y P B F X o N W L

--I-- Q C R y P B F X 0 N W A V K M U J 0 T H E S G I Z --- I-- 1-

s. L A V K M U J D T H E S G I Z Q C R Y P B F X 0 I..!!- --- I- G I Z Q C R y P B F X I..Q. J!. w L A V K M U J D T H E S --- I Z Q C R Y P B F X N W L A V K M U J 0 T H E S G

-1--

IX 0 N W L A V K M U J D T H E S G I Z Q C R y P B F

o

p

Q

R

S

T

U

V

W

X

Y

Z

A

B

C

o

E

F

G

H

I

J

K

L

.... 14

r:i:l

~N

106

TABLE XI

(1) ABC 0 E F G H I J K L 14 N 0 P Q R STU V WX Y Z ~) F B P Y R C Q ZIG S E H T D J U 14 K V A L WN 0 X

Components

PLAIN l'KX"T

ABC D E F G H I J K L M N 0 P Q R STU V W X Y Z

G ~I~ M P A R 0 S L }JfJ~_\EJ_2_ YIUIWIQ ~I~ H E B

I- -!-l-!.

H A W N Q B S P T M J G KED Z V XIR o L I F C y U

I B X 0 R C T U N K H LIF E A W Y S

-1-

P 14 J G D Z V

r-- ---- -

J C y P S 0 U R V 0 -----LIM G F B X Z T Q N K H E A W ---- fK D Z Q T E S W P 14 J N H G C Y A U R 0 L I F B X ---------/-- -1-

L E A R U F W T X J!. N K 0 I H D Z B V S P M J G C Y --------- M F B S V G X U Y R o L P J I E A C W -'!-~ N K H D Z - --------- -I-

-

N G C T W H Y V Z s ..!-.!.~~~ F BOX U R 0 L I ~.!. 0 H D U X I Z A ---- W T Q N R L K GeE Y V S P M J F B - --------- '--

P I E V Y J A X B U R 0 S M L H D F Z W T Q N K G C r-- --------- -1-

Q J F W Z K B Y C V S P T N M lEG A X U R 0 L H 0

R K G --------- X A L C Z D w.!.~~~!!. J F H B Y V S P M I E - ---- -IS L H Y B M D A E X U R V P 0 K G I C Z W T Q N J F --------~

T M I Z C N E B F Y~~~~~ L H J D A X U R 0 K G - ---- U N J A D 0 F C G Z ~-'!-~~~ M IKE B Y V S P L H I-- ---- -- V o K B E P G D H A X U Y S R N J L F C Z W T Q M I

r-- --------- r--

W P L C F Q H E I B -----

Y V Z T S ----

o K M G D A

-lX

IU R J!.,~ X Q M D G R I F J C

--1---

Z W AUT P L N H E B Y V S o K ---- -IY R N E H S J G K D A X B V U it.!-~-.!... F C Z W T P L ----- /-

-

Z S 0 F I T K H L E -----

B Y C W V----

R N P J G 0 A X U ~.!. A T P G J U L I M F C Z D X W~~~~ H E B Y V R N - ----- B U Q H K V M J N G D A E Y X T P R L I F C Z W S 0 I-- ---------- c V R I L W N K 0 H E B F Z Y ~it.!..!. J G D A X T P -'-- ----- D W S J M X 0 L P I F C G A Z - -- l-I-- ---------

V R T N K H E B Y~S E X T K N Y P M Q J G D H B A W SUO L I F C Z V R

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APPENDIX 2 1

ELEMI< NTARY STATISTICAL THEORY ApPLICABLE TO THE PHENOMENA OF REPETITION

IN CRYPTANAL1SIS

1 Introductory -a In Par qc It was stated that the phenomena of repetition in cryptanalyucs may be removed from the realm of mtumon and dealt With statistically The dISr ussion of the matter will here be confined to relutaveiy <nnple phases of the theory of probabihtv,

a defimtaon of which rmphes philosoplueul questions of no practical Interest to the student of

ctyptanalysis For his purposes, the following defimtaon of a priori probability will be sufflcient

The probability that an event will occur IS the rano of the number of "favorable cases" to the number of total possible eases, all cases bemg equally

likely to occur By a "favorable case" 1'\ meant one which will produce the

event m question

b In what follows, reference "Ill be made to random ossortmeni» of letters and especially to

random text By the latter \\-111 be meant merely that the text under consideration has been assumed to have been enciphered by "ome more or less complex cryptographic system so that for

all practical purposes the sequence of letters constitutmg tlus text IS a random assortment, that

I'!, the sequence IS Just about what would have been obtamed If the letters had been drawn at

random out of a box eontammg a large number of the 26 letters of the alphabet, all m equal

proportions, so that there ale exactly the same numbers of A's, B's, C's, Z's It IS assumed

that each time III rr-akmg a drawmg from such a box, the latter IS thoroughly shaken so that the

letter'! Me thoroughly mrxed and then a single letter If:> selected at random, recorded, and

replaced In the 'lame box In" hat follows, the word "bOA" WIll refer to the box a'l described

c A umhteral frequency distnbunon of a large volume of random text WIll be "flat,"

i e , lncl mg crests and troughs

d For purposes of statisncal analysis, the text of a monoalphabenc substituuon Cipher lCiI

equrvalent to plain tevt As a corollary, when a polvalphnbetic substrtutaon Cipher has been

reduced to the -ample terms of a set of monoalphabets, i e , when the letters constatutmg the

cipher text have been allocated mto their proper umhteral distnbutaons, the letters fallmg mto

the respectrve distnbutions are statisuoally equrvalent to plam tevt

2 Data pertauung to smgle letters -a (1) A single letter WIll be drawn at random Irom

the bOA What IS the probabihtv that It will be an A? According to the foregoing deflmtion of

probability, SInce the total number of possible casee IS 26 and the number of favorable eases IS

1 here only 1, the probablhty IS 1 26=26= 0385 Tins I"l the plobablht) of drawmg an A flom

the box The probablhty that the letter drawn will be a B, a C, a D, , a Z 1"1 thE' 'lame as fOl A

In OthOl word"!, the probabIlIty of drawmg any specifiedb'/,ngle letter I'> p= 0385

(2) The value p= 0385, 8S found above, may also be termed the probabilIty constant for

'3mglc lottE'rs m random te'\.t of It 26-letter alphabet For any language tbl"l constant IS mE'rely

the reCiprocal of the total umuLer of dlflerent Challl(tels whIch may be employed m wntmg thE'

text m que~tIOn

1 In the preparatIOn of thIS appendIX, the anthor has Iud the bt'nefit of the very helpful suggt''ltlOns of

Capt H G MIller, Signal Corp.., Mr F B Ro\\lett, DI S Knllhal'k, amI DJ A 8ml..O\, Assistaut Cryptanalysts,

o C BIg 0 CertalIl palto;; of 1)1 Kullhac.l,.'s IIl1pOltant }IJ.peJ "HtutlStll al Methods 111 ClyptanaIY~lb" form

the hll.!>lS of the dlSc.ulllilOn

(108)

109

(3) Another way of mterpretmg the notation p= 0385 IS to say that m a large volume of

random text, for example In 100,000 letters, aJ'ly letter that one may choose to specify may be

expected to occur about 3,850 tunes, In 10,000 letters It may be expected to occur about 385

tames, In 1/000 letters, about 385 times, and so on In every-day language It would be said

that "m the long run" or "on the average" m 1,000 letters of random text there will be about

38 5 occurrences of each of the 26 letters of the alphabet

(4) But unfortunately, In cryptanalysis It I"l not often the case that one has such a large

number of letters available for study In any single cipher alphabet More often the cryptanalyst

has a relatively small number of letters and these must be distributed over several Cipher

alphabets Hence It IS necessary to be able to dell! With smaller numbers of letters Consider

a specific piece of random text of only 100 letters It has been seen that "m the long run"

each letter may be expected to occur about 3 85 times m this amount of random te ....t, that IS,

the 26 letters will have an ave1age frequency of 3 85 But In reaching this average of 38,')

occurrences in 100 letters, It IS obv IOU~ that some letter 01' letters may not appear at all, some

may appear once, some twice, and so on How many will not appear at all, how many will

appear 1, 2, 3, times? In other words, how will the different categones of letters (different m respect to frequency of occurrence) be distributed, or what will the d'/,stnbutwn be like?

Will It follow any kind of law or pattern? The cryptanalyst also" ants to know the answer

to questions suc-h as these Wbat If> the probability that a specmed letter will not appeal at

all m a given piece of text? That It will appear eTactly 1,2, 3, times? That It WIll appear

at least 1, 2, 3, times? The "lamp sort of questions may he asked With respect to digraphs,

tngraphs, and so on

b (1) It may be stated at once that questrons of tills nature are not easily answered, and

a complete dISCUSSIOn falls quite outside the scope of this text However, It Will be sufficient

for the present purposes If the student IS provided with a more or less SImple and prac tical means

of finding the answers WIth this in VIew certain curves have been prepared fro~ data based

upon Poisson's exponential expansion, or the "law of small probabihnes' and their use wIII

now be explained Students WIthout a knowledge of the mathematical theory of probabihty

and statistaca Will have to take the curves lion farth" Those mterested m their denvanon are

referred to the following texts

Fisher, R A, Stahsttcal Methodsfor Research Workers, London, 1937

Fry, T C, Probabu'/,t'l} and Its Eng'/,nee1"Lng Uses, New York, 1928

(2) By means of these probab'/,hty curves, It IS possible to find, in a relativ ely easy manner,

the probability for 0, 1, 2, 11 occurrences of an event m n cases, If the mean (expected,

average, probable) number of occurrent es m these n eaaes IS known For example, given a cryptogram equivalent to 100 letters of undom te'\.t, wh'\t IS th€'probablhty that.my speCIfied sm~le

letter, whatever WIll not nppeol' at all m the cryptogram? Smce the probabIhty of the occurrence

of a specIfied SIngle letter IS io = 0385, and there nre 100 letters ill the cryptogram, the average

or expected or mean number of occurrences of an A, a B, a C, ,IS 0385X 100=385 Refer

now to that probability curvc ",inch IS marked "fo", meanmg "frequency zero", or "zero occurrences" On the horl1ontal 01 x aAlS of that curve find the POInt correspondmg to the value

3 85 and follow the vertical coordInate deternuned by thIs yalue up to the pomt of Intersection

WIth the curve Itself, then follo'\\ the hon,ontlll coordmate deteunmed by tm" mtersectIon pomt

over to the left and read the value on the vertiCal aXIs of the curve It IS appro"'Q.IIlately 021

Tws means that the probability that a spenfied smgle letter (an A, a B, a C, ) Will not appear

at allm the cryptogram: If It really were a pClfectly random assortmput of lOa lettet5, J.a 021-

1112018-38--&
110

That IS, according to the theory of probability, m 1,000 cases of random-text messages of 100

letters each, one may expect to find about 21 messages ill which a specified smgle letter WIll not

appear at all Another 'Way of sa.ymg the same tlun~ IS If 1 000 sets of 100 letters of random

text are ('xammed, m about 21 out of the 1,000 such sets any letter that one may choose to

name Will be absent 'Ihis, of course, IS merely 0. theoretical evpectaney , It indicates only

what probably will happen m the long run

(3) What IS the probability that a specified single letter will appear aactly once In 100

letters of random text? To answer tlns question, find on the curv e marked 111 tLe pomt of

mtersection of the vertical coordmate corresponding to the meen or average value 38") With

the curve, follow the honzontal coordmate thus doter mmed over to the vertical scale at the

left, read the value on thrs scale It IS 082, which means that ill 1,000 cases of random-text

messages of 100 letters each, one may expect to find about 82 messages in which any letter

one chooses to specify Wl11 occur exactly once, no more and no less

(4) In the same way, the probability that a specified single letter will appear exactly twice

IS found to be 158, exactly 3 times, 202, and so on, as shown m the table below

100 letteTB 0/ random text

Probabihtv th-rt

Frequency a spelll1led sm~le

(x) letter",ill OCCU\

eucth x tunes

0 o 021

1 082

2 158

3 202

4 195

5 150

6 096

7 053

8 026

9 011

10 004

11 001

(5) To find the probability that a specified single letter will occur at least 1, 2, 3, times

m a series of letters constrtutmg random text, one reasons as follows Since the concept "at least

1" unphes that the number specified 1<; to be considered only as the rmmmum, with no lmut

indicated as to maximum, occurrences of 2,3,4, are 0.1"10 "favorable" cases, the probabihtaes

for exactly 1, 2, 3, 4, occurrences should therefore be added and this v.III give tl-e probability

for "at least 1" Thus, in the case of 100 letters, the sum of the probabilities for exactly 1 to 11

occurrences, as set forth m the table directly above, IS 978, and the latter value approximates

the probability for at least 1 occurrence

(6) A more accurate result will be obtained by the following reasonmg The probability

Cor zero occurrences l~ 021 Since It 18 certain that a specified letter will occur either zero times

or 1, 2, 3, times, to find the probability for at least one tame It 10 merely necessary to subtract the probability for zero occurrences from umty That IS, 1- 021 = 979, which 18 001

greater than the result obtamed by the other method The reason It IS greater 151 that the value

979 includes occurrences beyond 11, whrch were excluded from the previous calculation Of

course, the probabilitiesfor these occurrences beyond 11 are very small, but taken all together they
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add up to 001, the difference between the results obtamed by the two methods The probal·Jhty for at least 2 occurrencea IS the difference between umty and the sum of the probability

for zero and exactly 1 occurrences, that IS, 1-(Po+Pt)=I-( 021+ 082)=1- 103-= 897 The

respectIve probabihtses for various numbers of occurrences of a specified single letter (from 0 to

11) are grven In the following table

100 letters oj random text

Probability that a ProbabIlity that a

Frequency speCIfiedsmilie speCIfied'111&1e

letter" 1'1 ooour letter WIlloccur (z) emctlyz at lea-t%

tImes tlDl88

0 o 021 1 000

1 082 979

2 158 897

3 202 739

\ 4 195 537

5 no 34~

6 096 192

7 053 096

8 026 043

9 011 017

10 004 006

11 001 002

(7) The foregoing calculations refer to random text composed of 100 letters For other

numbers of letters, It IS merely necessary to find the mean (multrply the probabihty for drawing

1

a specified single letter out of the box, which IS 26 or 0385, by the number of letters In the

assortment) and refer to the various curves, as before For example, for a random assortment

of 200 letters, the mean IS 200 X 0385, or 77, and tlns ISthe value of the POInt to be sought along

the horizontal or x axes of the curves, the mtersectaons of the respective vertical hnes correspondmg to thIs mean WIth the various curves fur 0, 1,2,3, occurrences give the probabihnes for

these occurrences, the reading bemg taken on the vertical or 'IJ axes of the curves

(8) The discussion thus far has dealt WIth the probabihtaes for 0, 1, 2, 3, occurrences

of specified SIngle letters It may be of more practrcal advantage to the student If he could be

shown how to find the answer to these questaons Given a random assortment of 100 letters

how many letters may be expected to occur elactly 0, 1,2, 3, times? How many may be

expected to occur at least 1, 2, 3, times? The curves may here again be used to answer

these questions, by a very Simple calculation multiply the probability value as obtamed above

for a specified SIngle letter by the number of dIfferent elements bemg considered For example,

the probability that 11 specified smgle letter "Ill occur exactly twice In a perfectly random assortment of 100 letters IS 158, since the number of different letters IS26, the absolute number of SIngle

letters that may be expected to occur exactly 2 times In this assortment IS 158)(26=4108

That IS,In 100 letters of randm text there should be about four letters WIDch occur exactly 2 times

The following table gives the data for variots numbers of occurrences
112

100 letters oj random te:U

PlObabl1lty tbat a Probablhty that a Probable number Probable number

Frequency specl1ledsmgle specified slnille

lettel WIll occur letter wJlI occur of letters appear of letters appear

(%) exactly % atleast% 109oxl\Ltly% mg at least %

tunes times times nmes

0 o 021 1000 0546 26 000

1 082 979 2 132 25 454

2 158 897 4. 108 23 322

3 202 739 5 252 19 214

4 195 537 5 070 13 962

5 150 342 3 900 8 892

6 096 192 2 496 4 992

7 053 096 1 378 2 496

8 026 043 b76 1 118

9 011 017 286 442

10 004 006 104 156

11 001 002 026 052

(9) Referring again to the curves, and specificallv to the tabulated results set forth directly

above, It WIll be seen that the probability that there will be exactly two occurrences of a specified

single letter ill 100 letters of random text ( 158), IS less than the probability that there will be

exactly three occurrences (202), in other words, the chances that a specified single letter will

occur exactly three times are better, by about 25 percent, than that It will occur only two times

Furthermore, there will be about five letters which will occur exactly 3 times, and about five

which \\-Ill occur exactly 4 tames, whereas there WIll be only about two letters which will occur

exactly 1 tune Other facts of a SImilarImport may be deduced from the foregomg table

c The dISCUSSIon thus far has dealt WIthrandom assortments of letters What about other

types of texts, for example, normal plain text? What IS the probability that E WIll occur 0, 1,

2, 3, times In ':;0 letters of normal Enghah? The relative frequency value or probability

that a letter selected at random from a large volume of normal English text will be E IS 12604

(In 100,000letters E occurred 12,604 times ) For 50 letters this value must be multiplied by 50,

grvmg 6 3 as the mean or POInt to be found along the x axes of the curves The probabihtaes for

0, 1, 2, 3, occunences are tabulated below

60 letters oj normal Engl~sh plaIn text

Frequency Probablhty that Probsbihtv th~tan Ewl1l be an EWlll be

(%) drawn exactly drawn at least % tunes % tunes

0 o 002 1 000

1 011 998

2 036 987

3 076 951

4 120 875

5 151 755

6 159 604

7 143 445

8 113 302

9 079 223

10 050 173

11 029 123

•

113

d (1) It has been seen that the probability of occurrence of a specified single letterIn random

1

text employing a 26-letter alphabet 18 P=26= 0385 If a considerable volume of such text 18

wntten on a large sheet of paper and a pencil IS directed at random toward thIS text, the probabilIty that the pencil POIntwill hit the letter A, or any otherletter wh~ch ma'lj be speC'Ljied ~n advance,

18 0385 Now suppose two pencils are directed simultaneously toward the sheet of paper The

probability that both pencil POInts will ~t two A's IS 2~X2~=2~1= 00148, since In tlus case

one 18 deahng WIth the probability of the simultaneoua occurrence of two events which are

Independent The probability of hrttmg two B's, two C's, , two Z's 18 likewise 2~ Hence,

If no particular letter IS specified, and merely this question IS asked "What IS the probability

that both pencil POInts will hit the same letter?" the answer must be the sum of the separate

probabihties for simultaneously hIttmg two A's, two B's, and so on, for the whole alphabet,

1 1 which IS 26X 261= 26= 0385 ThIS,then, IS the probability that any two lettersselected at random

In random text of a 26-letter alphabet will be rdentical or WIll eouunde SInce this value remams

the same so long as the number of alphabetic elements remains fixed, It may be sard that the

probabtltty of monographu cotnculence tn random text of a 26-element alphabet u 0385 The foregomg rtahcized expression 2 ISImportant enough to warrant assigmng a special symbol to It, mz,

Kr (read "kappa sub-r") For 8. 26-element alphabet, then, Kr = 0385

(2) Now If one asks "GIven a random assortment of 10 letters, what are the respectrve

probabihtiee of occurrence of 0, 1, 2, single-letter comcidences?" one proceeds as follows

As before, It 18 first necessary to find the mean or expected number of comcidences and then

refer to the various probability curves To find the mean, one reasons as follows GIven a

sequence of 10letters, one may begin WIththe 1st letter and compare It WIththe 2d, 3d, 10th

letter to see If any two letters coincide, 9 such comparisons may be made, or In other words there

are, begmmng WIth the 1st letter, 9 opportunities for the occurrence of a coincidence But

one may also start WIth the 2nd letter and compare It WIth the 3d, 4th 10th letter, thus

yieldmg 8 more opportunrnes for the occurrence of a coincidence, and so on ThIS process may

contmue until one reaches the 9th letter and compares It WIth the 10th, yielding but one opportumty for the occurrence m question The total number of compansons that can be made IS

therefore the sum of the senes of numbers 9, 8, 7, 1, which IS 45 cornpansons 8 SInce In

the 10 letters there are 45 opportumtres for comeidence of single letters, and SInce the probability

I The expression Itself may be termed a parameter, which In mathemataea IS often used to designate a constant

that charactenzes by each of ItS particular values some particular member of a system of values, functions, etc

The word 18 applicable In the case under diseussior because the value obtained for", IS 0385, for a 25~lement

alphabet, ",= 0400, for a 27-element alphabet, ",= 0370, etc

I The number of compansons may readilv be found by the formula n(n2 1) , where n IS the total number

of letters mvolved Tws formula IS merely a special case under the general formula for ascertammg the number

of eombmetions that may be made of n different tlungs taken r at a time wluch IS C,== n' In the , .. r'(n-r)'

present case, since only two letters arc compared at a time, r IS al\\ays 2, and hence the expression n' , r! (n-r)/

h h h n(n-1) (n-2)' n(n-1) w IC IS t e same as 2(n-2)' ' becomes by cancellation of the term (n-2)' reduced to --2-
114:

for monographie COIncidence In random text IIll 0385 the expected number of comcidenees IS

.0385X45=17325 Wlth m=l 7 one consults the vanous probability curves and an approximate distnbunon for exactly and for at least 0, 1,2, coincidences mayreadliy be ascertained 4

e (1) Now consider the matter of monographic eomeidence III Enghsh plain text 6 Follow.

mg the same reasonmg outlined III subpar d (1), the probability of coincidence of two A'sIn plam

text IS the square of the probability of occurrence of the SIngle letter A In such text The

probability of coincidence of two B's IS the square of the probability of occurrence of the SIngle

letter B, and so on The sum of these squares for all the letters of the alpha bet, as shown in

the following table, 18 found to be 066-7

Lettf'r Fn'qllellCYI III Probablllty of 'l'P Il~e ot probe

1,000letters arate occurrence Il1ty of separate of the letter occunepee

A 73 66 o 0737 o 0064

B 9 74 0097 0001

C_ - 30 68 0307 0009

D 42 44 0424 0018

E - -- 129 116 1300 0169

F 28 32 0283 0008

G - 16 38 0164 0003

H 33 88 0839 0012

I 73 52 0735 00640

J - - 1 64 0016 0000

K - - - - 2 96 00S{) 0000

L 36 42 0364 0013

M 24 74 0247 0006

N 79 50 0795 0063

0 75 28 0753 0057

P 26 70 0267 0007

Q - 3 50 0035 0000

R - 75 76 0758 0057

S 61 16 0612 0037

T 91 90 0919 0084

U 26 00 0260 0007

V 15 32 0153 0002

W 15 60 0156 0002

X 4 62 0046 0000

Y 19 34 0193 0004

Z 98 0010 0000

Total 1.000 00 1 0000 0667

I The data gIven are taken from Table 3 ApPl'ndl~ I l\tllltary Cryptallal)SIS Part I

ThIS then IS the probability that any two letters selected at random In a large volume of

normal EnglIsh telegraphic plain text will COInCIde Smce this value remsms the same so long

as the character of the language does not change radically, It may be said that the probab~l'IJy

of monograph'Lc comculence 'Ln ElIglt8h telegraphw pla'L"/, text 1,8 0667, or "p= 0667

t The approxrmation given by the Poisson distribution In the case of SIngle lettcrs IS not as good as that

III the case of digraphs, tl'lgraphs, etc , discussed In paragrapha 3, 4, below

5 The theory of monographic comeidcnee In plain text was onginally developed and applied by the author

In a technical paper written m 1925 dealing With his solution of messages enciphered by a cryptograph known

as the "Hebern Electric Super-Code" The paper was printed In 1934

i

>

~tI,

I

,

t~I

115

(2) GIven 10 letters of Enghsh plain text, whAt IS the probability that there wIll be 0, 1,

2, smgle~letter eomeidences? Following the hne of reasomng In subparagraph d (2), the

expeeted number of comcidences IS 0667X45=3 00, or m=3 The distnbuuon for exactly and

for at least 0, 1, 2, oomoidences may readily be found by reference to the various probabihty

curves (See footnote 4 )

f The fact that "p (for English) IS almost tWICe as great as "r IS of considerable unportance

1D cryptanalysI'3 It will be dealt Withm detail In a subsequent text At this POIntIt will merely be S8J.d that "p and Ie, for other languages and alphabets have been ealculated and show considerable vananon, as will be noted m the table shown In paragraph 3d

3 Data perta.uung to c4grapha,-a (1) The foregomg drseussioa has been restricted to

questaons coneermng SIngle letters, but by slight modification It can be applied to quesuons

concermng digraphs, tngraphs, and longer polygraphs

(2) In the preceding cases It was necessary, before refemng to the various probabihty

curves, to find the mean or expected number of oecurrences of the event m quesnon m the

total number of cases or trials being considered Given a piece of random text totalling 100

letters, tor example, what IS the mean (average, probable, expected) number of occurrences of

digraphs In this text? SIJlCe there are 67~ dIfferent digraphs, the probabihty of occurrence

1 of any specified digraph IS 676= 00148, SInceIn 100 letters there are 99 digiaphs (If the letters

are taken consecuuvelv In pears) the mean or average number of occurrences III this case IS

00148X99= 147 Having the mean number of occurrenees of the event under consideration,

one may now find the answers to these questaons What I'! the probability that any specified

digraph, say XY, will not occur? "'-nat IS the probability that It WIll occur exactly 1, 2,

3, times? At least 1, 2, 3, times?

(3) Agam the probability curves may be used as before, for the type of distnbunon IS the

same The following values are obtamable by reference to the various curves, usmg the mean

value 00148X99= 147

100 lelura oj random 1m

Probability tbat Probablltty that Probable number Probablll number

Frequeney a s8fCtf1e4 dlgraph a S~lfted dlgtapb of digraphs B~ of dlgl'aphs ap-

(r) w I occur exactly " I occur at least peIlI'mg exact y pearm4 at least

%times %tlmlllo %times % mes

0 o 86 100 581 36 676 00

1 13 14 8788 94. 6'

2 01 01 6 76 6 76

3 00 00 000 o 00

(4) Thus It.IS seen that In 100 letters of random te},.t the probability that a specified digraph

will occur exactly once, for example, IS 13, at least once, 14, at least twice, 01 The probabihty

that a specified digraph will occur at least 3 times IS negligible (By calculation, It IS found to

to be 0005)

b (1) The probability of digraphic coincidence In random text based upon a 26-element

alphabet IS of course quite SImply obtained since there are 262 different digraphs, the probability

1 of selectmg any specified digraph III random text I!> 262 The probability of selecting two iden1 1 1

troal dsgraphs In such text, when the d'Lgraph8 are 8pec'L:/i£d, IS 26lX262=264 Smce there are 26l

1 1

drlferent digraphs, the probability of digraphic comcidenoe In random text, ",:z, IS 262X2b4= 262=

00148
116

(2) Given a random assortment of 100 letters, what IS the probabihty of occurrence of

0, 1, 2, digrapluc eomcidences? Followmg the hne of reasoning m paragraph 2d (2), in

100 letters the total number of compansons that may be made to see If two digraphs coincide

IS 4 851 'I'his , number IS obtained as follows Consider the 1st and 2d letters m the senes of

100 letters, they may be combined to from a digraph to be compared With the digraphs formed

by eombmmg the 2d and 3d, the 3d and 4th, the 4th and 5th letters, and so on, grvmg a total of

98 comparisons Consider the digraph formed by oombmmg the 2d and 3d letters, It may be

compared WIth the digraphs formed by combmmg the 3d and 4th, 4th and 5th letters, and so on,

grvmg a total of 97 compansons ThIS process may be continued down to the digraph formed

by combmmg the 98th and 99th letters, WhICh yields only one eompanson, Since It may be

compared only With the digraph resultmg from eombmmg the 99th and 100th letters ThE'

total number of comparisons IS the sum of the sequence of numbers 98, 97, 96, 95, 1, which

IS 4,851 11

(3) Smce m the 100 letters there are 4,851 opportumtaes for the occurrence of a digraphic

coincidence, and since /(,2= 00148, the expected number of comcidences IS 00148X4851=

717948=72 The various probability curves may now be referred to and the followmg results

are obtamed ,

Dutnbut&on [or 100 letters oj random text

Probability for exllCtlyr Probab1llty for at 1e1l8ts Frequency (s) dlgJ'Bphlc eomeidenees dlgrllphlc eotnerdenees

0 o 001 1 000

1 005 999

2 019 994

3 046 975

4 083 929

5 120 846

6 144 726

7 148 582

8 134 434

9 107 300

10 077 193

11 050 116

c In this table It Will be noted that It IS almost certain that in 100 letters of random text

there WIll be at least one digraphic coincidence, despite the fact that there are 676 possible

digraphs and only 99 of them have appeared in 100 letters When one thmks of a total of 676

dIfferent digraphs from which the 99 digraphs may be selected It may appear rather incredible

that the chances are better than even ( 582) that one willfind at least 7 digraphic comcidences ill

100 letters of random text, yet that IS what the statistacal analysis of the problem shows to be

the case These are, of course, purely aCC7-dental repet'tt'tons It IS Important that the student

should fully realize that more comcidences or accidental repetraons than he feels mtuitrvely

should occur In random text WIll actually occur In the cryptograms he will study He must

therefore be on guard against puttmg too much reliance upon the surface appearances of the

phenomena of repention, he must calculate what may be expected from pure chance, to make

sure that the number and length of the repetmons he does see m a cryptogram are really better

than what may be expected m random text In studymg cryptograms composed of figures thIS

• The formula for findmg the number of eompansons that can be made IS as follows, where n=the total

en-t) en-t+1) number of letters in the sequence anet t 18 the length of the polYSl'aph NQ. of companlOns- 2

117

it V8Fy' Important, for as the number of drfferent symbols decreases the probability for purely

ehaaee comcldences mcreases

d. (1) For eonvemenee the followmg values of the reciprocals of vanous numbers from 20 to

36, and of the reeiprooals of the squares, cubes, and 4th powers of these numbers are hated:

r' 1" l/s' l/s' I1s1

20 o 0500 o 002500 o 000125 o 00000625

21 0476 002266 000108 00000514

22 0455 002070 000094 00000429

23 0435 001892 000082 00000358

24 0417 001739 000073 00000302

25 0400 001600 000064 00000256

26 0385 001482 000057 00000220

27 0370 001369 000051 00000187

28 0357 001274 000046 00000162

29 0845 001190 000041 00000142

30 0833 001109 000037 00000123

31 0323 001043 000034 00000109

32 0813 000980 000031 00000096

33 0803 000918 000028 00000084

34 0294 000864 000025 00000075

35 0286 000818 000023 000000b7

36 0278 000773 000021 00000060

(2) The followmg table grves the probabilities for monographic and digraphic comcidenea

for plain-text in several languages

Ianguap /Cp /CpI

Enghsh__________________________ 0 0667 0 0069 French__________________________ 0778 0093 German; ________________________ 0762 0112 Itallan __________________________ 073$ 0081 Spamsh__________________________ 0775 0093

4 Data pertauung to tngraphs, etc -a Enough has been shown to make clear to the student

how to calculate probability data concerning trigraphs, tetragraphs, and longer polygraphs

b (1) For example, in 100 letters of random text the value of m (the mean) for tngraphs

IS 00005689XlOO= 005689 With so small a value, the probability curves are hardly usable.

but at any rate they show that the probability of occurrence of a specified tngraph in so small

a volume of text IS so small as to be practically negligible The probability of a specified tngraph

occurrmg twice m that text IS an even smaller quantrty

(2) The calculation for findmg the probability of at least one tngraphic coincidence in 100

letters of random text IS as follows

m=e7;<98)(2~d)=4,753X 0000568912= 2704= 27

Referrmg to curvefo, Withm= 27 the probability of finding no tngraphic comoidenoe 18 76.

The probablhty of findmg at least one tngraphic comcidence IS therefore 1- 76= 24

c The calculation for a tetragraphie comcidence IS as follows

m=e6~97)(2~.)=4,656X 0000021888 .... 0101- 01

Referrmg to curvefo, with m= 01 the probabulty of flndmg no tetr'iiIapluc (l91JlCldeJ,lCe Ji

80 high as to amount almost to certamty Consequently, the probabilrty of findmg at least

162018-38-9
118

one tetragraplnc coincidence IS practically ml (It IS calculated to be 0094=approxnnately 01

Tins means that m a hundred cases of 100-letter random-text cryptograms, one might expect

to find but one cryptogram m winch a 4.-letter repetitaon IS brought about purely by chance, It

lS, m common parlance, a "hundred to one shot") Consequently, 1f a tetragraplnc repetmon

lS found m a cryptogram of 100 letters, the probability that It IS an accidental repetition IS

extremely small If not accidental, then It must be causal, and the cause should be ascertained

5 An example -a The message of Par Oa of the text proper will be employed FIrst, let

the repetitaons be sought and underlined, then the rcpetrnons are hsted for convenience Page

5,6

53

109

15

86

23

21

EquatIons, enciphering _

EqUIvalent pnmary components _

Expected number of occurrences _ Factonng _

Intervals _

Grvierge Footnote 1

Gronsfeld _

Identieal messages eneiphered by keywords of

dIfferent lengths __________________________ 89Jf

Idehtical aupemnposrtions, __________________ 86

Indexletter________________________________ 6

Indireet symmetry _ 9,52

Of posrtion 60,68,69,84,85

Interrelated alphebeta.; _____________ _____ 24

Latent symmetry__ __ 52

Law of small probabllitles____________________ 109 ~atchlng___________________ 47

I>IstnbutIons_____________ 94

]4eannuDlber______________________________ 109

Oomcidenees, of., _______________________ 114

Digraphs, of. ___ __________ 115

Mixed alphebeta.;, ______________________ 24

Monoalphebetac terms, conversion mto________ 46

Monographic comcidence, ____ ______ ____ 113Jf

Multiple alphabet system____________________ 3

Number of comparlsons_ __________ _____ 113-116

Parameter Footnote 2 113

Partial chams of eqUlvalents_________________ 85

Period, determination of; 10,15

Periodic systems____________________________ 2

85 Prunary classifieation, ___ _ __ 3

84 Phenomena of repetitIon_____________________ 108

2 Poe, Fdgar Allan Footnote L_ 2

POIsson's exponentral e'q)ansIon_______________ 109

Polyalpbabetec subetitutron

Distmgtnshed from monoalphabetIc_______ 1

Pnmary clasaifleation , __ 2

Sequence of study______________________ 3

Primary components________________________ 4,5

EqUIvalent____________ _____ ______ 53

Reconstruetron oL 27,52

Prmeiples of indirect symmetry of posmon

Appheatron of pnnClples_________________ 69Jf

Appheation to specifle example _____ _____ 60.1

Fund!unentaltheory 68,69

Probabihty

Deflmtion of aprlOM_____________________ 108

Of digraphic comcidenee • __._. 115-116

109

3

(119)

Pap

12

INDEX

ACCIdental repetrtions _

Alphabets ClassIficatIon of- __ ___________ ____ 4

I>enved_______________________________ 4

Interrelated____________________________ 24

]4IXed_________________________________ 24

Secondary___________________ 4

AnalytIcal key _____ __________ _______ 95

AperIodIC systems_________________ __ __ 2

AssumptIons for values, check________________ 76

Average_________________________ 109

Bazenes___________________________________ 23

Beaufort___________________________________ 9,19

Causal repetitIons____________ __ ______ 12

CJ.pherdisks; _____ ___ __ __ 5

ClassIficatIon of alphabets_ __________________ 4

Oomcideace I>lgrapluc 115-116

MonographIc______ _____ ____ ___ 113, 114

Tetragrapluc___________________________ 117

TngraphIC 117-118

Comparisons, number ot. 113,116

Oompletmg the plam componentsequence _19, 79, 82, 88

Component monoalphabets__________________ 15

Constant mtervals; __ __ ______ __ ______ __ 85

ConVersIon

Into monealphebetac terms 92,94

Into plaIn-component equrvalents 81,83

Cryptograms

In dIfferent keys, eontammg identieal plaIn text _

Wlth plain text _

Cychcphenomena _

Data pertammg to I>lgraphs 115-116

Tngraphs 117-118

Decunatron Footnote 1 53

I>elastelle___ 9

Derrved alphabets., __ _________________ __ 4

Digraphie comcidenee; ____________________ 115-116

Probabihty for________________________ 115

Digraphs data pertammg to___ 115

Direct symmetry___________________________ 9, 26

ApplIcatIOn of prmciples_________________ 32

Of pOBltIOn____ _ _____ __ __ __ 84

Distnbutron of different categories of letters in

respect to frequency of occurrence _

Double-key system _ ______ _ _ Footnote 2

.

f

•,

t

t

.,

,

o X U V D

G U R D P

L E J Y Q

G B MF I

X B W e S

U S Y E E

X V E U E

LED E e

Group Number of occurrenccs

Be 2

ex 2

Ee 2

LE 3

JY 2

PL 2

se 2

SY 2

US 3

YE 2

SYE 2

USY 2

USYE 2

E e P M eeL N

H X I P L I Bel J

o F P J W J E MG P

J Y MS G L LET A

U S YES

S e R H T

A Y ~x

MUS£..!

A

B

e

D

6 Referrmg to the table m Par 3a (3) above, It will be seen that m 100 letters of random

text one nught expect to find about 7 digraphs appeanng at least tWICe and no digraph appearing

3 times The hst of repetitions shows 8 digraphs occurnng twice and 2 occurring 3 tunes

c Agam, the hst of repetitions shows 10 digraphs each repeated at least twice, the table m

Par 36 (3) above shows that m 100 letters of random text the probabihty of findmg at least

that many digraphic comcrdencea lS only 193 That IS, the chances of this bemg an accident are

but 176m a thousand, or another way of expressmg the same thmg lS to say that the odds agamst

thJs phenomenon bemg an accident are as 807 IS to 193 or roughly 4 to 1

d The probability of findmg at least one tngraphic coincidence m 100 letters of random

text IS very small, as noted m Par 46, the probability of findmg at least one tetragraplnc COincidence IS still amaller (Par 4c) Yet tills CIphermessage of but 100 letters contams a repetition

of tlns length

e A considerauon of the foregomg leads to the conclusion that the number and length of the

repetitions manifested by the cryptogram are not accidental, such as might be expected to occur

m random text of the same length, hence they must be causal m their ongm The cause m tills

case lS not difficult to find repeated Isolated letters and repeated sequences of letters (digraphs,

tngraphs) In the plain text were actually enciphered by identacal alphabets, resultmg m producmg

repeated letters and sequences In the CIpher text
o

Page

120

Repetrtions Acctdental , 12, 116

Causal________________________________ 12

Phenomena oL_________________________ 108

Secondaryalphabets________________________ 4

Sequence reeonstructron skeleton _____________ 26

Square tables_____ ______ ________ _____ 5

Symmetry of position ______________________ 8, 9, 52

Tetragraphic comcidence, - - - - - - •• _- _. _ 117

Theory of factoring - - - - - - - - - -- - _•- 15, 8~

Trigraplnc comcidence • • __ 117

Trigraphs, data pertammg to_________________ 117

Types of Cipher squares ._ ___ 96ff

80 Vlgenere 9,19

a

lQ

Page

113ff

117

117

21.48

3

108

Probability-c-Contmued

Of monographic comcidence _

Of tetragraphic coincidence - - - - - -_

Of trigraphic coincidence -- __

Probable-word method - - - - ---

Progressive alphabet system _

Random text ---- ---- - - --

Reconstructaon of equivalent primary compo- nents • .,5,21,52. 5~

Reconstruction skeleton, • _ FQQtnOfe 1_. 2Q. 56

Relative frequencies - __._---__ W

Repeatmg-key ciphers

Primary components are different mixed

sequences _

Repeatmg-key system - __

AnalySls of', • --- - - -- __- __

Solution of subsequent messages enclp4ered

by the same primary components 78/.8QJ1