Breaking a substitution code

Objectives :

•

^{Study the mainmethodsto break a}substitution ode.

Materials :

•

^{Exerpt fromThe Gold Bug and tips for frequeny analysis.}

•

^Beamer.

•

^{Texts todeipher.}

1 – Breaking the Gold Bug cipher 55 mins

The method is explained by reading the relevant part in The Gold Bug, by Edgar Allan

Poe. Othertriks are extrated for The Code Book, by Simon Singh.

2 – Break a cipher on your own 55 mins

Working inteams, students have tobreak a substitution ode on ashort text.

Breaking a substitution code

Season 03

Episode 12 Document Document

Althoughitisnotknown whorst rea-

lized that the variation in the frequenies

of letters ould be exploited in order to

break iphers, the earliest known desrip-

tion of the tehnique isby the 9th entury

sientistAbuYusufYa'qub ibn Is-haq ibn

as-Sabbah ibn 'omran ibn Ismail al-Kindi.

KnownasthephilosopheroftheArabs',al-

Kindi was the author of 290 books on me-

diine,astronomy,mathematis,linguistis

and musi, but his greatest treatise, whih

was only redisovered in 1987 in the Su-

laimaniyyahOttoman Arhive in Istanbul,

is entitled "A Manusript on Deiphering

Cryptographi Messages."

The rst page of al-Kindi's manusript

"The Gold-Bug" isa short storyby Edgar AllanPoe,set onSullivan's Island, South

Carolina involving deiphering a seret message and nding buried treasure. The story

was rst published in the Philadelphia Dollar Newspaper in June 1843 after Poe had

won a ompetition held by the paper, reeiving a prize of US

100

. It inludes a detailed desription of a method forsolving asimple substitution ipherusing letter frequenies.

Excerpt from The Gold Bug by Edgar Allan Poe

Youobserve there are nodivisions between the words. Had there been divisions the

task would have been omparatively easy.In suhases I should have ommenedwith a

ollationandanalysis ofthe shorterwords,and, hadaword ofasingle letterourred, as

ismost likely(aorI,for example),Ishould haveonsidered the solutionasassured. But,

there being nodivision, my rst step was toasertain the predominant letters,aswellas

the least frequent. Counting all,I onstruted a tablethus :

8 ; 4

‡

⁾

∗

^{5 6}

†

^{1 0 9 2 : 3 ? $ .}

33 26 19 16 16 13 12 11 8 8 6 5 5 4 4 3 2 1 1

Now, inEnglish,the letter whih most frequently ours ise. Afterward, the sues-

sion runsthus:aoidhnrstuyfglmwbkpqx z.Epredominatessoremarkably,

that anindividual sentene of any length is rarely seen, in whih it is not the prevailing

harater.

Here, then,wehave,inthe verybeginning,the groundworkforsomethingmorethan

a mere guess. The general use whih may be made of the table is obvious but, in

this partiular ipher, we shall only very partially require its aid. As our predominant

Let us assume 8, then, as e. Now, of allwords in the language, 'the' is most usual;

let us see, therefore, whether there are not repetitions of any three haraters, in the

same order of olloation, the last of them being 8. If we disover a repetition of suh

letters, so arranged, they will most probably represent the word 'the.' Upon inspetion,

wendnoless thansevensuharrangements,the haratersbeing;48.Wemay,therefore,

assume that; represents t, 4 represents h, and 8 represents e the last being now well

onrmed. Thus a great step has been taken. But, having established a single word, we

are enabled toestablisha vastlyimportantpoint;that is tosay, several ommenements

and terminationsofotherwords.Letusrefer,forexample,tothe lastinstane butone,in

whihtheombination;48oursnot farfromtheend ofthe ipher.Weknowthatthe;

immediatelyensuingistheommenementofaword,and,ofthesixharaterssueeding

this 'the,'we areognizantofnoless thanve.Letussetthese haraters down,thus, by

the letterswe know them torepresent, leavinga spae for the unknown t.eeth.

Here we are enabled, atone, to disard the 'th,' as forming noportion of the word

ommening with the rst t; sine, by experiment of the entire alphabet for a letter

adapted tothe vaany, we pereive that noword an beformed of whih this than be

a part. We are thus narrowed into t.ee, and, going through the alphabet, if neessary, as

before, we arrive at the word 'tree,' as the sole possible reading. We thus gain another

letter, r,represented by (, with the words 'the tree' in juxtaposition.

Lookingbeyond these words, for ashort distane, we again see the ombination;48,

and employ it by way of termination to what immediately preedes. We have thus this

arrangement:

the tree;4(

‡

^{?34 the,}

or substituting the naturalletters, whereknown, it readsthus :

the tree thr

‡

^{?3h the,}

Now,if,inplaeoftheunknownharaters,weleaveblankspaes,orsubstitutedots,

we read thus :the tree thr...hthe, when the word 'through' makesitself evident at one.

But this disovery gives usthree new letters, o,u, and g,represented by and 3.

Lookingnow, narrowly,through the ipherfor ombinationsofknown haraters, we

nd, not very farfromthe beginning, thisarrangement,83(88,oregree,whih,plainly,is

the onlusionof the word 'degree,' and gives usanotherletter, d, represented by

†

^.

Four letters beyond the word 'degree,' we pereivethe ombination;46(;88.

Translatingthe known haraters, and representing the unknown by dots, asbefore,

we readthus:th.rtee, anarrangementimmediatelysuggestiveof theword 'thirteen,'and

again furnishing uswith two new haraters, iand n,represented by 6and

∗

^.

Referring,now, tothebeginningof theryptograph, wendthe ombination,53

‡‡†

^.

Translating as before, we obtain .good, whih assures us that the rst letter is A,

and that the rst two words are 'A good.'

It is now time that we arrange our key, as far as disovered, in a tabular form, to

avoid onfusion.It willstand thus :

Season 03 • Episode 12 • Breaking a substitution code

³

5 represents a

†

^d

8 e

3 g

4 h

6 i

∗

ⁿ

‡

^o

( (

; t

? u

We have, therefore, no less than eleven of the most important letters represented,

and it willbeunneessary to proeedwith the details of the solution.I have said enough

to onvine you that iphers of this nature are readily soluble, and to give you some

insight into the rationaleof their development. But be assured that the speimen before

usappertainstotheverysimplestspeiesofryptograph.Itnowonlyremainstogiveyou

the fulltranslation of the haraters uponthe parhment, as unriddled.Here it is:

'A good glass in the bishop's hostel in the devil's seat forty-one degrees and

thirteen minutes northeast and by north main branh seventh limb east side

shoot from the left eye of the death's-head a bee-line from the tree through

the shot fty feet out.'

Tips for frequency analysis

1

^{. Begin by ounting up the frequenies of all the letters in the iphertext. About}

ve of the letters should have a frequeny less than 1 per ent, and these probably

represent j, k, q, x and z. One of the letters should have a frequeny greater than

10 perent, and itprobably represents e.

The table belowshows the average frequeniesof all 26letters inEnglish.

0 2 4 6 8 10 12 14

a b d e f g

h i j

k l m n o p q

r s t u v w x y

z

2

^{. Ifthe plaintextdoesn'trevealitself}immediately,whihisoftenthe ase,then fous on pairs of repeated letters. In English the most ommon repeated letters are ss,

ee, tt,, ll,mmand oo.Ifthe iphertext ontains anyrepeated haraters, youan

assume that they represent one of these.

3

^{. More omplexuse ofstatistis anbeoneived, suhasonsideringountsof pairs}

ofletters,ortriplets(trigrams),and soon.Thisisdonetoprovidemoreinformation

totheryptanalyst, forinstane,q andunearlyalwaysourtogether inthat order

inEnglish,even thoughqitselfisrare.Also,th isthe mostommonbigram,and the

the most ommon trigram.

4

^{. One of the most useful skills for a} ryptanalyst is th eability to identify words, or even entire phrases, based on experiene or sheer guesswork. Any suh word or

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Breaking a substitution code

Season 03 Episode 12 Document Cipher 2

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Breaking a substitution code

Season 03 Episode 12 Document Cipher 4

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Breaking a substitution code

Season 03 Episode 12 Document Cipher 6

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Breaking a substitution code

Season 03 Episode 12 Document Cipher 8

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lxnsoslxnsxrsygxrvnsolkrzjrrm.

Season 03 • Episode 12 • Breaking a substitution code

¹³

Document 1

^Plaintexts

Cipher 1

CharlesBabbage wasborn Deember26,1791inLondon,England,and diedOtober18,

1871 in Marylebone, London, England. He was an English mathematiian, philosopher,

inventorand mehanialengineer who originatedthe onept of aprogrammableompu-

ter. Parts of hisunompletedmehanismsare ondisplay inthe LondonSiene Museum.

In 1991aperfetlyfuntioningdiereneenginewasonstrutedfromBabbage'soriginal

plans. Built to toleranes ahievable in the 19th entury, the suess of the nished en-

gine indiated that Babbage's mahine would have worked. Nineyears later, the Siene

MuseumompletedtheprinterBabbage haddesignedforthediereneengine,anastoni-

shingly omplexdevie forthe 19th entury. Babbage isredited with inventing the rst

mehanial omputer thateventuallyled tomore omplex designs.

Cipher 2

AlanMathison Turingwas born June 23,1912and died June 7,1954. Hewas anEnglish

mathematiian,logiianand ryptographer.Turingisoftenonsideredtobethe father of

modernomputer siene. He provided an inuential formalisationof the onept of the

algorithmandomputationwiththeTuringmahine.WiththeTuringtest,meanwhile,he

madeasigniantandharateristiallyprovoativeontributiontothe debateregarding

artiial intelligene : whether it willeverbepossible tosay that a mahine isonsious

and an think. He later worked atthe National Physial Laboratory, reating one of the

rst designs for a stored-program omputer, the ACE, although it was never atually

built in itsfull form.In 1948, he moved to the University of Manhester to work on the

Manhester Mark 1,then emerging asone of the world's earliesttrue omputers. During

the Seond World War Turing worked at Blethley Park, the UK's odebreaking entre,

and wasforatimeheadofHut8,the setionresponsibleforGermannavalryptanalysis.

Hedevised anumberof tehniques forbreakingGermaniphers,inludingthemethodof

thebombe,aneletromehanialmahinethatouldndsettingsfortheEnigmamahine.

Cipher 3

Philip R."Phil" ZimmermannJr.,born February 12,1954, isthe reatorof PrettyGood

Privay (PGP), the most widely used email enryption software in the world. He is also

knownforhisworkinVoIPenryptionprotools,notablyZRTPandZfone.Hewasbornin

Camden, NewJersey.Hisfather wasaonrete mixertrukdriver. Zimmermannreeived

a B.S. degree in omputer siene from Florida Atlanti University in Boa Raton in

1978, and urrently lives in the San Franiso Bay Area. In 1991, he wrote the popular

Pretty Good Privay (PGP) program, and made it available (together with its soure

ode) through publiFTPfor download,therst widelyavailableprogramimplementing

publi-keyryptography.Shortlythereafter, itbeameavailableoverseasviatheInternet,

though Zimmermann has said he had no part in its distribution outside the US. After

a report from RSA Data Seurity, In., who were in a liensing dispute with regard to

use of the RSA algorithm in PGP, the Customs Servie started a riminalinvestigation

of Zimmermann,for allegedly violating the Arms Export Control At. The investigation

lasted three years, but was nallydropped withoutling harges.

boratory (CSAIL). Ron Rivest is one of the inventors of the RSA algorithm (alongwith

AdiShamir and Len Adleman).He is the inventor of the symmetrikeyenryption algo-

rithms RC2, RC4, RC5, and o-inventor of RC6. The "RC" stands for "Rivest Cipher",

or alternatively, "Ron's Code". (RC3 was broken at RSA Seurity during development;

similarly, RC1 was never published.) He also authored the MD2, MD4, MD5 and MD6

ryptographi hash funtions. In 2006, he published his invention of the ThreeBallotvo-

ting system, an innovative voting system that inorporates the ability for the voter to

disernthat theirvotewasounted whilestillproteting theirvoterprivay. Mostimpor-

tantly, this system does not rely on ryptography at all. Stating "Our demoray is too

important",he simultaneously plaed ThreeBallotin the publi domain.

Cipher 5

Bailey Whiteld 'Whit' Die, born June 5, 1944, is a US ryptographer and one of

the pioneers of publi-key ryptography. He reeived a Bahelor of Siene degree in

mathematis from the Massahusetts Institute of Tehnology in 1965. Die and Martin

Hellman's paperNew Diretions inCryptography waspublished in 1976.It introdued a

radially new method of distributing ryptographi keys, whih went far toward solving

one of thefundamentalproblems of ryptography, key distribution.Ithas beomeknown

as Die-Hellman key exhange. The artile also seems to have stimulated the almost

immediatepublidevelopmentofanewlassofenryptionalgorithms,theasymmetrikey

algorithms. Diewas Managerof SeureSystems Researhfor NorthernTeleom, where

he designed the key management arhiteture for the PDSO seurity system for X.25

networks. In 1991 he joined Sun Mirosystems Laboratories (in Menlo Park, California)

as aDistinguished Engineer, working primarilyon publipoliy aspetsof ryptography.

As of May 2007 Die remains with Sun, serving as its Chief Seurity Oer, and as a

Vie President.

Cipher 6

Blaise de Vigenere was born April 5, 1523and died in 1596. He was a Frenh diplomat

and ryptographer. The Vigenère ipher is so named due to the ipher being inorretly

attributed tohiminthe 19thentury.Vigenèrewas borninthe villageofSaint-Pourçain.

At age 17 he entered the diplomati servie, and remained there for 30 years, retiring

in 1570. Five years into his areer he was sent to the Diet of Worms as a very junior

seretary. At age 24, he entered the servie of the Duke of Nevers. In 1549 he visited

Rome on a two-year diplomati mission, and again in 1566. On both trips, he ame in

ontat both with books on ryptography and ryptologists themselves. When Vigenere

retiredaged47,hedonated his1,000livresayearinometothepoorinParis.Hemarried

aMarieVare.Inhisretirement,hewasauthorofovertwentybooksinludingthe Traité

des Chires ouSerètes Manièresd'Esrire (1585).Inthis book he desribed anautokey

ipher hehad invented, itwasthe rst ipherofthis typeafterBellasonot tobetrivially

breakable.

Season 03 • Episode 12 • Breaking a substitution code

¹⁵

Cipher 7

Cliord Coks, born 28 Deember 1950, is a British mathematiian and ryptographer

at GCHQ who invented the widely-used enryption algorithmnow ommonly known as

RSA, about three years before it was independently developed by Rivest, Shamir, and

Adleman atMIT. He has not been generally reognised for this ahievement beause his

work was by denition lassied information,and therefore not released to the publi at

the time. At GCHQ, Coks was told about James H. Ellis' non-seret enryption" and

that no one had been able to nd a way to atually implement the onept. Coks was

intrigued, thought about it overnight, and invented, in 1973, what has beome known

as the RSA enryption algorithm,realising Ellis' idea. GCHQ appears not to have been

able tond a way touse the idea, and in any ase, treatedit as lassied, sothat when

it was reinvented and published by Rivest, Shamir, and Adleman in 1977, Coks' prior

ahievement remained unknown until1997.

Cipher 8

Al-Kindi, who lived approximately between 801 and 873, was an Arab polymath : an

Islamiphilosopher, sientist, astrologer, astronomer, osmologist,hemist, logiian,ma-

thematiian,musiian,physiian,physiist,psyhologist,andmeteorologist.Al-Kindiwas

a pioneerin ryptography, espeiallyryptanalysis. He gave the rst known reorded ex-

planation of ryptanalysis in A Manusript on Deiphering Cryptographi Messages. In

partiular, he is redited with developing the frequeny analysis method whereby va-

riations in the frequeny of the ourrene of letters ould be analyzed and exploited

to break iphers (i.e. ryptanalysis by frequeny analysis). This was detailed in a text

reently redisovered in the Ottoman arhives in Istanbul, A Manusript on Deiphe-

ringCryptographiMessages,whihalsooversmethodsofryptanalysis, enipherments,

ryptanalysisofertainenipherments,andstatistialanalysisoflettersand letterombi-

nationsinArabi.Al-Kindialsohad knowledgeofpolyalphabetiiphersenturiesbefore

Leon Battista Alberti.

Cipher 9

Noonringsout. Awasp,making anominoussound,asound akintoaklaxonoratosin,

itsabout.Augustus,whohashadabadnight,sitsupblinkingandpurblind.Ohwhatwas

thatword(ishisthought)thatranthroughmybrainallnight,thatidiotiwordthat,hard

asI'd trytopunit down,wasalwaysjustaninhortwo outof my grasp-fowlorfoulor

VoworVoyal?-awordwhih,by assoiation,broughtintoplayaninongruousmassand

magma of nouns, idioms,slogans and sayings, a onfusing,amorphous outpouringwhih

I sought in vain to ontrolor turn o but whih wound aroundmy mind a whirlwind of

a ord, a whiplash of a ord, a ord that would split again and again, would knit again

and again, of words without ommuniation or any possibility of ombination, words

without pronuniation, signiation or transription but out of whih, notwithstanding,

wasbroughtforthaux, aontinuous,ompatandluid ow:anintuition,availlating

frisson of illuminationas if aught in a ash of lightning or in a mist abruptly rising to

unshroud an obvioussign - but asign, alas, that would lastan instant onlyto vanish for

good.