ClassicalCryptographyCourse, VolumesIandIIfromAegeanParkPress ByRandyNichols(LANAKI) PresidentoftheAmericanCryptogramAssociationfrom19941996. ExecutiveVicePresidentfrom19921994 TableofContentsl l l l l l

Lesson1 Lesson3 Lesson5 Lesson7 Lesson9 Lesson11

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Lesson2 Lesson4 Lesson6 Lesson8 Lesson10 Lesson12

CLASSICALCRYPTOGRAPHYCOURSE BYLANAKI December05,1995 LECTURE4 SUBSTITUTIONWITHVARIANTS PartIII MULTILITERALSUBSTITUTIONSUMMARYWelcomebackfromtheThanksgivingholidaybreak.ThegoodnewsisthatthislecturewillcometoyouaboutChristmas,therefore,nohomework.Thenotsogood newsisthatthisconcludingLecture4onSubstitutionwithVariantscoverssomedifficultmaterialofwidepracticallyinthefield. InLecture4,wecompleteourlookintoEnglishmonoalphabeticsubstitutionciphers,bydescribingmultiliteralsubstitutionwithdifficultvariants.TheHomophonicand GrandPreCipherswillbecovered.Theuseofisologsisdemonstrated.AsynopticdiagramofthesubstitutionciphersdescribedinLectures14willbepresented.

MULTILITERALSUBSTITUTIONWITHMULTIPLEEQUIVALENTCIPHERALPHABETS aka"MONOALPHABETICSUBSTITUTIONWITHVARIANTS"EachEnglishletterinplaintexthasacharacteristicfrequencywhichaffordsdefinitecluesinthesolutionofsimplemonoalphabeticciphers.Associationswhichindividual lettersformincombiningtomakeupwords,andthepeculiaritieswhichcertainofthemmanifestinplaintext,affordfurtherdirectcluesbymeansofwhichordinary monoalphabeticsubstitutionenciphermentsofsuchplaintextmaybereadilysolved.[FR1] Cryptographershavedevisedmethodsfordisguising,suppressing,oreliminatingtheforegoingcharacteristicsinthecryptogramsproducedbymethodsdescribedin Lectures13.Onecategoryofmethodscalled"variantsorvariantvalues"isthatinwhichthelettersoftheplaincomponentofacipheralphabetareassignedtwoormore cipherequivalents. Systemsinvolvingvariantsaregenerallymultiliteral.Insuchsystems,therearealargenumberofequivalentsmadeavailablebycombinationsandpermutationsofalimited numberofelements,eachletteroftheplaintextmayberepresentedbyseveralmultiliteralcipherequivalentswhichmaybeselectedatrandom.Forexample,if3letter combinationsareemployedasmultiliteralequivalents,thereare26 3or17,576availableequivalentsforthe26lettersoftheplaintext. Theymaybeassignedinequalnumbersofdifferentequivalentsforthe26letters,inwhichcaseeachletterwouldberepresentableby676different3letterequivalentsor theybeassignedonsomeotherbasis,forexampleproportionatelytotherelativefrequenciesoftheplaintextletters.[FR1] Theprimaryobjectofsubstitutionwithvariantsisagaintoprovideseveralvalueswhichmaybeemployedatrandominasimplesubstitutionofcipherequivalentsforthe

plaintextletters. Asaslightdiversion,thereadermayaskaboutuniliteralsubstitutionwithvariants.Itisbutnotverypractical.NotethefollowingcipheralphabetconstructedinFrenchby CaptainRogerBaudouininreference[BAUD]: Plain a b c d e f g h i l m n o p q r s t u v x z Cipher L G O R F Q A H C M B T I D N P U S Y E W J K V (NotethattheCaptainwasnotanACAmember.TheH=Hcombinationisnotallowed.) BaudouinproposedthattheJandYplainbereplacedbyIplainandKplainbyCplainorQplainandWplainbyVVplain.Fourcipherletterswouldbeavailableas variantsforthehighfrequencyplaintextlettersinFrench. MixedalphabetsformedbyincludingallrepeatedlettersofthekeywordorkeyphraseintheciphercomponentwerecommoninEdgarAllenPoe'sdaybutare impracticalbecausetheyareambiguous,makingdeciphermentdifficultforexample: EncipheringAlphabet: Plain a b c d e f g h i j k l m n o p q r s t u v w x y z Cipher N O W I S T H E T I M E F O R A L L G O O D M E N T Inverseformfordeciphering: Cipher A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Plain p v h m s g d l x j q k a b r w y n t u TheaveragecipherclerkwouldhavedifficultyindecryptingaciphergroupsuchasTOOET,eachletterhaving3ormoreequivalents,fromwhichplaintextfragments(n) inth,ftthi(s),itthi,etc.canbeformedondecipherment.[FR1] o e f i z c X Z

THEORETICALDISTINCTIONSInsimpleorsingleequivalentmonoalphabeticsubstitutionwithvariants,twopointsareevident: 1)thesameletteroftheplaintextisinvariablyrepresentedbybutoneandalwaysthesamecharacterorcipherunitofthecryptogram. 2)Thesamecharacterorcipherunitofthecryptograminvariablyrepresentsoneandalwaysthesameletteroftheplaintext. Inmultiliteralequivalentmonoalphabeticsubstitutionwithvariants,twopointsarealsoevident: 1)thesameletteroftheplaintextmayberepresentedbyoneormoredifferentcharactersorcipherunitsofthecryptogram.But, 2)Thesamecharacterorcipherunitofthecryptogramneverthelessinvariablyrepresentsoneandalwaysthesameletteroftheplaintext.

SIMPLETYPESOFCIPHERALPHABETSWITHVARIANTSFigure41 6 7 8 9 0 1 2 3 4 5 * * * * * * * * 6 1 * A B C D E 7 2 * F G H IJ K 8 3 * L M N O P 9 4 * Q R S T U 0 5 * V W X Y Z Figure42 V W X Y Z

V W X Y Z Q R S T U * * * * * * * * * L F A * A B C D E M G B * F G H IJ K N H C * L M N O P O I D * Q R S T U P K E * V W X Y Z Figure43 A E I O U * * * * * * T N H B * A B C D E V P J C * F G H IJ K W Q K D * L M N O P X R L F * Q R S T U Z S M G * V W X Y Z Figure44 V W X Y Z Q R S T U L M N O P F G H I K A B C D E * * * * * * V Q L F A * A B C D E W R M G B * F G H IJ K X N S H C * L M N O P Y T O I D * Q R S T U Z U P K E * V W X Y Z Figure45 O M N J K L F G H I A B C D E * * * * * * O M J F A * E N A L U N K G B * T R S F W L H C * O IJ H Y X I D * D C M V K E * P G B Q Z Figure46 Z W X Y S T U V N O P Q R * * * * * * M J F A * E N A L U K G B * T R S F W L H C * O IJ H Y X

I D * D C M V K E * P G B Q Z Figure47 1 2 3 4 5 6 7 8 9 0 * * * * * * * * * * * 7 4 1 * A B C D E F G H I J 8 5 2 * K L M N O P Q R S T 9 6 3 * U V W X Y Z . , : Figure48 1 2 3 4 5 6 7 8 9 * * * * * * * * * * 7 4 1 * A B C D E F G H I 8 5 2 * J K L M N O P Q R 9 6 3 * S T U V W X Y Z * Figure49 1 2 3 4 5 6 7 8 9 * * * * * * * * * * 5 1 * A B C D E F G H I 6 2 * J K L M N O P Q R 7 3 * S T U V W X Y Z 1 8 4 * 2 3 4 5 6 7 8 9 0 Figure410 1 2 3 4 5 6 7 8 9 * * * * * * * * * * 0 8 5 1 * T E R M I N A L S 9 6 2 * BC D F G H K J K 7 3 * P Q U V W X Y Z 1 4 * 2 3 4 5 6 7 8 9 0 ThematricesinFigures41to410representsomeofthesimplermeansforaccomplishingmonoalphabeticsubstitutionwithvariants.Thematricesareextensionsofthe basicideasofmultiliteralsubstitutionpresentedinLecture3. Thevariantequivalentsforanyplaintextlettermaybechosenatwillthus,inFigure41,e=10,15,60,or65inFigure42,e=AU,AZ,FU,FZ,LUorLZ. EnciphermentbymeansofmatricesshowninFigures42,43,46iscommutative.Thecoordinatesmaybereadrowbycolumnorvisaversa.Thereisnocryptographic ambiguity.Theremainingmatricesarenoncommutative.Thegeneralconventionistoreadrowbycolumn. InFigures45and46,thelettersinthesquarehavebeeninscribedinsuchamannerthat,coupledwiththeparticulararrangementoftherowandcolumncoordinates,the numberofvariantsavailableforeachplaintextletterisroughlyproportionaltothefrequenciesofthelettersintheplaintext.Figure35incorporatesakeywordontopof thisidea.[FR1]

HOMOPHONICTheHomophonicCipherisasimplevariantsystem.Itisa4level(alphabets)dinomecipher.ConsiderFigure411. Figure411 A B C D E F G H IJ K L M N 08 09 10 11 12 13 14 15 16 17 18 19 20 35 36 37 38 39 40 41 42 43 44 45 46 47 68 69 70 71 72 73 74 75 51 52 53 54 55 87 88 89 90 91 92 93 94 95 96 97 98 99 O P Q R S T U V W X Y Z 21 22 23 24 25 01 02 03 04 05 06 07

48 49 50 26 27 28 29 30 31 32 33 34 56 57 58 59 60 61 62 63 64 65 66 67 00 76 77 78 79 80 81 82 83 84 85 86 ThekeywordTRIPisfoundbyinspectingdinomes01,26,51,and76.(Thelowestnumberineachofthefoursequences.)[FR1][FR5] TheRussiansaddedaninterestinggimmickcalledtheDisruptionArea.ConsiderFigure412andnotetheslashesunderUXforthefourthlevelofdinomes.Thefamous VICcipherusedthisfeatureveryeffectively.[NIC4] Figure412 A B C D E F G H I J K L M 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 58 59 60 61 62 63 64 65 66 67 68 69 70 81 82 83 84 85 86 87 88 89 90 91 92 93 N O P Q R S T U V W X Y Z 01 02 03 04 05 06 07 08 09 10 11 12 13 40 41 42 43 44 45 46 47 48 49 50 51 52 71 72 73 74 75 76 77 78 53 54 55 56 57 94 95 96 97 98 99 00 // // // // 79 80 ThekeywordNAVYisrepresentedbydinomes01,27,53,and79. SecurityforHomophonicsystemsisgreatlyimprovedifthedinomesandthefoursequencesareassignedrandomly.However,theeasymnemonicfeatureofthe keywordedfoursequencesislost. TheMexicanCipherdeviceisaHomophonicconsistingoffiveconcentricdisks,theouterdiskbearing26lettersandtheotherfourbearingsequences0126,2752,53 78,7900.Thecipherdiskenhancesfrequentkeychanges.Figure412showsthematrixwithoutthedisruptionarea.[FR5][NIC4]

HOMOPHONICCRYPTANALYSISLetssolvethefollowingcryptogram.6832109022480576511188648420364523509144 0576422684002255700397357140748252440768 5105893074921884726409328042550618679882 8514445886325745513656019457227684468350 4521971649905286510611886440448966970553 1849106985485793368450957706120979529148 5610908546620626550932800325689721644282 3403184989685645378912530774016849438544 1136887616569052071058864674722249009136 6285124551351801423050886440840623112876 0557958980295039971332720364338268904516 5226321175064457225568951869577609567215 53049085679730

AssumingwedidnotknowthattheabovecryptogramwasaHOMOPHONIC,wemightmakeapreliminaryanalysistoseeifwearedealingwithacipheroracode. Wewillcovercodesystemslaterinthecourse,butafewintroductoryremarksmightbeinorder.Thefivelettergroupscouldindicateeitheracipheroracode. Ifthecryptogramcontainsanevennumberofdigits,asforexample494inthepreviousmessage,thisleavesopenthepossibilitythatthemessageisaciphercontaining 247pairsofdigitswerethenumberofdigitsanexactoddmultipleoffive,suchas125,135,etc.,thepossibilitythatthecryptogramisincodeofthe5figuregrouptype mustbeconsidered. Wenextstudythemessagerepetitionsandwhattheircharacteristicsare.Iftheciphertextisof5figurecodetype,thensuchrepetitionsasappearshouldgenerallybein wholegroupsoffivedigits,andtheyshouldbevisibleinthetextjustasthemessagestands,unlessthecodemessagehasbeensuperenciphered.Ifthecryptogramisa cipher,thenrepetitionsshouldextendbeyondthe5digitgroupingsiftheyconformtoanydefiniteatalltheyshouldforthemostpartcontainevennumbersofdigitssince eachletterisprobablyrepresentedbyapair(dinome)ofdigits. Westartwith4partfrequencydistribution.Wenextassumea25characteralphabetfrom0100.Thisisthecommonschemeofdrawingupthealphabets.Breakingthe textintodinomes(2digit)pairsyields: 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 /// /// /// / ///// ////// /// / / / ////// / //// //// ///// /// / / / ///// / / / / /// /// //// ////// / 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 19 20 21 22 23 24 25 / // ///// // / 44 45 46 47 48 49 50 ////// ////// /// /// ///// /////

51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 ///// ///// /// ////// / / //// ///// ////// // /// //// / ////// ////// /// // ////// / 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 //// ///// ////// /// / /

69 70 71 72 73 74 75 / /// / //// ////// / //// / // 94 95 96 97 98 99 00 /

// /////// //

Whatwehavebeforeusarefoursimple,monoalphabeticfrequencydistributionssimilartothoseinvolvedinamonoalphabeticsubstitutioncipherusingstandardcipher alphabets.Thenextstepistofitthedistributiontothenormal.SinceI=Jforthe25letteralphabet,wefindthattheKeywordisJUNEandthefollowingalphabetsresult: 01IJ26U51N76E 02K27V52O77F 03L28W53P78G 04M29X54Q79H 05N30Y55R80IJ 06O31Z56S81K 07P32A57T82L 08Q33B58U83M 09R34C59V84N 10S35D60W85O 11T36E61X86P 12U37F62Y87Q 13V38G63Z88R 14W39H64A89S 15X40IJ65B90T 16Y41K66C91U 17Z42L67D92V 18A43M68E93W 19B44N69F94X 20C45O70G95Y 21D46P71H96Z 22E47Q72IJ97A 23F48R73K98B 24G49S74L99C 25H50T75M00D

Thefirstgroupsofthecryptogramdecipherasfollows: 68 32 10 90 22 48 05 76 51 11 88 64 84 20 36 45 23 e a s t e r n e n t r a n c e o f

Ifa26elementalphabetwereusedonlythedistributionanalysiswouldhavebeenchangedtobeonabasisof26,theprocessoffittingthedistributiontothenormal wouldbethesame.

PLAINCOMPONENTCOMPLETIONMETHODSupposeweknowthattwocorrespondentshavebeenusingthesamevariantsystemasinthepreviousHomophonic.Themessageinterceptedis:

4822688423520999360476059056513668352267 971145446676

971145446676

Avariationoftheplaincomponentcompletionmethodcanbeusedtocrackthenewmessage.Wecopythemessageintodinomesandseparatebylevels. 48 22 68 84 23 52 09 99 36 04 76 05 90 56 51 36 68 35 22 67 97 11 45 44 66 76 2 1 3 4 1 3 1 4 2 1 4 1 4 3 3 2 3 2 1 3 4 1 2 2 3 4 Levels:(1)22230904052211 (2)483636354544 (3)68525651686766 (4)849976909776

Thesedinomesareconvertedintotermsofplaincomponentbysettingeachoftheciphersequencesagainsttheplaincomponentatanarbitrarypointofcoincidence,such asthefollowing: A B C D E F G H IJ K L M N O P Q R S T U V W X Y Z 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 So: Levels:(1)22=W23=X09=I04=D05=E22=W11=L (2)48=X36=L36=L35=K45=U44=T (3)68=S52=B56=F51=A68=S67=R66=Q (4)84=I99=Y76=A90=P97=W76=A

Thismethodworksbecauseboththeplaincomponent(A,B..)andtheciphercomponent(01,02..)areknownsequences. TheplaincomponentsequenceiscompletedonthelettersofthefourlevelsbyCaesarRundown,asfollows: Level1 Level2 Level3 SBFASRQ Level4 IYAPWA KZBQXB LACRYC MBDSZD ODFUBF QFHWDH RGIXEI SHKYFK WXIDEWL XLLKUT YZLFGYN ABNHIAP BCOIKBQ CDPKLCR EFRMNET FGSNOFU

XYKEFXM YMMLVU TCGBTSR ZNNMWV UDHCUTS BPPOYX CQQPZY DRRQAZ FTTSCB GUUTDC ZAMGHZO AOONXW VEIDVUT XGLFXWV ZINHZYZ AKOIAZY BLPKBAZ

WFKEWVU NCETAE YHMGYXW PEGVCG

DEQLMDS ESSRBA

GHTOPGV HVVUED HIUPQHW IWWVFE IKVQRIX LMXSTLZ

CMQLCBA TILZGL DNRMDCB UKMAHM VLNBIN WMOCKO XNPDLP YOQEMQ ZPRFNR AQSGOS BRTHPT

KXXWGF EOSNEDC MZZYIH GQUPGFE HRVQHGF ISWRIHG KTXSKIH

KLWRSKY LYYXHG FPTOFED MNYTUMA NAAZKI NOZUVNB OBBALK OPAVWOC PCCBML QRCXYQE REEDON RSDYZRF STEZASG TUFABTH SFFEPO TGGFQP

PQBWXPD QDDCNM LUYTLKI

MVZUMLK CSUIQU NWAVNML DTVKRV OXBWONM EUWLSW FVXMTX

UHHGRQ PYCXPON

UVGBCUI

VIIHSR

QZDYQPO RAEZRQP

GWYNUY HXZOVZ

VWHCDVK WKKITS

Thegeneratriceswiththebestassortmentofhighfrequencylettersforthefourlevelsare:Level1 Level2 Level3 Level4

EFRMNETREEDONEOSNEDCNCETAE

Arrangingthelettersofthesegeneratricesinorderofappearanceoftheirdinomeequivalents,accordingtolevelswehave: 48 22 68 84 23 52 09 99 36 04 76 05 90 56 51 36 68 35 22 67 97 E F R M N E A R E E D

E

O

S N

E

D

N

C

E

T

Theplaintextreads"Reinforcementsneededa[tonce]". Lookingattheequivalents01,26,51,76werevealthekeywordJUNE. Inevaluatinggeneratrices,thesumofthearithmeticfrequenciesofthelettersineachrowmaybeusedasanindicationoftherelative"goodness".Astatisticallybetter procedureusesthelogarithmoftheprobabilitiesoftheplaintextlettersformingthegeneratrices.See[FR2] TheHomophonicisapopularcipherandhasbeendiscussedinseveralissuesofTheCryptogramaswellasLEDGES'NOVICENOTES.Seereferences[HOM1 HOM6]and[LEDG]. Forourcomputerbugs,TATTERSHomophonicsolverisveryeasytouseandavailableontheCryptoDropBox.

MORECOMPLICATEDTYPESOFCIPHERALPHABETSWITHVARIANTS GRANDPREConsidertheciphermatricesshowninfigures411to413.Thesearecalledfrequentialmatrices,sincethenumberofciphervaluesavailableforanygivenplaintextletter closelyapproximatesitsrelativeplaintextfrequency. Figure411 A B C D E A * * T G A U R B * * S L I E Y C * * C N D O M D * * R A P T F E * * N T X N E V * * N O A T E W * * I H R O Q X * * O I E T A Y * * F T L O S Z * * I S N D R (676cellmatrix) Infigure411,thenumberofoccurrencesofaparticularletterwithinthematrixisproportionaltothefrequencyinplaintextthelettersareinscribedinrandommanner,in ordertoenhancethesecurityofthesystem. Figure412 6 8 9 1 5 4 3 7 2 0 7 * * A A A C D E E I L N 1 * * A A C D E E H K N O 3 * * A B D E E H J N O R V W X Y Z I E C A P F R N S T E L T I H O Y S O V C E R E D A L E Z H E T R T B C N P E S A M T I U I E D O N

8 * * A D E E H I N O R S 9 * * C E E G I N O R S T 2 * * E E F I M O Q S T T 0 * * E F I M O P R T T U 5 * * F I L N P R S T U X 6 * * I L N P R S T U W Y 4 * * L N O R S T T V Y Z Infigure412,thesameideaas411ispresentedinreducedformfrom26x26to10x10.Thelettershavebeeninscribedbyasimplediagonalroute,fromlefttoright, withinthesquare,andthecoordinatesscrambledbymeansofakeywordorkeynumber. Figure413 "Grandpre" 0 1 2 3 4 5 6 7 8 9 0 * * E N T R U C K I N G 1 * * Q U A R A N T E E N 2 * * U N E X P E C T E D 3 * * I M P O S S I B L E 4 * * V I C T O R I O U S 5 * * A D J U D I C A T E 6 * * L A B O R A T O R Y 7 * * E I G H T E E N T H 8 * * N A T U R A L I Z E 9 * * T W E N T Y F I V E Figure413illustratesthefamousGrandpreCipherinthissquaretenwordsareinscribedcontainingallthelettersofthealphabetandlinkedbyacolumnkeyword ("equivalent")asamnemonicforinscriptionoftherowwords.ACAliteraturealsocoversthiscipher.Seereferences[LEDG]and[GRA13]forsolutionhintsforthe Grandprecipher.

SACCOGeneralLuigiSaccoproposedafrequentialtypesystemthatusesbothencipheringanddecipheringmatrices.Theinscribeddinomeswerecompletelydisarrangedby applyingadoubletranspositiontosuppresstherelationshipsbetweenletters.References[SACC]and[FR1]bothgiveagooddescriptionoftheprocess.Thenumberof variantvaluesinthissystemarereflectiveoftheItalianlanguage.

BACONIANTheBaconianciphersfoundintheCryptogramareavariantsystem.The"a"elementsmayberepresentedbyanyoneof20consonantsasvariants,whilethe"b"elements mayberepresentedbyanyoneof6vowelsorthelettersAMmaybeusedtorepresentthe"a"elementsandthelettersNZforthe"b"elementsdigitsmaybeusedfor eitherthe"a"or"b"elements,eitheronthebasisoffirstfiveorlastfivedigits,oroddversusevendigits,orthefirst10consonants(BM)andthelast10consonants(NZ)

SUMMINGTRINOMEFriedmandescribesacomplexvariantknownasthesummingtrinomesystem.Eachplaintextletterisassignedavaluefrom126thisvalueisexpressedasatrinome,the digitsofwhichsumtothedesignatedvalueoftheletter.Theletterassignedthevalueof4mayberepresentedbyanyof15permutationsandcombinations.Friedman discussesfurtherwaysofcomplicationincludingdisarrangement,additionofpunctuationandnulls.See[FR1]pages109110.Notetheinvertednormaldistribution representationofthiscipher.

ANALYSISOFASIMPLEVARIANTEXAMPLEThefollowingcryptogramisavailableforstudy: QMDCV PLFNF DHNWJ WLKDK NHBPV PQKJR PVFLM RLTVM BKLWD WVHVK SHBCL WGTJH QKXFN ZVFDM LTBPL

VWSML KGCNR LRNKV MGFXW JRGMV DCNWN HBCVZ NMLWQ FDHDW VZBRV KLCVC VRDHL RVTLF DTSCB CLRXR CLZLR LMVTS DCNKV PBTNT NCDKG MXWXM ZNKBW VPBRN GHJZL FQFVK

BWDZX PNHSP PQRNZ LZVTV HLZLM FNCBZ LZDTB FDFVR

GHLKL GCLPQ

FVZLT

VMLKD

MNTGM NZVFX KSFDC PNCDW VRJTN RJKVM SFQWB FLPJM DHJCX

VWNPV PDZDW JPNWL DKVWG ZSHBH

XMDTS MGFDR DKLWJ

NotethetotalabsenceofA,E,I,O,U,andY.Remarkableanddefinitelynonrandomevent.Sinceauniliteralsubstitutionalphabetwith6lettersmissingishighlyunlikely, thenextguessiswearedealingwithamultiliteralsubstitution.Closerinspectionshowsthattenconsonantsareinitials(BDGJLNQSVX)andtheremainingten consonantsareusedasterminals(CFHKMPRTWZ).Thisimpliesbothbipartiteandbiliteralcharacter. Weconstructadigraphicdistribution: C F H K M P RTW Z B * 3 1 1 1 1 2 2 1 2 1 D * 4 1 3 3 1 1 1 3 4 2 G * 2 2 2 0 3 0 0 1 0 1 J * 1 1 1 1 1 1 2 1 1 1 L * 1 4 0 4 3 4 5 3 3 4 N * 4 1 4 3 1 1 1 2 3 3 Q * 0 2 0 2 1 1 1 0 1 1 S * 1 2 2 0 2 1 0 0 0 1 V * 1 4 1 3 4 4 4 3 4 3 X * 0 1 0 1 2 1 1 0 2 0 Weassumetheuseofasmallencipheringmatrixwithvariantsforrowsandcolumns.Weassumethatthevariouspossibleciphervariantsareofapproximatelyequal frequencythecolumnindicatorspairequallyoftenwiththerowindicatorsoftheencipheringmatrix.Welookforsimilarrowprofilesandcolumnprofiles.Wematchfirst therowsandthenthecolumns. RowLandVdistributionshavepronouncedsimilarities.Theyare"heavy"intheirfrequencydistributionsinthesameplaces.SoarerowsDandN.Theyhave homologousattributesinappearance. C F H K M P RTW Z L * 1 4 0 4 3 4 5 3 3 4 V * 1 4 1 3 4 4 4 3 4 3 D * 4 1 3 3 1 1 1 3 4 2 N * 4 1 4 3 1 1 1 2 3 3 Findingthenextrowsarenotobvious.Weusea"goodnessofmatch"proceduretoequateinterchangeablevariants.Wecalculatethecrossproductsumsforeachtrial. ThenextheavyrowisG.WetestGagainsttheremainingrows. G B * 2 2 2 0 3 0 0 1 0 1 * 3 1 1 1 1 2 2 1 2 1

G*B * 6 2 2 0 3 0 0 1 0 1 =15 Wecomparethebalanceofrows: G*B + 6 2 2 0 3 0 0 1 0 1 =15 G*J + 2 2 2 0 3 0 0 1 0 1 =11 G*Q + 0 4 0 0 3 0 0 0 0 0 =7 G*S + 2 4 4 0 6 0 0 0 0 1 =17 G*X + 0 2 0 0 0 6 0 0 0 0 =8 TheresultsaremostprobablymatchGandS. ThenextheaviestrowisB.Testingagainsttheremainingthreerowswehave: B*J + 3 1 1 1 1 2 4 1 2 1 =17 B*Q + 0 2 0 2 1 2 2 0 2 1 =12 B*X + 0 1 0 1 2 2 2 0 4 0 =12

ThecorrectpairingsareBwithJandQwithX.Sincewehavenotfoundmorethantworowsforanyonesetofinterchangeablevaluestheoriginalmatrixhasonlyfive rows. C F H K M P RTW Z B J 4 2 2 2 2 3 4 2 3 2 D N 8 2 8 7 2 2 2 5 7 5 G S 3 4 4 0 5 1 0 1 0 2 L V 2 8 1 7 7 8 9 6 7 7 Q X 0 3 0 3 3 2 2 0 3 0 Valuesrepresentthesumsofthecombinedrows.Weapplythesameprocesstomatchingcolumns.CandHareamatchedpair.FwithMandPwithR.Weusethe crossproductsumsforthebalanceofthecolumns. K*T+ 4 35 42 K*Z+ 4 35 49 T*W+ 6 35 42 T*Z+ W*Z+ 6 35 49 Combinations: KT,WZ+ 81 + 90 = 171 KW,TZ+ 111 + 73 = 186 KT,TW+ 88 + 83 = 171 WewouldexpectthattheproperpairingsareKwithWandTwithZ. C F K P T H M W R Z B J 6 4 5 7 4 PHI(p)= 1962 D N 16 4 14 4 10 PHI(r)= 1132 G S 7 9 Q X 1 3 PHI(o)= 1670 4 L V 3 15 14 17 13 6 6 81 88 83 90

K*W+ 4 49 49 9 111

4 25 2 42 73

Weconvertthemultiliteraltexttouniliteralequivalentsusinganarbitrarysquareforreductiontoplaintext. C F K P T H M W R Z B J A B C D E D N F G H IJ K G S L M N O P L V Q R S T U Q X V W X Y Z TheconvertedcryptogramissolvedviatheprincipalsinLecture2andLecture3.ThebeginningofthemessagereadsWeatherforecast.Theoriginalkeyingmatrixis recoveredwithakeywordofATMOSPHERIC. C F K P T H M W R Z B J A T M O S D N P H E R I G S C B D F G L V K L N Q U Q X V W X Y Z Themethodofmatchingrowsandcolumnsappliesequallywellforallthematricesshownpreviously.Itiskeytostartwiththebestrowsandcolumnsfromnotonly heavinessstandpointbutthedistinctivecrestsandtroughs.Asecondkeyisthelowfrequencyletters.Novariantsystemcanadequatelydisguiselowfrequencylettersand

theywillhavethesamefrequencyintheciphertext.Friedmandescribesamoregeneralsolutiontovariantanalysis.[FRE1,p119ff] Chapter10ofreference[FRE1]coversthedisruptionprocessassociatedwithmonomedinomealphabetsofIrregularLengthciphertextunits.Figures414andFigure 415showencipheringmatriceswheretheenciphermentisdisruptedandcommutative.Thenormalrowconventionsareusedtoencipherexceptwhentherowindicator wasthesamefortheimmediatelyprecedingletter.InFigure414,EIGHTcouldbeencrypted10297849andthenrearrangedintostandardgroupsof5letters (numbers).InFigure415,E=24or42,T=621or162.Figure416isanexampleoftheRussiandisruptionprocessaddedforsecurity.

ISOLOGSCryptogramsproducedusingidenticalplaintextbutsubjectedtodifferentcryptographictreatment,andyieldingdifferentciphertextsarecalledisologs.(isos=equaland logos=wordinGreek).Isologsareusuallyequalornearlyequalinlength.Isologs,nomatterhowthecryptographictreatmentvaries,areamongthemostpowerfultools availabletothecryptanalysttosolvedifficultcryptosystems. TaketwomessagesAandBsuspectedofbeingisologsandwritethemoutundereachother.Wethenexaminethesimilaritiesanddifferences.Assumethemessagesboth startwith"Referenceyourmessage..."Iwillarrangethemessagesinaspecialtabletofacilitatethestudy. GroupNo. 510 A822656310374839698423252970115 A'301508749714511973604967650106 B802778910694000138285408240065 B'456479918169672538894156325203 C636293391843158810482645845039 C'906287753620351105708927775011 D817135253873309207496175216476 D'351999013899974502320411589216 E387289114799926414681336533881 E'384631754714648006468586453898 F896979381651750570741180443255 F'261218387894889337281127220504 G281202773031199799622786560653 G'064843210398715426628076089880 H908704086746594198551082222987 H'441055290059728228558730070893 J4672936245 J'5968246253

15

Thedinomedistributionsforthesetwomessagesareasfollows: MessageA 1 2 3 4 5 6 7 8 9 0 1 * 2 1 1 1 2 1 1 1 2 2 * 1 1 1 1 2 2 2 1 3 * 2 2 1 1 5 2 2 4 * 1 1 1 1 2 3 1 1 1 5 * 1 1 2 1 2 2 1 1 6 * 1 3 1 2 1 1 1 1 7 * 1 2 1 2 2 1 1 1 1 1 8 * 2 2 1 1 1 2 1 2 2 9 * 1 2 2 1 1 2 2 2 1 0 * 2 1 1 1 1 2 1 2 2 MessageB 1 2 3 4 5 6 7 8 9 0 1 * 4 1 2 1 1 1 2 1 2 * 1 1 1 1 2 2 2 1 1 3 * 1 2 2 1 1 5 2 4 * 1 1 1 3 2 1 1 1 5 * 1 1 1 1 2 1 1 2 1 6 * 3 2 1 2 1 1 1 7 * 1 1 1 1 2 1 1 1 1 1 8 * 1 1 1 1 2 1 2 3 9 * 1 1 2 1 2 3 2 1 0 * 2 1 2 2 2 3 1 3 1

Bothdistributionsaretooflatnocrestsortroughs.Weassumeavariantsystemofamonoalphabeticcryptosystem.[FRE3]showsushowtouseaPoissonexponential distributiontoevaluaterandomtext.Thegistofthestatisticsisthattheexpectednumberofblanksistoolow.Thechitestindicatesextremenonrandomnessforboth messages.Thechitestappliedtobothdistributionsimpliesthattheybothhavebeenencipheredbythesamecryptosystembecausethereexistsaclosecorrelation betweenthepatternsofthetwodistributions.[FR1,p123}discussesthepotentialitiesofthecryptomathematicsasasupportingsciencetocryptography. Thereareseveralidenticalvaluesbetweenthemessages.Thisimpliesthatnotonlyhasthesamecryptosystembeenusedbutalsothesameencipheringmatrix.Thevalues 38and62mustrepresentverylowfrequencylettersbecausenovariantsareevenprovidedforthisletter. Wenowformisologchainsbetweenthemessages.

(06141526283135737481899899) (0207202243446390) (1237485169708394) (03304154658297) (05102432498793) (16183676787986) (274553648092) (11397588) (21587784) (46596872) (005267) (045561) (082956) (197196) (0125) (1385)SingleDinomes: (4260)(38)(47)(50)(62)(91)

Thesechainsofciphervaluesrepresentidenticalplaintextpairs.Beginningwiththefirstvalueinthemessage82and30apartialchainofequivalentvariantsisformed nowlocatingtheotheroccurrencesofeithervaluewenotethevaluethatcoincideswithitintheothermessage.Wethereforeextendthechain.Theplaintextvaluesare arbitrarilyfitinto10x10square:1234567890 ................... 1.DNHEEAACO 2.ITOMESEFT 3.EOEANBDR 4.RYTTSLVNO 5.NUSRPFILX 6.PWTSRULNY 7.CLEEDAIAAN 8.ERNIHAODES 9.GSONCREET 0.MTRPOETFU

Manipulatingtherowsandcolumnswithaviewtouncoveringthekeysorsymmetry,wefindalatentdiagonalpatternwithoutkeyword.Wesetupthefollowing encipheringmatrix:

6891543720 ................... 7.AAACDEEILN 1.AACDEEHKNO 3.ABDEEHJNOR 8.ADEEHINORS 9.CEEGINORST 2.EEFIMOQSTT 0.EFIMOPRTTU 5.FILNPRSTUX 6.ILNPRSTUWY 4.LNORSTTVYZ

Icannotoveremphasizethevalueofisologs.Thevaluegoesfarbeyondsimplevariantsystems.Isologsproducedbytwodifferentcodebooksortwodifferent encipheredcodeversionsofthesameplaintextortwoencryptionsofidenticalplaintextatdifferentsettingsofaciphermachine,mayallproveofinestimablevalueinthe attackonadifficultsystem.

SYNOPTICCHARTOFCRYPTOGRAPHYPRESENTEDINLECTURES15

Cryptograms . . CipherCodeEncipheredCode . . SubstitutionTranspositionCombined .Substitution .Transposition . . MonoalphabeticMultiplePolyalphabetic .Alphabetic

.Systems . . Uniliteral.........................Multiliteral .. .. .. Standard...Mixed. AlphabetsAlphabets. .. .. Keyword...Random. MixedMixed. . . . ............................... .. SingleEquivalentVariant........ .. .. ..................... ... FixedLengthMixedLength. CipherGroupsCipherGroups. ... ......................... Biliteral...Nliteral... MonomeDinomeOthers. . . . ................................... . . .......................... .. MatriceswithNonBipartite Coordinates (Bipartite)

Hereisthetentativeplanforthebalanceofthecourse.Justaplansubjecttorevision.

LECTURES57WewillcoverrecognitionandsolutionofXENOCRYPTS(languagesubstitutionciphers)indetail.

LECTURES812WewillinvestigateandcrackPolyalphabeticSubstitutionsystems.

LECTURES1318WewillinvestigateandcrackCipherExchangeandTranspositionsproblems.

LECTURE19WewilldevotethislecturetoInternationalLaw.

LECTURES2023WewillwalkthroughthemathematicalfieldstosolveCryptarithms.

LECTURES2425Wewillintroducemoderncryptographicsystemsandfieldspecialtopics.WewilldoaprimeronPGP. SOLUTIONSTOHOMEWORKPROBLEMSFROMLECTURE3 ThankstoJOEOforhisconcisesols.Mv1.FromMartinGardner.

8518519119913 16125112168125 2093315452081 209225145225 1819551425615 18513125252515 21311421195920 91425152118315 12211314 13118209147118414518 8514451819151422912125 141518208311815129141

IpresentedMv1inastrangeformat.Itfooledsomebutnotall.TheKeyis01=1=a,02=2=b,...26=z.thealphabetisstandard.Messagereads:"Here'sasimple alphabeticcodethatI'veneverseenbefore.Maybeyoucanuseitinyoucolumn.MartinGardner,Hendersonville,NorthCarolina. Solveandreconstructthecryptographicsystemsused.

Mv2. 0602100501010515220206082 3251008040221090804082211 0804171513142221022402012 2020201081906151708011122 1402011906051002021122140 6231905150122130205061302 0501100523062102221406020 2221406020226020605211902 0211220302172402190206150 5110602190506220105050119 0521152215050122051805060 60503

Dividetheoriginalcipherintopairs,notingthateachpairstartedwith0,1,or2andendedwith09.ConstructamatrixsimilartoFigure32.(3x10)Fillinthematrix withA=01,endingwithZ=26.Used00=blank.Reducebyconvertingdinomestoletters.ApplythePhitestandfoundmonalphabetic.Usedfrequency,VOCcount, andconsonantlinetoidentifyB,H,EasvowelsandN,D,X,C,I,Y,R,J,aspossibleconsonants.Markingthemessagewiththeseassumptions,foundlasteightcharacters tobeapatternwordinCryptodictasTOMORROW.Workingbetweenciphertextandkeyalphabetmatrix,restfell. Messagereads:ReconnoiterAuysCayesBayatdaylightseventeenAprilandthenproceedthroughpointGeorgeoncoursethreethreezerospeedtwelveperiodreport noonpositiontomorrow. Key=NEWYORK,3X10matrix,Rows0,1,2,columns09and00blank.Mv3.

5324154532244325124324231 5444545325143441415214115 4345352123351251142153334 5324423154545244324144432 1253244344241544452443352 1533313144415454451432515 2324155224431531331331455 3241345212533522434131245 4452334433223335334521352 4444445321513155224431531 2451131424443343152235242 5352133133123121314334533 1213444124433312143224333 1324512253512532335125114 4415454143244424134515221 2514512132445321251441513 1425242445

Notedallentrieswerenumbered15.Assumeda5x5matrixfilledwithastraightalphabet,substitutedlettersforthedinomes.Usedfrequencycount,contactcountand phitesttoconfirmmonoalphabeticity.Identified8consonantsand2vowels.MadetheE,Tassumptionbasedonfrequency.Firstworddroppedasweather.Restof messagefellapartwithadditionofW,A,Rtothematrix. Messagereads:WeatherforecastThursdaypartlycloudy...atpresentaboutonethousandfeet.

Key=Beginningcolumn1=MONDAY,in5x5matrix. Mylasttwoproblemsweretakenfromreference[OP20]course.

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