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Elementary Cryptanalysis
A Mathematical Approach
Second Edition
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Second Edition
©2009 by
The Mathematical Association of America (Incorporated)
Library of Congress Catalog Card Number 2009927623
Print ISBN 978-0-88385-647-5
Electronic ISBN 978-0-88385-937-7
Printed in the United States of America
Current Printing (last digit):
10 9 8 7 6 5 4 3 2 1
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Elementary Cryptanalysis
A Mathematical Approach
Second Edition
Abraham Sinkov
Revised andUpdatedby
Todd Feil
®
PublishedandDistributedby
TheMathematicalAssociationofAmerica
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ANNELILAXNEWMATHEMATICALLIBRARY
PUBLISHEDBY
THEMATHEMATICALASSOCIATIONOFAMERICA
EditorialBoard
HaroldP.Boas,Editor
SteveAbbott
MichaelE.Boardman
GailA.Kaplan
KatherineS.Socha
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ANNELILAXNEWMATHEMATICALLIBRARY
1. Numbers:RationalandIrrationalbyIvanNiven
2. WhatisCalculusAbout?byW.W.Sawyer
3. AnIntroductiontoInequalitiesbyE.F.BeckenbachandR.Bellman
4. GeometricInequalitiesbyN.D.Kazarinoff
5. The Contest Problem Book I Annual High School Mathematics Examinations
1950–1960.CompiledandwithsolutionsbyCharlesT.Salkind
6. TheLoreofLargeNumbersbyP.J.Davis
7. UsesofInfinitybyLeoZippin
8. GeometricTransformationsIbyI.M.Yaglom,translatedbyA.Shields
9. ContinuedFractionsbyCarlD.Olds
10. (cid:1) ReplacedbyNML-34
11. HungarianProblemBooksIandII,BasedontheEo¨tvo¨sCompetitions
12. 1894–1905and1906–1928,translatedbyE.Rapaport
13. EpisodesfromtheEarlyHistoryofMathematicsbyA.Aaboe
14. GroupsandTheirGraphsbyE.GrossmanandW.Magnus
15. TheMathematicsofChoicebyIvanNiven
16. FromPythagorastoEinsteinbyK.O.Friedrichs
17. The Contest Problem Book II AnnualHighSchoolMathematicsExaminations
1961–1965.CompiledandwithsolutionsbyCharlesT.Salkind
18. FirstConceptsofTopologybyW.G.ChinnandN.E.Steenrod
19. GeometryRevisitedbyH.S.M.CoxeterandS.L.Greitzer
20. InvitationtoNumberTheorybyOysteinOre
21. GeometricTransformationsIIbyI.M.Yaglom,translatedbyA.Shields
22. ElementaryCryptanalysis—AMathematicalApproach,SecondEditionby
A.Sinkov,revisedandupdatedbyToddFeil
23. IngenuityinMathematicsbyRossHonsberger
24. GeometricTransformationsIIIbyI.M.Yaglom,translatedbyA.Shenitzer
25. TheContestProblemBookIIIAnnualHighSchoolMathematicsExaminations
1966–1972.CompiledandwithsolutionsbyC.T.SalkindandJ.M.Earl
26. MathematicalMethodsinSciencebyGeorgePo´lya
27. International Mathematical Olympiads—1959–1977. Compiled and with
solutionsbyS.L.Greitzer
28. TheMathematicsofGamesandGambling,SecondEditionbyEdwardW.
Packel
29. TheContestProblemBookIVAnnualHighSchoolMathematicsExaminations
1973–1982. Compiled and with solutions by R. A. Artino, A. M. Gaglione,
andN.Shell
30. TheRoleofMathematicsinSciencebyM.M.SchifferandL.Bowden
31. InternationalMathematicalOlympiads1978–1985andfortysupplementary
problems.CompiledandwithsolutionsbyMurrayS.Klamkin
32. RiddlesoftheSphinxbyMartinGardner
33. U.S.A.MathematicalOlympiads1972–1986.Compiledandwithsolutions
byMurrayS.Klamkin
34. Graphs and Their Uses by Oystein Ore. Revised and updated by Robin J.
Wilson
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35. ExploringMathematicswithYourComputerbyArthurEngel
36. GameTheoryandStrategybyPhilipD.Straffin,Jr.
37. Episodes in Nineteenth and Twenthieth Century Euclidean Geometry by
RossHonsberger
38. TheContestProblemBookVAmericanHighSchoolMathematicsExaminations
andAmericanInvitationalMathematicsExaminations1983–1988.Compiledand
augmentedbyGeorgeBerzsenyiandStephenB.Maurer
39. OverandOverAgainbyGengzheChangandThomasW.Sederberg
40. TheContestProblemBookVIAmericanHighSchoolMathematicsExaminations
1989–1994.CompiledandaugmentedbyLeoJ.Schneider
41. The Geometry of Numbers by C. D. Olds, Anneli Lax, and Giuliana P.
Davidoff
42. Hungarian Problem Book III Based on the Eo¨tvo¨s Competitions 1929–1943
translatedbyAndyLiu
43. MathematicalMiniaturesbySvetoslavSavchevandTituAndreescu
44. GeometricTransformationsIVbyI.M.Yaglom,translatedbyA.Shenitzer
Othertitlesinpreparation.
Booksmaybeorderedfrom:
MAAServiceCenter
P.O.Box91112
Washington,DC20090-1112
1-800-331-1622 fax:301-206-9789
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Contents
PrefacetotheFirstEdition ix
PrefacetotheSecondEdition xiii
1 MonoalphabeticCiphersUsingAdditiveAlphabets 1
1.1 TheCaesarCipher. . . . . . . . . . . . . . . . . . . . . 1
1.2 Modulararithmetic . . . . . . . . . . . . . . . . . . . . 3
1.3 Additivealphabets. . . . . . . . . . . . . . . . . . . . . 6
1.4 Solutionofadditivealphabets . . . . . . . . . . . . . . . 9
1.5 Frequencyconsiderations . . . . . . . . . . . . . . . . . 12
1.6 Multiplications . . . . . . . . . . . . . . . . . . . . . . 17
1.7 Solutionofmultiplicativealphabets . . . . . . . . . . . 22
1.8 Affineciphers . . . . . . . . . . . . . . . . . . . . . . . 26
2 GeneralMonoalphabeticSubstitution 31
2.1 Mixedalphabets . . . . . . . . . . . . . . . . . . . . . . 31
2.2 Solutionofmixedalphabetciphers . . . . . . . . . . . . 34
2.3 Solutionoffive-lettergroupings . . . . . . . . . . . . . 40
2.4 Monoalphabetswithsymbols . . . . . . . . . . . . . . . 47
3 PolyalphabeticSubstitution 51
3.1 Polyalphabeticciphers . . . . . . . . . . . . . . . . . . 51
3.2 Recognitionofpolyalphabeticciphers . . . . . . . . . . 54
3.3 Determinationofnumberofalphabets . . . . . . . . . . 62
3.4 Solutionsofadditivesubalphabets . . . . . . . . . . . . 66
3.5 Mixedplainsequences . . . . . . . . . . . . . . . . . . 70
3.6 Matchingalphabets . . . . . . . . . . . . . . . . . . . . 73
3.7 Reductiontoamonoalphabet . . . . . . . . . . . . . . . 83
3.8 Mixedciphersequences . . . . . . . . . . . . . . . . . . 84
3.9 Generalcomments . . . . . . . . . . . . . . . . . . . . 99
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viii Contents
4 PolygraphicSystems 103
4.1 LinearTransformations . . . . . . . . . . . . . . . . . . 103
4.2 Multiplicationofmatrices—inverses . . . . . . . . . . . 110
4.3 Involutorytransformations . . . . . . . . . . . . . . . . 115
4.4 Recognitionofdigraphicciphers . . . . . . . . . . . . . 118
4.5 Solutionofalineartransformation . . . . . . . . . . . . 119
4.6 HowtomaketheHillSystemmoresecure . . . . . . . . 127
5 Transposition 129
5.1 Columnartransposition . . . . . . . . . . . . . . . . . . 129
5.2 Completelyfilledrectangles . . . . . . . . . . . . . . . 135
5.3 Incompletelyfilledrectangles . . . . . . . . . . . . . . . 138
5.4 Probablewordmethod . . . . . . . . . . . . . . . . . . 140
5.5 Generalcase . . . . . . . . . . . . . . . . . . . . . . . . 145
5.6 Identicallengthmessages . . . . . . . . . . . . . . . . . 153
6 RSAEncryption 159
6.1 Public-keyencryption . . . . . . . . . . . . . . . . . . . 159
6.2 TheRSAmethod . . . . . . . . . . . . . . . . . . . . . 160
6.3 CreatingtheRSAkeys . . . . . . . . . . . . . . . . . . 162
6.4 WhyRSAworks—Fermat’sLittleTheorem . . . . . . . 163
6.5 Computationalconsiderations. . . . . . . . . . . . . . . 166
6.6 MapleandMathematicaforRSA . . . . . . . . . . . . . 171
6.7 BreakingRSAandsignatures . . . . . . . . . . . . . . . 175
7 PerfectSecurity—One-timePads 179
7.1 One-timepads . . . . . . . . . . . . . . . . . . . . . . . 179
7.2 Pseudo-randomnumbergenerators . . . . . . . . . . . . 181
AppendixA:Tables 185
AppendixB:ASCIICodes 191
AppendixC:BinaryNumbers 193
SolutionstoExercises 197
FurtherReadings 205
Index 207
AbouttheAuthors 211
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Preface to the First Edition
Toaverygreatextent,mankindowesitsprogresstotheabilitytocommu-
nicate,andakeyaspectinthisabilityisthecapabilityofcommunicatingin
writing.Fromtheearliestdaysofwriting,therehavebeenoccasionswhen
individuals have desired to limit their information to a restricted group of
people.Theyhadsecretstheywantedtokeep.Tothisend,suchindividuals
developed ideas by means of which their communications could be made
unintelligibletothosewhohadnotbeenprovidedwiththespecialinforma-
tion needed for decipherment. The general techniques used to accomplish
such a purpose, i.e. the hiding of the meaning of messages, constitute the
studyknownascryptography.
Before the development of postal systems and electrical transmission
ofinformation,theusualmannerofsendingacommunicationwasbypri-
vatemessenger.Evenunderthesecircumstancestheuseoftheconcealment
tactics of cryptography was often advisable because of the possibility that
the messenger might be apprehended or prove disloyal. In recent times a
messagetransmittedbyradiocouldbecopiedbyanyonehavingappropri-
ateequipmentandlisteningtotherightfrequencyattherighttime.Insuch
a case, a sender desiring privacy of communication would be required to
employsomemethodofcryptographicconcealment.
Now, just as the sender of the message attempted to conceal informa-
tion from any but the desired recipient, there would be individuals very
muchinterestedindeterminingwhatthemessagesaid—mostprobablythe
veryindividualsfromwhomthesenderwastryingtokeeptheinformation.
Shouldsuchindividualsobtain—inonewayoranother—acopyofthecryp-
tographedmessage,theywouldattempttounravelthesecretitcarried.But
ofcoursetheirattemptwouldhavetobemadewithoutaknowledgeofthe
cryptographic details employed tohide the content. Efforts aimed at read-
ingasecretmessageinthiswaycomeundertheheadingofthestudycalled
cryptanalysis.
Historyaboundswithaccountsofsituationswheresuccessfulcryptanal-
ysis proved a most important element in achieving diplomatic successes,
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