Table Of ContentSPRINGER BRIEFS IN MATHEMATICS
Sara Díaz Cardell
Amparo Fúster-Sabater
Cryptography with
Shrinking Generators
Fundamentals
and Applications of
Keystream Sequence
Generators Based on
Irregular Decimation
123
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Sara Díaz Cardell • Amparo Fúster-Sabater
Cryptography with Shrinking
Generators
Fundamentals and Applications of Keystream
Sequence Generators Based on Irregular
Decimation
123
SaraDíazCardell AmparoFúster-Sabater
DepartmentofMathematics InstituteofPhysicalandInformation
InstituteofMathematics,Statisticsand Technologies(ITEFI)
ScientificComputing(IMECC) C.S.I.C.
UniversityofCampinas(UNICAMP) Madrid,Spain
Campinas,Brazil
ISSN2191-8198 ISSN2191-8201 (electronic)
SpringerBriefsinMathematics
ISBN978-3-030-12849-4 ISBN978-3-030-12850-0 (eBook)
https://doi.org/10.1007/978-3-030-12850-0
LibraryofCongressControlNumber:2019933100
MathematicsSubjectClassification:94A60,68Q80,94A55
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ToHenrique,Aida,Jose,Leticia,Fran,
GonzaloandJimena.
ToAngelMiguel,Isabel,Miguel,Irene
andLydia.
Preface
Cryptography is a hot research area which affects all of us whether or not we
are aware of it on a daily basis: connecting to the internet, unlocking a car door
with a remote control device, sending or receiving WhatsApp messages or paying
with a credit card. Other emerging applications, including e-health, the so-called
Internet of Things (IoT) and smart buildings, are making cryptography even more
ubiquitous.
Therearetwokindsofcryptosystems:symmetricandasymmetric.Asymmetric
systems use a public key to encrypt a message and a private key to decrypt it. In
symmetricsystems,thesamesecretkeyisusedtoencryptanddecryptamessage.
Furthermore, symmetric key ciphers are implemented as either block ciphers or
stream ciphers (even though sponge-based constructions are also emerging). After
theirruptionofpublic-keycryptography,withitsnewandfascinatingpossibilities,
it seemed that secret-key cryptography had been confined to a few irrelevant
applications. Nothing is further from the truth, since there are many examples of
secret-key cryptography hardware devices and stream ciphers are simple and the
fastestamongallencryption/decryptionprocedures.
The ECRYPT stream cipher project, called eSTREAM, revitalized the field
of stream ciphers after the widespread deployment of AES and highlighted the
importance of stream ciphers in many technological areas. It is not surprising that
theconjunctionofsimplicityandspeedinasingleprocesspreservesitsleadingpart
inanyapplication.
This book addresses a particular class of stream ciphers known as irregular
decimation-based sequence generators. This family has given rise to a wide
collectionofdesigns,high-speedimplementationsandthesubsequentdevelopment
of cryptanalytic attacks. Our main purpose is to gather in a comprehensive survey
all the research literature on this topic. In particular, we shall focus on the four
mostimportantdecimation-basedgenerators:theshrinking,self-shrinking,modified
and generalized self-shrinkinggenerator. Moreover, parallelmodelling intermsof
linear cellular automata reveals the implicit linearity of such generators, which
were paradoxically designed and conceived to exhibit non-linearity. Indeed, these
vii
viii Preface
results confirm the subtle thought by J. L. Massey: “Linearity is the curse of the
cryptographer”(Crypto’89).
Our approach is based on traditional linear feedback shift registers (one bit per
LFSRstage),meaningthatthedecimationcriteriondependsonasinglebinarydigit.
Duetothenoveldesignofsuchregistersoverextendedfields(morethanonebitper
LFSR stage), new decimation criteria have to be designed. As a consequence, the
familyofdecimation-basedsequencegeneratorsmustberedefinedaccordingtothis
newscenario.Therefore,thenewchallengeistodesignsecureirregulardecimation-
basedgeneratorsoverextendedfields.
We have tried to make this text self-contained and accessible to graduate
or advanced undergraduate students, as well as engineers and enthusiasts. The
mathematicalbackgroundisnotverydemanding,andweclarifythemainconcepts
withmanyillustrativeexamples,tablesandfigures.
The authors wish to thank Professor Carlile Lavor for the opportunity to write
this book and Professor Henrique Sá Earp for many valuable corrections and
suggestions to the manuscript. We also thank the Springer team for their support
andencouragement,withspecialgratitudetooureditorsVinodhiniSrinivasanand
RobinsondosSantos.
Campinas,Brazil SaraDíazCardell
Madrid,Spain AmparoFúster-Sabater
February2018
Contents
1 IntroductiontoStreamCiphers............................................. 1
1.1 StreamCipher............................................................ 3
1.1.1 A Basic Structure in Stream Cipher: The Linear
FeedbackShiftRegister(LFSR)................................ 4
1.2 LFSR-BasedSequenceGenerators...................................... 7
1.2.1 Non-linearCombinationGenerators............................ 8
1.2.2 Non-linearFilters................................................ 10
1.2.3 Clock-ControlledGenerators.................................... 13
1.2.4 Decimation-BasedGenerators .................................. 15
1.2.5 DynamicLFSRGenerators...................................... 16
1.2.6 OtherTypesofKeystreamSequenceGenerators............... 18
1.3 eSTREAM................................................................ 22
2 KeystreamGeneratorsBasedonIrregularDecimation.................. 25
2.1 ShrinkingGenerator ..................................................... 26
2.1.1 DefinitionandBasicFeatures................................... 26
2.1.2 CharacteristicPolynomialandInterleavedPN-Sequences .... 27
2.1.3 ShrunkenSequencesandDifferenceEquations................ 29
2.1.4 ObtainingtheSecondPN-SequencefromtheShrunken
Sequence ......................................................... 30
2.1.5 ObtainingtheFirstPN-SequencefromtheShrunken
Sequence ......................................................... 31
2.2 Self-ShrinkingGenerator................................................ 33
2.3 ModifiedSelf-ShrinkingGenerator ..................................... 34
2.4 GeneralizedSelf-ShrinkingGenerator.................................. 35
2.4.1 DefinitionandFeatures.......................................... 35
2.4.2 GeneralizedSelf-ShrunkenSequencesandDifference
Equations......................................................... 38
2.4.3 RelationshipwiththeModifiedSelf-ShrinkingGenerator..... 40
ix
x Contents
3 ModellingThroughLinearCellularAutomata............................ 45
3.1 TheConceptofCellularAutomaton.................................... 45
3.2 ModellingaPN-Sequence............................................... 48
3.2.1 Cattell–MuzioAlgorithm........................................ 48
3.2.2 OtherCAsthatGeneratePN-Sequences........................ 53
3.3 ModellingtheShrinkingGenerator..................................... 55
3.3.1 TheFúster–CaballeroAlgorithm................................ 55
3.3.2 OtherCAsthatGeneratetheShrunkenSequence.............. 57
3.3.3 ComparisonofbothFamilies ................................... 60
3.4 ModellingtheGeneralizedSelf-ShrinkingGenerator.................. 60
3.4.1 Characterizationofthe150/90-CA ............................. 60
3.4.2 Characterizationofthe102-CA................................. 61
4 Cryptanalysis.................................................................. 65
4.1 AnAlgebraicAttack:StrategicBitsoftheShrinkingGenerator...... 65
4.1.1 TheShrunkenSequenceasanInterleavedSequence .......... 66
4.1.2 ComputationoftheR InitialState............................. 68
2
4.1.3 ComputationoftheR InitialState............................. 69
1
4.1.4 AnIllustrativeExample ......................................... 70
4.2 LinearConsistencyTestAgainsttheShrinkingGenerator............. 73
4.3 102-CARecoveryAttackAgainsttheShrinkingGenerator........... 79
4.3.1 OverlappingAttack.............................................. 79
4.3.2 CA-BasedCryptanalysisAgainsttheShrinkingGenerator.... 80
4.4 DifferentAttacksAgainsttheShrinkingGenerator:
AComparison............................................................ 83
4.5 AProbabilisticAttackAgainsttheSelf-ShrinkingGenerator ......... 84
4.5.1 OtherProbabilisticAttacksAgainsttheSelf-Shrinking
Generator......................................................... 87
4.6 CA-BasedLinearizationAttackAgainsttheSelf-Shrinking
Generator................................................................. 88
4.6.1 ComputationoftheCAInitialState ............................ 89
4.6.2 ReconstructionoftheSelf-ShrunkenSequence:The
NumberofXOROperations .................................... 91
4.6.3 102/60vs.90/150CAProposals................................ 92
4.7 DifferentAttacksAgainsttheSelf-ShrinkingGenerator:
AComparison............................................................ 93
References.......................................................................... 95
