Table Of ContentSynthesis Lectures on
Engineering, Science, and Technology
Rafael Martínez-Guerra
Juan Javier Montesinos-García
Juan Pablo Flores-Flores
Encryption and
Decryption Algorithms
for Plain Text and
Images using
Fractional Calculus
Synthesis Lectures on Engineering, Science,
and Technology
The focus of this series is general topics, and applications about, and for, engineers
and scientists on a wide array of applications, methods and advances. Most titles cover
subjects such as professional development, education, and study skills, as well as basic
introductory undergraduatematerial and other topics appropriate for a broader and less
technicalaudience.
Rafael Martínez-Guerra •
Juan Javier Montesinos-García (cid:129)
Juan Pablo Flores-Flores
Encryption and Decryption
Algorithms for Plain Text
and Images using Fractional
Calculus
RafaelMartínez-Guerra JuanJavierMontesinos-García
AutomaticControl InstituteofElectronicsandMechatronics
CenterforResearchandAdvancedStudiesof TechnologicalUniversityoftheMixteca
theNationalPolytechnicInstitute HuajuapandeLeón,Mexico
(CINVESTAV-IPN)
MexicoCity,Mexico
JuanPabloFlores-Flores
AutomaticControl
CenterforResearchandAdvancedStudiesof
theNationalPolytechnicInstitute
(CINVESTAV-IPN)
MexicoCity,Mexico
ISSN2690-0300 ISSN2690-0327 (electronic)
SynthesisLecturesonEngineering,Science,andTechnology
ISBN978-3-031-20697-9 ISBN978-3-031-20698-6 (eBook)
https://doi.org/10.1007/978-3-031-20698-6
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2023
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In memoryofmyfather,
CarlosMartínezRosales.
To mywifeandsons,
Marilen,Rafael,andJuanCarlos.
To mymotherandbrothers,
Virginia,Victor,Arturo,Carlos,
Javier,and Marisela.
Rafael Martínez-Guerra
To mymotherIrma andmyfatherJavier,
whoseloveandsupportallowedme
toreach allmygoals.
To myauntRosarioand UncleClemente,
theirteachings,loveandsupportwere
instrumentalinmyformationup totoday.
To mycousinOscar,forall theseyears
ofsupportandassertivecounsel.
In lovingmemoryofmygrandmotherSofia,
auntMargaritaand cousinDaniel.
Juan JavierMontesinos-García
To myfamily,
Eladio,Candelaria,Edy,Abril,
Mario,Amaia,AkaneandNajmeh.
Juan PabloFlores-Flores
Preface
Manypeopleknowthatitis possibleto interceptandmodifydataif an applicationdoes
not protect it when travels on an untrusted network, and the application then becomes
a disaster when it comesto security.In this book,we offeran alternativeto encryptand
decryptmessagesusingobjectscalledintegerandfractional-orderestimatorsorobservers,
bymeansofsecuritycodes.Weestablishaclassofobserverscapableofcarryingoutthis
work, by means of security codes where finally, since an observer is nothing more than
a mathematical model represented through nonlinear differential equations that can be
of integer or fractional type that serve as means to send messages either of the plain-
text type or of the image type whose key or security code to encrypt or decrypt is
nothingmorethanasetofinitialconditionswhereitmakessensetospeakofthismeans
of transporting the message either plain-text or image for specific attacks for chaotic
cryptosystemsofthestreamciphertype.Inthisbook,wementionthetypeofobserversto
treateithertheintegerorfractionalordertypeandtheirmaincharacteristics.We discuss
an importantpropertyof some systems such as Liouville systems that is very important
for the encryption and decryption of messages in integer and fractional order nonlinear
systemsbyusingthesynchronizationpropertyofchaoticsystemswhereweaddresssome
logisticmapssuchasMandelbrotsetsincludingJuliaandfractalsetstakingadvantageof
theircharacteristicstoencryptorrecovermessages.Wediscusssomeissuesaboutstream
andblockciphersand somestate observers.Varioustypesof observersare proposedfor
nonlinearsystemsofintegerandfractionalorderfromthesimplest(LuenbergerObserver)
tothemostsophisticatedsuchastheSupertwistingObserverformessagereceiversaswell
astheirvulnerabilitytoattacks.Observersoftheexponentialpolynomialtypeareproposed
togetherwiththepropertyoftheLiouvilletype.Wealsoproposetheusefulnessofrobust
fractionalsystemsofslidingmodeswithLiouvillecharacteristicsasmeansoftransmission
and reception of plain-text and image messages. Of all the alternatives for encryption
anddecryptionofmessagesshownhere,avulnerabilityanalysistocryptographicattacks
(cryptoanalysis)is made,this is a security analysis, an importanttopic on the subjectof
secure communications. The book is self-contained, that is to say, the necessary tools
to address the issues such as fractionalcalculus are given in the same book and several
vii
viii Preface
examplesarepresented.Moreover,thisbookincludesexercisesthatarelefttothereader.
The book is directed to an audience such as professionals in the areas of mathematics,
physics and engineering and researchers in general and related areas with a minimum
ofknowledgeinhighermathematics.However,italsocontainsadvancedresearchtopics
forpeopleinterestedinencryptionanddecryption,observers,synchronizationandsecure
communications areas. The book is organized as follows. In Chap.1, a brief overview
of the main topics covered is presented giving an introduction to the state of the art on
encryptionanddecryptionalgorithms,synchronizationofchaoticsystems,securitykeysor
codes,securityanalysissuchascryptographicattacks,linearanddifferentialcryptanalysis,
inadditiontospecificattacksforchaoticcryptosystemsoftypestreamcipher.InChap.2,
some definitionsare given aboutthe Lyapunovexponents, stability, and state observers;
also fractalsandsynchronizationarebrieflyintroduced.Chapter3 showsthestream and
block ciphers and observers, binary representations as well as some conversions from
binaryto decimaland viceversa,representationsofplain textand imagesin integerbits
andcipherswithgeneralizedsynchronization.Chapter4dealswiththestudyofLiouville
systems and cryptography, and a supertwisting observer is addressed as a receiver as
wellasitsvulnerabilitytocryptanalysis.Chapter5presentssomebasicconceptsofstate
observers,theexponentialpolynomialobserverisusedasareceptor,andthereceptorsare
based on properties related to Liouville systems. Chapter 6 shows some basic elements
of fractional calculus and some observers. Chapter 7 deals with the implementation of
systemswiththepropertyofLiouvilleandfractionalordersystemsusedfortheencryption
anddecryptionofplain-textandimagemessages.InChap.8,wepresentrobustfractional
orderstateobserversasmeansofencryptionanddecryption,presentingasecurityanalysis
andsituationsthatleadtodecryptionfailures.Finally,inChap.9,anewtopicisdescribed
insecurecommunications,andwepresentencryptionanddecryptionalgorithmsbyusing
stateobserversthatarerepresentedbymeansoffractional-orderchaoticsystemswiththe
Atangana-Baleunufractionalderivative.Additionally,thereaderwillfindthroughoutthis
materialsomeexercisestostrengthentheknowledgeacquired.
MexicoCity,Mexico RafaelMartínez-Guerra
JuanJavierMontesinos-García
JuanPabloFlores-Flores
Contents
1 Introduction ......................................................................... 1
1.1 ChaoticSystemSynchronizationandEncryptionAlgorithms............... 1
1.1.1 EncryptionThroughChaoticSystems................................ 2
1.2 KeyorSecurityCode.......................................................... 3
1.3 SecurityAnalysis............................................................... 4
1.3.1 CryptographicAttacks(Cryptanalysis)............................... 4
1.3.2 DifferentialCryptanalysis............................................. 5
1.3.3 LinearCryptanalysis................................................... 6
1.4 SpecificAttacksforStreamCipher-TypeChaoticCryptosystems........... 6
1.4.1 MessageExtraction.................................................... 6
1.4.2 ParametricEstimation................................................. 7
1.4.3 BruteForceAttacks.................................................... 7
References............................................................................ 7
2 SynchronizationofChaoticSystems.............................................. 9
2.1 ChaoticSystems................................................................ 9
2.1.1 LyapunovExponents .................................................. 10
2.2 Stability......................................................................... 12
2.2.1 NonlinearSystems..................................................... 13
2.2.2 StabilityandLinearization ............................................ 15
2.2.3 Lyapunov’sDirectMethod............................................ 24
2.3 StateObservers................................................................. 30
2.3.1 LuenbergerObserver .................................................. 30
2.4 FractalsandSynchronization.................................................. 38
References............................................................................ 46
3 StreamCyphersandBlockCyphers.............................................. 47
3.1 MessageandDataCarrierSignals ............................................ 47
3.1.1 DecimalandBinaryNumbers......................................... 48
ix
x Contents
3.1.2 BinarytoDecimalandDecimaltoBinaryConversions............. 53
3.1.3 RepresentationofPlaintextwith8Integers .......................... 55
3.1.4 RepresentationofPlainImageswith8BitIntegers.................. 58
3.1.5 DataCarrierSignal .................................................... 69
3.2 StreamCiphersandStateObservers.......................................... 72
3.2.1 PseudorandomNumberGenerator.................................... 73
3.2.2 TheLuenbergerObserverinaStreamCipher........................ 77
3.3 BlockCiphersandObservers.................................................. 83
3.3.1 BlockCipher........................................................... 85
References............................................................................ 95
4 LiouvillianSystemsandCryptography .......................................... 97
4.1 Introduction .................................................................... 97
4.2 Transmitter ..................................................................... 98
4.3 Receiver ........................................................................ 101
4.3.1 Super-TwistingBasedReceiver....................................... 101
4.3.2 ProofofStability ...................................................... 103
4.3.3 ReconstructionoftheStatesBasedReceiver......................... 107
4.4 NumericalSimulation.......................................................... 108
4.5 VulnerabilitytoCryptanalysis................................................. 113
4.6 ConcludingRemarks........................................................... 114
References............................................................................ 115
5 StateObserversandCryptography............................................... 117
5.1 Introduction .................................................................... 117
5.2 Encryption...................................................................... 118
5.2.1 GeneratingPseudo-RandomNumbers................................ 118
5.2.2 EncryptionAlgorithm ................................................. 119
5.3 DataRecovery.................................................................. 123
5.3.1 ExponentialPolynomialReceiver..................................... 123
5.3.2 Stability ................................................................ 124
5.3.3 LiouvillianSystemPropertiesBasedReceiver....................... 126
5.4 NumericalSimulation.......................................................... 128
5.5 ConcludingRemarks........................................................... 129
References............................................................................ 132
6 FractionalSystems.................................................................. 133
6.1 GammaFunction............................................................... 133
6.1.1 SomePropertiesoftheGammaFunction............................. 135
6.2 BetaFunction .................................................................. 138
6.3 Euler’sNumberandItsRelationtotheGammaFunction ................... 141
6.4 MiscellaneousExamples....................................................... 144
