Table Of ContentFinancial Mathematics and Fintech
Zhiyong Zheng
Kun Tian
Fengxia Liu
Modern
Cryptography
Volume 2
A Classical Introduction to Informational
and Mathematical Principle
Financial Mathematics and Fintech
SeriesEditors
ZhiyongZheng,RenminUniversityofChina,Beijing,Beijing,China
AlanPeng,UniversityofToronto,Toronto,ON,Canada
This series addresses the emerging advances in mathematical theory related to
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Thekeywordsinthisseriesincludebutarenotlimitedto:
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· ·
Zhiyong Zheng Kun Tian Fengxia Liu
Modern Cryptography
Volume 2
A Classical Introduction to Informational
and Mathematical Principle
ZhiyongZheng KunTian
SchoolofMathematics SchoolofMathematics
RenminUniversityofChina RenminUniversityofChina
Beijing,China Beijing,China
HenanAcademyofSciences
Zhengzhou,China
FengxiaLiu
ArtificialIntelligenceResearchInstitute
BeihangUniversity
Beijing,China
ISSN 2662-7167 ISSN 2662-7175 (electronic)
FinancialMathematicsandFintech
ISBN 978-981-19-7643-8 ISBN 978-981-19-7644-5 (eBook)
https://doi.org/10.1007/978-981-19-7644-5
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Preface
Forintegerfactorizationanddiscretelogarithmcalculation,P.W.Shorpublishedan
effective quantum calculation in SIAM Journal on Computing in 1997, which is
called the Shor algorithm in academic circles. Classical public key cryptosystems
suchasRSA,ECCandsooncouldnotresisttheattackoftheShoralgorithm,sothe
majorsecurityrisksofpublickeycryptosystemsarecompletelyexposedtotheShor
algorithmandquantumcomputer.
Inthepast20years,theriseanddevelopmentofpost-quantumcryptographyhave
close relation with the lattice cryptosystems. The academic community believes
that the hard problems on lattice, such as the shortest vector problem (SVP), the
continuous shortest vector problem (SIVP) and the determination of the shortest
vector problem (GapSVP) can resist quantum computing effectively, so the public
key cryptosystems based on the hard problems on lattice become the core theory
andtechnologyofthepost-quantumcryptography.Atpresent,therearesixkindsof
publishedpost-quantumcryptosystems:
1. Ajtai-Dwork cryptosystem (1997). Ajtai constructed a collision-resistant Hash
functionbythecirculantmatrixandidealmatrix,andconvertedthecollisionpoint
intotheshortestvectorproblemonq-aryintegerlattice.Ajtaifirstproposedthe
concept of random lattice (Gauss lattice) in 1996, and established the famous
reduction principle ‘from the worst case to the average case’. The security of
Ajtai-Dworkcryptosystemcouldbefullyprovedbythisreductionprinciple.
2. GGH/HNF cryptosystem (1997). In 1997, Goldereich, Goldwasser and Halevi
constructed a public key cryptosystem based on the closest vector problem on
the q-ary integer lattice, which was further improved by Micciancio using the
Hermitenormalbasisin2005.TheideaofMicciancioisverysimple.Sincethe
HNFbasisofanylatticecanbeeasilycomputedfromitsgeneratedmatrix,the
GGHcryptosystemusestheHNFbasisasthepublickeydirectly.
3. NTRU cryptosystem (1998). Number Theory Research Unit (NTRU) is a
quantum-resistantcomputingpublickeycryptosystemdevelopedbyJ.Hoffstein,
J.PipherandJ.H.SilvermaninBrownUniversityin1998,whichhasbecome
themostattractivepost-quantumcryptosystemduetoitssimplealgorithm,fast
v
vi Preface
calculation speed and small storage space. In 2009, the National Institute of
StandardsandTechnologywroteasurveyreport:thereisnocryptosystemcould
consider both public key encryption and digital signature, and resist the Shor
algorithmsimultaneously.TheNTRUencryptionalgorithmseemstobethemost
likelychoiceamongmanylattice-basedencryptionschemes.ThePQCRYPTO
program(Horizon2020ICT-645622)byEuropeanUnionhopestodevelopanew
EuropeanencryptionstandardbasedontheNTRUimprovedbyStehle-Steinfeld.
4. MacElience/Niderreitercryptosystem(1998).Linearcodesaretheearliesterror-
correcting codes in coding theory. Later, algebraic coding developed based on
theidealtheorygreatlyenrichedandimprovedthelinearcodingtheory.Cycle
codeandGoppacodearethemostimportanterror-correctingcodesinalgebraic
coding.MacElienceandNiderreiterconstructedanewpublickeycryptosystem
by using the asymmetry of encoding algorithm and decoding algorithm of the
error-correctingcodeindependently,whichwecallMacElience/Niderreitercryp-
tosystem.Sinceacode(linearcodeoralgebraiccode)canberegardedasalattice
onafinitefield,thesecurityofthiscryptosystemiscloselyrelatedtotheclosest
vectorproblemontheq-aryintegerlattice.Recentstudieshaveshownthatcoding
theoryplaysanimportantroleinlattice-basedcryptosystems.
5. LWE cryptosystem (2005). In 2005, O. Regev of Tel Aviv University in Israel
proposedthefamousLWEcryptosystembasedontheLWEdistribution.Because
ofthiswork,Regevwonthehighestawardinthetheoreticalcomputerscience
in 2018—the Godel Award. The LWE distribution (Learning With Errors) is
a random linear system with errors having Gauss distribution. Regev’s cryp-
tosystemencryptsasinglebitofplaintexteachtime.SincethesecurityoftheLWE
problemhasbeenclearlyproved(seeChap.3ofthisbook),LWEcryptosystem
iscurrentlythemostactiveandmainstreamresearchtopic.
6. Fully homomorphic encryption (FHE). In 1985, R.Rivest, C.Adleman and
M.Dertouzosfirstproposedtheconceptofdatabankandtheconjectureoffully
homomorphicencryption.Someindividualsandorganizationsencrypttheorig-
inaldataandstoretheminthedatabankforprivacyprotection,whichisobviously
ahugewealth.Howtocomputetheseencrypteddataeffectively?R.Rivest,C.
AdlemanandM.Dertouzospresentedthefullyhomomorphicencryptionconjec-
ture.In2009,C.GentryofStanfordUniversitypartiallysolvedtheRADconjec-
ture.Gentry’sworkisbasedontheideallattice,thatis,anintegerlatticewhich
hasaone-to-onecorrespondencetotheidealofpolynomialring.Butthecryp-
tosystem of Gentry isa finite-time fully homomorphic encryption, and infinite
fullyhomomorphicencryptionisstillanunsolvedpublicproblem.In2012and
2013,thesecondandthirdfullyhomomorphicencryptionalgorithmsbasedon
the LWE distribution were proposed one after another. Gentry won the 2022
GodelAwardforhiscontributions.
In the book Modern Cryptography, we give a detailed introduction to the basic
theory of lattice and the first four kinds of lattice-based cryptosystems. The main
purpose of this book is to discuss the computational complexity theory of lattice
Preface vii
cryptosystems, especially Ajtai’s reduction principle, and fill the gap that post-
quantum cryptography focuses on the encryption and decryption algorithms, and
the theoretical proof is insufficient. In Chaps. 3, 4 and 6, we introduce the LWE
distribution,LWEcryptosystemandfullyhomomorphicencryptionindetail.When
using stochastic analysis tools, there are many ‘ambiguity’ problems in terms of
definitionsandalgorithms,suchasthe‘≈’notationappearedinalargenumberof
papersandbooks,whichisunprecisemathematically.Thebiggestcharacteristicof
this book is to use probability distribution to provide rigorous mathematical defi-
nitionsandproofsforvariousunclearexpressions,makingitarigoroustheoretical
systemtofacilitateteachinganddisseminationinclass.Chapters5and7arebased
ontwopaperspublishedbytheauthorsinthejournalJournalofInformationSecurity
(seereferences[63,64]).Thesematerialscanberegardedassomeimportanttopics,
suchas thefurther extension and improvement of cyclic lattices,ideallattices and
generalizedNTRUcryptosystems.
This book contains the most cutting-edge and hottest research topics in post-
quantum cryptography. Reading all the chapters requires a lot of mathematical
knowledgeandagoodmathematicalfoundation.Therefore,thisbookcanbeused
asatextbookforgraduatestudentsinmathematicsandcryptography,orareference
book for researchers in cryptography area. Due to the rush of time, all the mate-
rialsaresummarizedfromdomesticandforeignresearchpapersinthelast20years,
andshortcomingsandmistakesareinevitable.Wewelcomereaderstocriticizeand
correctthem.
Zhengzhou,China ZhiyongZheng
September2022
Contents
1 RandomLatticeTheory ......................................... 1
1.1 FourierTransform .......................................... 3
1.2 DiscreteGaussMeasure ..................................... 7
1.3 SmoothingParameter ....................................... 13
1.4 SomePropertiesofDiscreteGaussDistribution ................. 25
2 ReductionPrincipleofAjtai ..................................... 33
2.1 RandomLinearSystem ...................................... 33
2.2 SISProblem ............................................... 35
2.3 INCGDDProblem .......................................... 39
2.4 ReductionPrinciple ......................................... 46
3 LearningwithError ............................................ 53
3.1 CirculantMatrix ........................................... 53
3.2 SISandKnapsackProblemonRing ........................... 61
3.3 LWEProblem .............................................. 72
3.4 ProofoftheMainTheorem .................................. 80
3.4.1 FromLWEtoDGS ................................... 81
3.4.2 FromDGStoHardProblemsonLattice ................. 93
3.4.3 FromD-LWEtoLWE ................................ 97
4 LWEPublicKeyCryptosystem .................................. 99
4.1 LWECryptosystemofRegev ................................. 99
4.2 TheProofofSecurity ....................................... 104
4.3 PropertiesofRoundingFunction .............................. 108
4.4 GeneralLWE-BasedCryptosystem ........................... 112
4.5 ProbabilityofDecryptionErrorforGeneralDisturbance ......... 115
5 CyclicLatticesandIdealLattices ................................ 119
5.1 SomeBasicPropertiesofLattice ............................. 119
5.2 IdealMatrices .............................................. 123
ix
x Contents
5.3 φ-CyclicLattice ............................................ 129
5.4 ImprovedUpperBoundforSmoothingParameter ............... 137
6 FullyHomomorphicEncryption ................................. 143
6.1 DefinitionsandExamples .................................... 144
6.2 GadgetMatrixandGadgetTechnique ......................... 148
6.3 BoundedFullyHomomorphicEncryption ...................... 154
6.4 ConstructionofGentry ...................................... 165
6.5 Attribute-BasedEncryption .................................. 170
7 AGeneralizationofNTRUencrypt ............................... 175
7.1 φ-CyclicCode ............................................. 176
7.2 AGeneralizationofNTRUencrypt ............................ 182
References ........................................................ 189
