Partially Homomorphic Encryption

148 Pages·2021·1.621 MB·English

by  Çetin Kaya Koç, Funda Özdemir, Zeynep Ödemiş Özger

Çetin Kaya Koç
Funda Özdemir
Zeynep Ödemiş Özger
Partially
Homomorphic
Encryption
Partially Homomorphic Encryption
Çetin Kaya Koç • Funda Özdemir
Zeynep Ödemiş Özger
Partially Homomorphic
Encryption
Çetin Kaya Koç Funda Özdemir
Department of Computer Science Faculty of Engineering
University of California and Natural Sciences
Santa Barbara, CA, USA Istinye University
Istanbul, Turkey
Zeynep Ödemiş Özger
Faculty of Engineering and Architecture
Kâtip Çelebi University
İzmir, Turkey

ISBN 978-3-030-87628-9 ISBN 978-3-030-87629-6 (eBook)
https://doi.org/10.1007/978-3-030-87629-6

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Preface
The potential applications of homomorphic operations over encryption functions
were recognized and appreciated almost about the same time as the first public-
keycryptographicalgorithmRSAwasinvented.TheRSAalgorithmismultiplica-
tively homomorphic. The ensuing 30 years have brought on several additively or
multiplicativelyhomomorphicencryptionfunctionswithincreasingalgorithmicin-
ventiveness and complicated mathematics. However, until 2009, it was not clear
whether a fully homomorphic encryption algorithm, one that allows both additive
andmultiplicativehomomorphisms,wouldexist.ThiswasresolvedbyCraigGen-
try,andfollowedupbyseveralauthorswhohaveproposedfullyhomomorphicen-
cryption algorithms and addressed issues related to their formulation, arithmetic,
efficiencyandsecurity.Whileformidableefficiencybarriersremain,wenowhavea
varietyoffullyhomomorphicencryptionalgorithmsthatcanbeappliedtovarious
privatecomputationproblemsinhealthcare,financeandnationalsecurity.
Thisbookstartedwiththedesireandmotivationtodescribeandimplementpar-
tiallyhomomorphicencryptionfunctionsusingaunifiedmathematicalnotation.We
hope that we accomplished our projected goal, and you will find our exposition
helpfultounderstandandimplementthesealgorithms.Ifahighlyefficientfullyho-
momorphicencryptionfunctionformultipleapplicationsweretobecomeavailable,
the usefulness of partially homomorphic encryption functions would be in ques-
tion.Butthisloftyendisnotyetinsight.Studyingpartiallyhomomorphicencryp-
tionfunctionsmayhelpustounderstandthedifficultiesaheadandperhapstoavoid
blindalleysanddeadends.Moreover,partiallyhomomorphicencryptionalgorithms
mayhavecertainperhapslimitedapplicationsforwhichsignificantlymoreefficient
implementationscanbeobtained.Ifnot,itisalwaysenjoyabletolearninteresting
algorithmsandtheirunderlyingbeautifulmathematics.
Theauthorsthanktheirfamiliesanddedicatethisbooktothem.
C¸etinKayaKoc¸
FundaO¨zdemir
ZeynepO¨demis¸O¨zger
v
Contents
1 Introduction................................................... 1
2 MathematicalBackground ...................................... 3
2.1 NumberTheory ............................................ 3
2.1.1 DivisibilityandFactorization .......................... 3
2.1.2 GreatestCommonDivisor(GCD) ...................... 4
2.1.3 ModularArithmetic .................................. 6
2.1.4 ModularExponentiation .............................. 8
2.1.5 ModularInverse ..................................... 9
2.1.6 Euler’sTheorem..................................... 11
2.1.7 Carmichael’sTheorem................................ 12
2.1.8 Generators.......................................... 13
2.1.9 ChineseRemainderTheorem .......................... 14
2.1.10 QuadraticResidues................................... 15
2.1.11 Higher-OrderResidues ............................... 19
2.1.12 ResidueClasses ..................................... 19
2.1.13 RandomNumberGenerators........................... 22
2.2 GroupTheory ............................................. 23
2.2.1 BasicAxioms ....................................... 23
2.2.2 MultiplicativeGroups ................................ 23
2.2.3 AdditiveGroups ..................................... 24
2.2.4 OrderofaGroup .................................... 24
2.2.5 CyclicGroupsandSubgroups.......................... 25
2.2.6 DiscreteLogarithmProblem........................... 26
2.2.7 Diffie-HellmanProblem .............................. 26
2.2.8 CosetsandLagrange’sTheorem........................ 27
2.2.9 Sylow’sTheorem .................................... 28
2.2.10 DirectProducts...................................... 29
2.2.11 Homomorphisms .................................... 29
2.3 FieldTheory............................................... 30
2.4 EllipticCurves............................................. 31
vii
viii Contents
2.4.1 GroupLaw ......................................... 32
2.4.2 EllipticCurvePointMultiplication ..................... 34
2.4.3 EllipticCurvesoverFiniteFields....................... 34
3 Rivest-Shamir-AdlemanAlgorithm .............................. 37
3.1 KeyGeneration ............................................ 37
3.2 Encryption ................................................ 38
3.3 Decryption ................................................ 38
3.4 HomomorphicProperties .................................... 39
3.5 Security .................................................. 40
3.6 Example .................................................. 41
4 Goldwasser-MicaliAlgorithm ................................... 43
4.1 KeyGeneration ............................................ 43
4.2 Encryption ................................................ 44
4.3 Decryption ................................................ 45
4.4 HomomorphicProperties .................................... 45
4.5 Security .................................................. 46
4.6 Example .................................................. 47
5 ElGamalAlgorithm ............................................ 51
5.1 MultiplicativelyHomomorphicElGamalAlgorithm.............. 51
5.1.1 KeyGeneration...................................... 51
5.1.2 Encryption.......................................... 51
5.1.3 Decryption.......................................... 52
5.1.4 HomomorphicProperties.............................. 53
5.1.5 Security ............................................ 55
5.1.6 Example............................................ 56
5.2 AdditivelyHomomorphicElGamal............................ 58
5.2.1 Encryption.......................................... 58
5.2.2 Decryption.......................................... 59
5.2.3 HomomorphicProperties.............................. 59
5.3 TheEllipticCurveElGamalAlgorithm ........................ 60
5.3.1 KeyGeneration...................................... 60
5.3.2 Encryption.......................................... 60
5.3.3 Decryption.......................................... 61
5.3.4 HomomorphicProperties.............................. 61
6 BenalohAlgorithm ............................................. 63
6.1 KeyGeneration ............................................ 63
6.2 Encryption ................................................ 64
6.3 Decryption ................................................ 64
6.4 HomomorphicProperties .................................... 65
6.5 Security .................................................. 67
6.6 Example .................................................. 67
Contents ix
7 Naccache-SternAlgorithm ...................................... 71
7.1 TheDeterministicVersion ................................... 71
7.1.1 KeyGeneration...................................... 71
7.1.2 Encryption.......................................... 72
7.1.3 Decryption.......................................... 72
7.1.4 Security ............................................ 74
7.2 TheProbabilisticVersion .................................... 74
7.2.1 KeyGeneration...................................... 74
7.2.2 Encryption.......................................... 75
7.2.3 Decryption.......................................... 75
7.2.4 HomomorphicProperties.............................. 75
7.2.5 Security ............................................ 77
7.2.6 Example............................................ 77
8 Okamoto-UchiyamaAlgorithm .................................. 83
8.1 KeyGeneration ............................................ 83
8.2 Encryption ................................................ 84
8.3 Decryption ................................................ 84
8.4 HomomorphicProperties .................................... 87
8.5 Security .................................................. 89
8.6 Example .................................................. 91
9 PaillierAlgorithm.............................................. 95
9.1 KeyGeneration ............................................ 95
9.2 Encryption ................................................ 95
9.3 Decryption ................................................ 98
9.4 HomomorphicProperties ....................................101
9.5 Security ..................................................102
9.6 Example ..................................................103
10 Damga˚rd-JurikAlgorithm ......................................107
10.1 FormingthePlaintextSpace..................................107
10.2 KeyGeneration ............................................110
10.3 Encryption ................................................110
10.4 Decryption ................................................111
10.5 HomomorphicProperties ....................................115
10.6 Security ..................................................116
10.7 Example ..................................................118
11 Boneh-Goh-NissimAlgorithm ...................................123
11.1 KeyGeneration ............................................125
11.2 Encryption ................................................125
11.3 Decryption ................................................127
11.4 HomomorphicProperties ....................................127
11.5 Security ..................................................130
11.6 Example ..................................................131
x Contents
12 Sander-Young-YungAlgorithm ..................................135
12.1 KeyGeneration ............................................135
12.2 Encryption ................................................135
12.3 Decryption ................................................136
12.4 HomomorphicProperties ....................................138
12.5 Security ..................................................139
12.6 Example ..................................................139
References.....................................................141
FurtherReading ...................................................142
Index .............................................................143
Chapter 1
Introduction
Homomorphic encryption is the last chapter for now in the 3000-year history of
cryptographyandabout50yearsofpublic-keycryptography.Performingmeaning-
ful computations with encrypted text without decrypting it seems magical at first.
Imagineadeckofcardswithunknownnumbersoneach,facingdownonthetable:
Can you add these numbers and obtain the total value with 100% accuracy, with-
out peeking at any of them? Homomorphic encryption can do this for you; so can
asuccessfulillusionistusingsomeelaboratetricks,buthomomorphicencryptionis
honest and straightforward, and involves no tricks whatsoever; it is just beautiful
mathematics.
Answers to the question of how homomorphic cryptography accomplishes this
feat are found in tens of doctoral theses and in hundreds of journal, conference
and electronic archival articles. This book brings together several of these algo-
rithms with their descriptions, algorithmic steps, examples, and security analyses.
Wearelimitingourscopehowevertopartiallyhomomorphicencryptionfunctions,
whicharethosethatallowoneorafewarithmeticoperationstobeperformedhomo-
morphically,ratherthanallarithmeticoperations.Wealsoincludedtwosomewhat
homomorphic encryption functions; however, focused on their partial homomor-
phismsonly.Thereadershouldnotethat,inthisbook,weusethetermshomomor-
phicencryptionandhomomorphiccryptographyinterchangeably.
The general concept of homomorphic encryption was introduced by Rivest,
AdlemanandDertouzousin[22]usingthetermprivacyhomomorphism,soonafter
theRSAalgorithmwasinvented[21].Themultiplicativehomomorphicpropertyof
theRSAalgorithmwasfirstnotedinprintinthispaper,as(xe)(ye)=(xy)e.Unfor-
tunately,theRSAalgorithmdoesnotallowadditivehomomorphism,andtherefore
homomorphicallyevaluatingapolynomialisnotpossibleusingtheRSAencryption
function.
The breakthrough in the search for fully homomorphic encryption functions
camewiththedoctoraldissertationofCraigGentryin2009.Heintroducedthefirst
construction for a fully homomorphic encryption scheme [8]. Furthermore, since
1978,wehaveseenveryproductiveresearchonpartialhomomorphismsofexisting
ornewencryptionfunctions.
© Springer Nature Switzerland AG 2021 1
Ç. K. Koç et al., Partially Homomorphic Encryption,
https://doi.org/10.1007/978-3-030-87629-6_1