Notes on Cryptography Paulo Mateus, Amılcar Sernadas, Andr´e Souto and Luıs Antunes 2012

333 Pages·2012·4.55 MB·English

Notes on Cryptography
Paulo Mateus, Am´ılcar Sernadas,
1
Andre´ Souto and Lu´ıs Antunes
2012
1Parasefazerolivroa` posteriori
Totheguardian
Caution
The themes presented in this work are in a very preliminary phase of maturation. It contains
lotsoftypos,examplesthatarenotyetcomplete,definitionsthatarenotwrittenwithaunified
notations and theorems that are not properly written. In fact, there are lots of topics still to be
written and others that will deserve a profound reflection before publishing. Many exercises
and examples are part of other books are meant to be used in classes. They will be substituted
in mean time by others that fulfill our purposes of presentations and focus on crucial aspects
thatwehaveinmind.
Readingthislecturenotesrequiresthereadertobepreparedtofindsomeproblemsofcompre-
hension and we suggest that it should be complemented with the reading of standard books
used to teach the materials presented here. We suggest for example the readings of [Sti06],
[MVO96],[KL07]andtheotherreferencesthatweputonalongthetextforafullunderstanding
ofcryptography.
3

Preface
Thisworkismeanttocompileinabooktheprogrammaticcontentsoftheintroductorycourse
Criptografia e Protocolos de Seguranc¸a of the Master of Bologne in Mathematics and appli-
cations and also from the Doctoral program me on Security of Information lectured in the
mathematical department of Instituto Superior Te´cnico of Universidade Te´cnica de Lisboa. It
alsousesmateriallecturedinthecourseCriptografialecturedatcomputersciencedepartment
ofFaculdadedeCieˆnciasofUniversidadedoPorto.
The idea of this work is provide a very useful self contained book about cryptography using
a perspective of a mathematician and also a perspective of a computer scientist in order to be
abletoatoolthatotherscanusetoteachasimilarcourse.
For financial support the author Andre´ Souto is deeply thankful to FCT through the grant
SFRH/BPD/76231/2011.
WearegratefultotheSQIGmembersfortheniceworkingambientandalltheencouragement
thattheyhaveus.
Aspecialthanksisduetoseveralpeoplewithwhichwehadveryhelpfuldiscussions,feedback
andadvisingonthethemesandtheapproacheschoosen.
2012.
MathematicalDepartment
5
InstitutoSuperiorTe´cnico,UniversidadeTe´cnicadeLisboa
ComputerScienceDepartment
FaculdadedeCieˆnciasdaUniversidadedoPorto
SecurityandQuantumInformationGroup
InstitutodeTelecomunicac¸o˜es
Contents
Caution 3
Preface 5
1 Intro 3
I Basic concepts 5
2 Algebraicstructuresandnumbertheory 9
2.1 Groups,ringsandfields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Thenaturalnumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Congruencesandmodularalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.1 FiniteFields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 ProbabilitiesandShannonentropy 29
3.1 Probabilitytheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
7
3.1.1 Thenotionofbiasofadistribution . . . . . . . . . . . . . . . . . . . . . . . 32
3.2 Entropyandinformationtheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4 Notionofcomputationalcomplexity 41
4.1 Pseudo-codeasbaseforcomputation . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2 Measuringcomplexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2.1 Thebig-Ohnotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2.2 Complexityclasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3 One-wayfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3.1 Thecandidatestoone-wayfunctions . . . . . . . . . . . . . . . . . . . . . 68
II Classical Cryptography 71
5 Classicalcryptographicsystems 73
5.1 Cryptographicsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.1.1 Steganography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.1.2 Thesubstitutioncipher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.1.3 TheVermanciphersystem . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.1.4 TheVigene`recipher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.1.5 TheHillcipher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.1.6 Streamciphers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.1.6.1 LinearFeedbackShiftRegisters . . . . . . . . . . . . . . . . . . . 90
5.2 BlockCipherSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.2.1 ECBmode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2.2 CBCmode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.2.3 CFBmode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.2.4 OFBmode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.3 Breakingdownthecryptographicsystem . . . . . . . . . . . . . . . . . . . . . . . 101
5.3.1 Breakingdownthesubstitutioncipher . . . . . . . . . . . . . . . . . . . . . 106
5.3.2 BreakingtheVigene`recipher . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.3.3 BreakingtheHillcipher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.3.4 BreakingtheLFCS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6 Perfectsecrecy 121
6.1 Definitionofperfectsecurityandresults . . . . . . . . . . . . . . . . . . . . . . . . 123
6.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7 Blockciphers: DESandAES 133
7.1 Substitution-PermutationNetworks . . . . . . . . . . . . . . . . . . . . . . . . . . 134
7.2 DES–DataEncryptionStandards . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7.2.1 DescriptionofDES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.2.2 BreakingdowntheDES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.2.2.1 LinearApproximationof S-boxes . . . . . . . . . . . . . . . . . . 145
7.2.2.2 Thedifferentialattack . . . . . . . . . . . . . . . . . . . . . . . . . 150
7.2.2.3 Analyticattack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7.3 AES–AdvancedEncryptionStandards . . . . . . . . . . . . . . . . . . . . . . . . 155
7.3.1 DescriptionofAES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
III The Public Key Cryptography 163
8 ThestorybehindPublicKeyCryptography 165
8.1 Describingapublickeycryptographicsystem . . . . . . . . . . . . . . . . . . . . 168
9 TheRSAcryptographicsystem 173
9.1 TheRSAcryptographicsystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
9.2 EuclideanAlgorithmforthegcdandthemodularexponentiation . . . . . . . . . 178
9.3 Checkingfastprimalityofnumbers . . . . . . . . . . . . . . . . . . . . . . . . . . 183
9.3.1 ThequadraticresidueproblemandtheLegendreandJacobisymbols . . . 187
9.3.2 Solovay–Strassenalgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
9.3.3 Miller-Rabin’salgorithmforprimalitytest . . . . . . . . . . . . . . . . . . 197
9.3.4 TheAKSalgorithmprovingthatPrimes ∈ P . . . . . . . . . . . . . . . . 198
9.4 AttackingtheRSA-Factorizing n . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
9.4.1 Pollard’s p−1method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
9.4.2 Dixon’srandomsquarealgorithms . . . . . . . . . . . . . . . . . . . . . . . 209
9.4.3 Shor’salgorithmforfactorization . . . . . . . . . . . . . . . . . . . . . . . 212
9.4.3.1 quantummechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 212
9.4.3.2 Thealgorithmanditsexplanation . . . . . . . . . . . . . . . . . . 212
9.5 AttackingtheRSA-otherattacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
9.6 AttackingRSAbasedonpartialinformationleaked . . . . . . . . . . . . . . . . . 215
9.7 TheRabin’scryptographicsystemanditsrelationshipwithRSA . . . . . . . . . . 218
9.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
10 Cryptographicschemesbasedonthediscretelogarithmicproblem 235
10.1 Thediscretelogarithmicproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235