Introduction to Cryptography

354 Pages·2004·9.61 MB·English

by  Johannes Buchmann

Undergraduate Texts in Mathematics
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Introduction.
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Cederberg:ACourse inModem
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Johannes Buchmann
Introduction to
Cryptography
Second Edition
, Springer
JohannesA.Buchmann
DepartmentofComputerScience
'TechnicalUniversity, Darmstadt
Hochschulstr, 10
64289Darmstadt
Germany
EditorialBoard
S.Alder F.w.Gehring K.A.Ribet
MathematicsDepartment MathematicsDepartment MathematicsDepartment
San FranciscoState East Hall UniversityofCalifornia
University UniversityofMichigan Berkeley
San Francisco,CA94132 Ann Arbor,MI48109 Berkeley, CA94720-3840
USA USA USA
[email protected] [email protected] [email protected]
Cover:The factorizationofRSA-576,a576-bitor 174-digitprimenumber,wasthe goalofan
openchallengesponsoredby RSALaboratories(Bedford,Mass.).RSA-576wasfactoredbya
teamofresearchersinGermanyand othercountriesin December, 2003.
MathematicsSubjectClassification(2000):94-01,94A60,11T7l
Library ofCongressCatalogingin PublicationData
Buchmann,Johannes.
IntroductiontocryptographyI JohannesBuchmann.- [2nded.].
p.ern.- (Undergraduatetexts in mathematics)
Includesbibliographicalreferencesandindex.
ISBN0-387-21156-X(hardcover: alk.paper)- ISBN0-387-20756-2(softcover: alk.paper)
1.Codingtheory. 2.Cryptography. I.Title. 11Series.
QA268.B832004
003'.54-dc22 2004041657
ISBN0-387-21156-X Printedonacid-freepaper.
Germanedition:EinfurungindieKryptographieII.'Springer-Verlag, Heidelberg,1999.11.'2004,
2001Springer-VerlagNY,LLC
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For Almut, Daniel, andJan

Contents
Preface for the Second Edition xiii
Preface xv
1 Integers 1
1.1 Basics. . . . . .. . . . .. 1
1.2 Divisibility . . . . . . . . . 3
1.3 Representation of Integers 4
1.4 0-and Q-Notation . . . . . 6
1.5 Cost ofAddition, Multiplication, and Division with
Remainder . . . . . . . . . 7
1.6 Polynomial Time . . . . . 9
1.7 Greatest Common Divisor 9
1.8 Euclidean Algorithm ... 12
1.9 Extended Euclidean Algorithm . 16
1.10 Analysis ofthe Extended Euclidean Algorithm 18
1.11 Factoring into Primes 22
1.12 Exercises . . . . . . . . . . . . . . . 24
2 Congruences and Residue Class Rings 29
2.1 Congruences . 29
..
Vll
V111 Contents
2.2 Semigroups . 32
2.3 Groups . 34
2.4 Residue Class Ring . 35
2.5 Fields . 36
2.6 Division in the Residue Class Ring . 36
2.7 Analysis ofthe Operations in the Residue Class Ring 38
2.8 Multiplicative Group ofResidues mod m 39
2.9 Order ofGroup Elements 41
2.10 Subgroups . 42
2.11 Fermat's Little Theorem 44
2.12 Fast Exponentiation . . . 45
2.13 Fast Evaluation of Power Products 48
2.14 Computation ofElement Orders . 49
2.15 The Chinese RemainderTheorem 51
2.16 Decomposition ofthe Residue Class Ring 53
2.17 AFormula for the Eulercp-Function 55
2.18 Polynomials . . . . . . . . . 56
2.19 Polynomials over Fields. . . . . . . 58
2.20 Construction ofFinite Fields . . . . 61
2.21 The Structure of the Unit Group of Finite Fields 65
2.22 Structure ofthe Multiplicative Group of Residues
Modulo a Prime Number 66
2.23 Exercises . 67
3 Encryption 71
3.1 Encryption Schemes . 71
3.2 Symmetric and Asymmetric Cryptosystems 73
3.3 Cryptanalysis . . . . 74
3.4 Alphabets and Words 77
3.5 Permutations . . . . . 80
3.6 BlockCiphers . . . . 81
3.7 Multiple Encryption . 82
3.8 The Use of BlockCiphers 83
3.9 Stream Ciphers ... .. 93
3.10 The Affine Cipher . . . . 95
3.11 Matrices and Linear Maps 97
3.12 Affine Linear BlockCiphers 102
3.13 Vigenere, Hill, and Permutation Ciphers 103
Contents
3.14 Cryptanalysis ofAffine Linear BlockCiphers . 104
3.15 Secure Cryptosystems . 105
3.16 Exercises . .. . . ... .. . III
4 Probability and Perfect Secrecy 115
4.1 Probability . ... ... 115
4.2 Conditional Probability 117
4.3 Birthday Paradox . . . . 118
4.4 PerfectSecrecy . . . . . 119
4.5 Vernam One-Time Pad 123
4.6 Random Numbers . . . 124
4.7 Pseudorandom Numbers 124
4.8 Exercises . . . ... ... 125
5 DES 127
5.1 Feistel Ciphers . 127
5.2 DESAlgorithm. 128
5.3 An Example . . 134
5.4 Security of DES 136
5.5 Exercises . . .. 137
6 AES 139
6.1 Notation . . . 139
6.2 Cipher .... 140
6.3 KeyExpansion 145
6.4 An Example 146
6.5 InvCipher 148
6.6 Exercises . . 148
7 Prime Number Generation 151
7.1 ThaI Division . . . . . 151
7.2 FermatTest . . . . . . 153
7.3 Carmichael Numbers 154
7.4 Miller-RabinTest . 156
7.5 Random Primes 159
7.6 Exercises ... . . 160