Mathematical foundations of public key cryptography

230 Pages·2015·5.226 MB·English

by  Meng, Xianmeng, Wang, Mingqiang, Wang, Xiaoyun, Xu, Guangwu

Mathematical
Foundations of
Public Key
Cryptography
© 2016 by Science Press
© 2016 by Science Press
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Mathematical
Foundations of
Public Key
Cryptography
Xiaoyun Wang
Guangwu Xu
Mingqiang Wang
Xianmeng Meng

© 2016 by Science Press
CRC Press
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Version Date: 20150813
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© 2016 by Science Press
Contents
Foreword ix
Preface xi
Acknowledgments xv
hapter
C 1(cid:2) Divisibility of Integers 1
1.1 THECONCEPTOFDIVISIBILITY 1
1.2 THEGREATESTCOMMONDIVISORAND
THELEASTCOMMONMULTIPLE 6
1.3 THEEUCLIDEANALGORITHM 12
1.4 SOLVINGLINEARDIOPHANTINEEQUATIONS 16
1.5 PRIMEFACTORIZATIONOFINTEGERS 18
hapter
C 2(cid:2) Congruences 27
2.1 CONGRUENCES 27
2.2 RESIDUECLASSESANDSYSTEMSOFRESIDUES 31

2.3 EULER'STHEOREM 37
2.4 WILSON'STHEOREM 40
hapter
C 3(cid:2) Congruence Equations 47
3.1 BASICCONCEPTSOFCONGRUENCESOF
HIGHDEGREES 47
3.2 LINEARCONGRUENCES 50
3.3 SYSTEMSOFLINEARCONGRUENCEEQUATIONSAND
THECHINESEREMAINDERTHEOREM 52
v
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3.4 GENERALCONGRUENCEEQUATIONS 55
3.5 QUADRATICRESIDUES 57
3.6 THELEGENDRESYMBOLANDTHEJACOBISYMBOL 61
hapter
C 4(cid:2) Exponents and Primitive Roots 71
4.1 EXPONENTSANDTHEIRPROPERTIES 71
4.2 PRIMITIVEROOTSANDTHEIRPROPERTIES 76
4.3 INDICES,CONSTRUCTIONOFREDUCEDSYSTEM
OFRESIDUES 79
4.4 NTHPOWERRESIDUES 85
hapter
C 5(cid:2) Some Elementary Results for Prime
Distribution 91
5.1 INTRODUCTIONTOTHEBASICPROPERTIESOF
PRIMESANDTHEMAINRESULTSOFPRIME
NUMBERDISTRIBUTION 91
5.2 PROOFOFTHEEULERPRODUCTFORMULA 95
5.3 PROOFOFAWEAKERVERSIONOFTHEPRIME
NUMBERTHEOREM 97
5.4 EQUIVALENTSTATEMENTSOFTHEPRIME
NUMBERTHEOREM 105
hapter
C 6(cid:2) Simple Continued Fractions 109
6.1 SIMPLECONTINUEDFRACTIONSANDTHEIR
BASICPROPERTIES 109

6.2 SIMPLECONTINUEDFRACTIONREPRESENTATIONS
OFREALNUMBERS 113
6.3 APPLICATIONOFCONTINUEDFRACTIONIN
CRYPTOGRAPHY—ATTACKTORSAWITHSMALL
DECRYPTIONEXPONENTS 118
hapter
C 7(cid:2) Basic Concepts 121
7.1 MAPS 121
7.2 ALGEBRAICOPERATIONS 125
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7.3 HOMOMORPHISMSANDISOMORPHISMSBETWEEN
SETSWITHOPERATIONS 128
7.4 EQUIVALENCERELATIONSANDPARTITIONS 129
hapter
C 8(cid:2) Group Theory 133
8.1 DEFINITIONS 133
8.2 CYCLICGROUPS 135
8.3 SUBGROUPSANDCOSETS 137
8.4 FUNDAMENTALHOMOMORPHISMTHEOREM 141
8.5 CONCRETEEXAMPLESOFFINITEGROUPS 146
hapter
C 9(cid:2) Rings and Fields 151
9.1 DEFINITIONOFARING 151
9.2 INTEGRALDOMAINS,FIELDS,ANDDIVISIONRINGS 154
9.3 SUBRINGS,IDEALS,ANDRINGHOMOMORPHISMS 159
9.4 CHINESEREMAINDERTHEOREM 165
9.5 EUCLIDEANRINGS 168
9.6 FINITEFIELDS 170
9.7 FIELDOFFRACTIONS 172
hapter
C 10(cid:2) Some Mathematical Problems in Public
Key Cryptography 177
10.1 TIMEESTIMATIONANDCOMPLEXITYOF
ALGORITHMS 177

10.2 INTEGERFACTORIZATIONPROBLEM 184
10.3 PRIMALITYTESTS 185
10.4 THERSAPROBLEMANDTHESTRONGRSAPROBLEM 188
10.5 QUADRATICRESIDUES 189
10.6 THEDISCRETELOGARITHMPROBLEM 192
hapter
C 11(cid:2) Basics of Lattices 195
11.1 BASICCONCEPTS 195
11.2 SHORTESTVECTORPROBLEM 196
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11.3 LATTICEBASISREDUCTIONALGORITHM 197
11.4 APPLICATIONSOFLLLALGORITHM 200
References 207
Further Reading 209
Index 211

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Foreword
Asacornerstoneofinformationsecurity,cryptographyisasubjectwith
an ancient history, but it also is an emerging discipline that is widely
and effectively used in many areas of modern society. One of the two
fundamental issues of cryptography is to securely encrypt information
so that a third party will not get the content from its encrypted form.
The other one, in contrast, is how to break the encryption and get the
information from its encrypted form. There are many methods that
can be used in cryptography to protect information and break codes.
Mathematicshasalwaysbeenanimportanttoolincryptography.With
the invention of public key cryptosystem, mathematics has been play-
ing an indisputably important role in cryptography, so cryptography
has also become a special subject of mathematics.
Professor Xiaoyun Wang has always attached great importance to
training students in information security and building its core curric-
ular structure. “Number Theory and Algebraic Structures” is a core
course for cryptography, and she started to write lecture notes for the
courseasearlyas2003.Theselecturesweresuccessfullytaughtatboth
Shandong University and Tsinghua University, China. In collaboration
with Guangwu Xu, Minqiang Wang, and Xianmeng Meng, Professor
Wang revised and extended these lecture notes and published them in

this book. Since the content of this book has been described in detail
bytheauthorsinthepreface,Iamnotgoingtogooverthatagainhere.
However, I do want to point out that this book has a distinctive fea-
ture;namely,itcloselyintegratesbasicnumbertheoryandalgebrainto
cryptographic algorithms and complexity theory in a well-organized
manner throughout this book. This is of great importance for training
students in cryptography to have a unique way of thinking. My feel-
ing is that the “formalization,” which has been a quite effective way
of thinking in mathematics, is hard to work directly in cryptography
as it seems that the “formalization” thinking does not lead one to the
essence of cryptographic problems. This is a key problem that needs
ix
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x (cid:2) Foreword
special attention for mathematicians who decide to shift their research
to cryptography. Therefore, this book will definitely play a very posi-
tiveroleinimprovingthequalityofcorecoursesininformationsecurity
curriculum.
Professor Xiaoyun Wang studied with my brother, Professor
Chengdong Pang, a strong advocate in applying mathematics to sci-
ence and technology. He participated in the preliminary research of
seepage theory, thin shell theory, spline theory and application, direc-
tional blasting, and so on. Around 1990, he spearheaded the effort to
establish a cryptography research group and train graduate students
at Shandong University, China. Professor Xiaoyun Wang was the most
accomplished researcher in this rather successful group. In the field of
cryptanalysis, she has proposed and developed a theory and technique
for collision attack of hash functions, successfully breaking some major
popular cryptographic hash functions, such as MD5 designed by the
Turing award recipient Rivest and SHA-1 designed by the National
Institute of Standards and Technology (NIST) and the National Secu-
rity Agency (NSA) of the United States. These hash functions were
the core components of commonly used digital signatures and digital
certificates. Professor Wang’s research has surprised the cryptography
community, prompting the NIST to initiate a 5-year project for the
new standard hash function SHA-3 in 2007. She has also made great
contributions in analyzing message authentication codes and designing
cryptographic algorithms. She led the design of hash algorithm SM3,
which has been adopted as the national commercial hash algorithm.
Although I am not an expert in cryptography, I am delighted
to write the Foreword to Xiaoyun Wang’s book, as I feel it is my
responsibility and obligation to do so. I hope that she will continue

to passionately dedicate herself to research and teaching without being
distracted by fame and gain.
Chengbiao Pan
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