Modern Cryptography and Elliptic Curves: A Beginner's Guide (Student Mathematical Library)

265 Pages·2017·3.166 MB·English

by  Thomas R. Shemanske (author)

STUDENT MATHEMATICAL LIBRARY
Volume 83
Modern
Cryptography
and Elliptic
Curves

A Beginner's Guide
Thomas R. Shemanske
STUDENT MATHEMATICAL LIBRARY
Volume 83
Modern
Cryptography
and Elliptic
Curves

A Beginner's Guide
Thomas R. Shemanske
American Mathematical Society
Providence, Rhode Island
Editorial Board
SatyanL. Devadoss John Stillwell (Chair)
Erica Flapan Serge Tabachnikov
2010 Mathematics Subject Classification. Primary 11-01, 68-01, 11Axx,
14G50,11T71, 68P25, 11Y05, 91A66, 11G05, 81P68.
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Contents
Preface vii
Introduction ix
Chapter 1. Three Motivating Problems 1
§1.1. Fermat’s Last Theorem 3
§1.2. The Congruent Number Problem 5
§1.3. Cryptography 6
Chapter 2. Back to the Beginning 9
§2.1. The Unit Circle: Real vs. Rational Points 10
§2.2. Parametrizing the Rational Points on the Unit Circle 12
§2.3. Finding all Pythagorean Triples 16
§2.4. Looking for Underlying Structure: Geometry vs.
Algebra 27
§2.5. More about Points on Curves 34
§2.6. Gathering Some Insight about Plane Curves 38
§2.7. Additional Exercises 43
Chapter 3. Some Elementary Number Theory 45
§3.1. The Integers 46
§3.2. Some Basic Properties of the Integers 47
iii
iv Contents
§3.3. Euclid’s Algorithm 52
§3.4. A First Pass at Modular Arithmetic 56
§3.5. Elementary Cryptography: Caesar Cipher 63
§3.6. Affine Ciphers and Linear Congruences 66
§3.7. Systems of Congruences 70
Chapter 4. A Second View of Modular Arithmetic: Z and U 73
n n
§4.1. Groups and Rings 73
§4.2. Fractions and the Notion of an Equivalence Relation 77
§4.3. Modular Arithmetic 79
§4.4. A Few More Comments on the Euler Totient Function 93
§4.5. An Application to Factoring 95
Chapter 5. Public-Key Cryptography and RSA 101
§5.1. A Brief Overview of Cryptographic Systems 102
§5.2. RSA 107
§5.3. Hash Functions 114
§5.4. Breaking Cryptosystems and Practical RSA Security
Considerations 123
Chapter 6. A Little More Algebra 127
§6.1. Towards a Classification of Groups 128
§6.2. Cayley Tables 128
§6.3. A Couple of Non-abelian Groups 131
§6.4. Cyclic Groups and Direct Products 134
§6.5. Fundamental Theorem of Finite Abelian Groups 138
§6.6. Primitive Roots 141
§6.7. Diffie–Hellman Key Exchange 143
§6.8. ElGamal Encryption 144
Chapter 7. Curves in Affine and Projective Space 147
§7.1. Affine and Projective Space 147
§7.2. Curves in the Affine and Projective Plane 153
Contents v
§7.3. Rational Points on Curves 156
§7.4. The Group Law for Points on an Elliptic Curve 159
§7.5. A Formula for the Group Law on an Elliptic Curve 179
§7.6. The Number of Points on an Elliptic Curve 185
Chapter 8. Applications of Elliptic Curves 189
§8.1. Elliptic Curves and Factoring 190
§8.2. Elliptic Curves and Cryptography 196
§8.3. Remarks on a Post-Quantum Cryptographic World 198
Appendix A. Deeper Results and Concluding Thoughts 203
§A.1. The Congruent Number Problem and Tunnell’s
Solution 203
§A.2. A Digression on Functions of a Complex Variable 209
§A.3. Return to the Birch and Swinnerton-Dyer Conjecture 211
§A.4. Elliptic Curves over C 212
Appendix B. Answers to Selected Exercises 219
§B.1. Chapter 2 219
§B.2. Chapter 3 231
§B.3. Chapter 4 233
§B.4. Chapter 5 236
§B.5. Chapter 6 238
§B.6. Chapter 7 241
Bibliography 245
Index 249

Preface
A question implicit in many conversations with undergraduates at
first-yearorientationis,“IfIweretotakeonlyonemathematicsclass,
what should it be?” Of course, there is no one correct answer. A
moreinterestingquestion(notthatitgetsasked)mightbe,“Istherea
classwhichintroducesmetothesubjectofmathematics?” Inanswer,
somemightproffervariousdiscrete/finitemath/introductiontoproof
courses, but I think these are not really an answer to that question.
A number of years ago, our department was interested in providing
a wide variety of courses which attempted to answer that question,
and this book details one such offering.
Therefore, one goal of thisbookis topresent some of the vistaof
modern mathematics to undergraduates who have just started their
mathematical careers, with the intent of enticing (and guiding) them
into taking mathematics courses beyond the typical calculus regime.
As mathematicians, we have acquired the perspective that mathe-
matics represents not only a broad collection of tools which can be
broughttobeartosolveamyriadofproblems,butmoreinterestingly,
that the various subdisciplines in mathematics are interconnected of-
teninsurprisingways,addingimmenselytotherichnessandallureof
thedisciplineitself. Undergraduatesbeginningtheircareerscertainly
do not have that perspective, and if they are lucky, they begin to
acquireitonlyneartheendoftheirundergraduatemajors. Thehope
vii
viii Preface
inwritingthisbookistoprovidesomeofthatperspectivetostudents
at an early stage in their careers so as both to (re)excite them about
mathematical exploration, and to help inform their choices as they
go forward in their undergraduate experience.
Thefocalpointforthistextistoleadstudentstounderstandthe
arithmetic of elliptic curves over a finite field and some applications
of elliptic curvestomoderncryptography. Assuming only calculus as
prerequisite, there is a great deal of ground to cover, but a wonder-
ful opportunity to demonstrate how many areas of mathematics are
intertwined.
That said, this book is not (nor is it intended to be) a typical
textbook in many respects. While the topics introduced include ma-
terial on elementary number theory, abstract algebra, cryptography,
affineandprojectivegeometry,theintentisnottopresentathorough
introduction to any of those subjects; it is meant to generate interest
in exploring those subjects in more detail. Excellent books devoted
individuallytothosetopicsareplentiful,buttypicallytheyareaimed
at a more mathematically sophisticated audience.
This book aims at a mathematically young audience, one that
morelikelythannothasneverseenasubstantivemathematicalproof.
Indeed, the only real prerequisite for this book is some one-variable
calculus; the rest of the mathematical topics are introduced on-the-
fly. This is also not a standard textbook in another important sense.
Insteadofpresentingsuccinctproofsofresults(asisdoneinatypical
textbook), many proofs are presented more as explorations, includ-
ing (on occasion) some intentional peeks down blind alleys. That is
to say, this book makes a significant effort to teach students about
how to produce or discover a proof, by presenting mathematics as an
exploration. Indeed, while somewhat of a clich´e, the book is a great
deal more about the journey than the destination, and it is intended
to point to the many branches off the main path to be explored in
the future.
Intheend,thebookseemstoserveseveralpurposes. Itserves,as
initially conceived, as a means of introducing many topics in modern
mathematicswithinterconnectionsamongthemtomotivatestudents
to take more mathematics. It also seems well suited to serve as an
Introduction ix
alternate course introducing proofs and abstract mathematics, which
occupies a prominent place in many mathematics programs. And
finally,giventhatonecannotreallyunderstandmoderncryptography
without some of its mathematical underpinnings, this book is well
suited to computer science programs which desire to offer a course
investigatingthepracticalandimplementationsidesofcryptography,
butwhichneedtheirstudentstohavesomesemblanceofitsnecessary
mathematical background.
Introduction
This book is written to introduce a student with only single-variable
calculus as background to enough mathematics to understand the
basics of elliptic curves over finite fields and their applications to
modern cryptography. Topics include basic notions in elementary
number theory and abstract algebra, aspects of affine and projective
geometry, as well cryptography and cryptanalysis. The goal of this
book is not so much to provide complete answers to the questions we
raise as it is to show the many connections between those questions
and areas of mathematics whose further study will provide deeper
answers.
In Chapter 1 we give a cursory exposition of three problems in
number theory which are connected to elliptic curves: Fermat’s Last
Theorem,thecongruentnumberproblem,andapplicationsofnumber
theory to cryptography.
Chapter2isquitebroad,recastingproblemsinnumbertheoryas
problemsamenable togeometricoralgebraicinterpretation. Welook
at connections of congruent numbers to Pythagorean triples, and at
connections between Pythagorean triples and rational points on the
unit circle. We explore in detail how to parametrize the rational
points on the unit circle and use the parametrization to produce a
simple algorithm toenumerate square-free congruent numbers. Then
we begin to look for structure inherent in certain sets. For example,
we know the set of points in R3 that satisfy x+y+z = 0 has the
geometricstructureofaplane throughtheorigin. Theset ofrational
pointsthatsatisfythesameequationdoesnotseemtohavegeometric
structure,butitdoesstillhavealgebraicstructureoncewedefinethe