Algebra for Cryptologists

311 Pages·2016·1.39 MB·English

by  Meijer

Springer Undergraduate Texts
in Mathematics and Technology
Alko R. Meijer
Algebra for
Cryptologists
Springer Undergraduate Texts
in Mathematics and Technology
SeriesEditors:
J.M.Borwein,Callaghan,NSW,Australia
H.Holden,Trondheim,Norway
V.H.Moll,NewOrleans,LA,USA
EditorialBoard:
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A.Iske,Hamburg,Germany
P.E.T.Jorgensen,IowaCity,IA,USA
S.M.Robinson,Madison,WI,USA
Moreinformationaboutthisseriesathttp://www.springer.com/series/7438

Alko R. Meijer
Algebra for Cryptologists
123
AlkoR.Meijer
Tokai,CapeTown
SouthAfrica
ISSN1867-5506 ISSN1867-5514 (electronic)
SpringerUndergraduateTextsinMathematicsandTechnology
ISBN978-3-319-30395-6 ISBN978-3-319-30396-3 (eBook)
DOI10.1007/978-3-319-30396-3
LibraryofCongressControlNumber:2016945761
MathematicsSubjectClassification(2010):94A60,08-01,11A,11T
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Foreword
Itissometimes claimed thattheworld’s population falls intotwoclasses: those whobelieve
thatthepopulationcanbedividedintotwoclasses,andthosewhodon’t.Wewon’tenterinto
that argument, but it is true that the reading population can be divided into those who read
forewords and those who don’t. Since you have read this far, you clearly belong to the first
class,whichgivesmeachancetoexplainwhatthisbookisabout.
Cryptology is a subject that lies in the intersection of three major fields of science: Com-
puterScience,ElectricalandElectronicEngineeringandMathematics—giveninalphabetical
order,toavoidfruitlessdiscussionsabouttheirrelativeimportance.Mathematics,intheform
of(“abstract”)algebra,numbertheoryanddiscretemathematicsingeneral,providesmuchof
its logical foundation, but computer scientists provide many of the opportunities for its use,
while the electrical and electronic engineers supply the platforms on which implementation
cantakeplace.
The primary purpose of this book is to provide individuals with Electronic Engineering
or Computer Science backgrounds, who find themselves entering the world of Cryptology,
either as practitioners or as students, with some essential insights into the Algebra used.
Here “cryptology” encompasses both the secret key/symmetric and public key/asymmetric
aspects of the field, but the emphasis is perhaps more on the symmetric side than is usual
in the textbook literature. Partial justification for this, if justification is needed, is given in
Sect.1.6.3, and the importance of symmetric cryptography in practice will, I hope, become
apparent inthecourseofreadingthisbook.
Thisshouldgiveanindicationofthemathematicalknowledgeyou,thereader,areassumed
to have: firstly, that mysterious quality which used to be called “mathematical maturity”,
meaning you don’t run away screaming in fright when you encounter a † sign. More
constructively, you need to have some idea of what a mathematical proof is (e.g. proof
by induction), and have a very basic understanding of propositional logic and Boolean
algebra. You are assumed to be familiar with linear algebra (or at least with vectors and
matrices). Familiarity with elementary probability theory would be useful, but the concepts
mostessential forourpurposes aredealtwithintheAppendix.
On the other hand, the reader is not expected to have had any previous exposure to
(“abstract” or, as itwas called in the middle decades of the last century, “modern”) Algebra
v
vi Foreword
as such, and I believe that this book could, in fact, be used as an introduction to Algebra,
where,unusually,theemphasisisonitsapplications, whilethepurelyalgebraiccontentitself
isseverelylimited.
It is assumed that your “mathematical maturity” will allow for the development of the
material at a fairly rapid pace and allow for explanations of how and where these concepts
appear in the field. If the reader already has some, or even a great, knowledge of Algebra, I
apologiseforboringhimorherbuthopethattheapplicationsmayneverthelessbeinteresting.
Myadvice tosuch readers is to scan the relevant sections very quickly. Thesame applies to
readers who, say, have previous acquaintance with Coding Theory or Information Theory:
justscanthebriefintroductions tothesefields.
Just three final comments (and thank you for reading this far): Firstly, many sections are
followed by ashort sequence of generally not very demanding exercises. Please do not skip
them,evenifyoudon’tactuallysolvealltheproblems.Someofthemaresimpleapplications,
a few others demand considerable thought. Give at least some thought to the latter kind.
There are also expressions like “the diligent reader is invited to prove this” in the text, not
explicitly labelled “Exercises”. Such statements mean that you are welcome to accept the
relevantstatementastrue,withoutactually constructing aproofyourself.
Secondly:WhilethisbookisaimedatthoseinterestedinCryptology,Imuststressthatthe
book concentrates on the Mathematics involved: our descriptions blithely ignore implemen-
tation issues, including their security, and all matters concerned with the complexity of any
algorithms thatwedealwith.
Finally, you will notice that there are many footnotes.1 Most of these refer to original
sources of definitions, theorems, etc., but some take the form of more or less parenthetical
remarks, which are relevant to the topic discussed, but don’t fit into the argument being
developed.2 If footnotes get you down, blame the LATEX typesetting language, which makes
inserting themfartooeasy.
James Thurber quoted with approval the schoolgirl who reviewed a book she had been
told toread and review inthesingle sentence “Thisbook toldmemoreabout penguins than
I wanted to know.” I hope that in reading this book you won’t find yourself muttering the
equivalent of “Perish these pestilential penguins!” to yourself. But there will be occasions
where the subject matter is already familiar to you, in which case you should merely scan
the relevant section. This may apply in particular to the early sections of Chap.1; you are
probably familiar with the ideas considered there, but the material is included in an attempt
tomakethebookasselfcontainedaspossible.Itismypleasuretothankthereviewersofthe
manuscript, and in particular Dr Christine Swart of the University of Cape Town, for their
encouragement andconstructive criticism. Thanksarealsoduetovariousgroups ofaspiring
cryptologists onwhomIhavetestedmuchofthematerialinthisbook.
ButImustacceptresponsibility foranyremaining errorsandothershortcomings.
1Ioncereadtheacademicworkofanauthordescribedas“laboriouslyhobblingalongonasetoffootnotes”.
Ihopethisdescriptiondoesnotfithere.
2Andsomeofthefootnotesservenopurposeatall,likethetwoonthispage.
Foreword vii
I hope you will get as much enjoyment out of this book as I have out of my involvement
withCryptology.
Tokai,CapeTown,SouthAfrica AlkoR.Meijer
December2015

Contents
1 PrerequisitesandNotation .......................................................... 1
1.1 Sets.............................................................................. 1
1.2 ProductsofSets ................................................................ 4
1.3 Relations........................................................................ 5
1.4 Functions ....................................................................... 8
1.5 BinaryOperations.............................................................. 9
1.6 Cryptography................................................................... 10
1.6.1 Encryption Mechanisms.............................................. 11
1.6.2 Confusion andDiffusion ............................................. 11
1.6.3 SymmetricandAsymmetricEncryption ............................ 12
1.7 Notational Conventions ........................................................ 13
1.7.1 FloorandCeiling ..................................................... 13
1.7.2 Fractional Part ........................................................ 13
1.7.3 ExclusiveOr .......................................................... 14
1.7.4 MatrixMultiplication................................................. 14
2 BasicPropertiesoftheIntegers..................................................... 17
2.1 Divisibility...................................................................... 17
2.2 IdealsandGreatestCommonDivisors ........................................ 19
2.3 TheEuclideanAlgorithm ...................................................... 23
2.3.1 Stein’sgcdAlgorithm ................................................ 27
2.4 Congruences.................................................................... 28
2.5 Fermat’sFactoringMethod .................................................... 31
2.6 SolvingLinearCongruences................................................... 33
2.7 TheChineseRemainderTheorem............................................. 36
2.8 SomeNumber-Theoretic Functions ........................................... 38
2.8.1 Multiplicative Functions ............................................. 38
2.8.2 TheMöbiusFunction................................................. 41
2.8.3 Euler’s(cid:2)-Function.................................................... 42
2.8.4 TheCasenD p(cid:2)q.................................................... 44
ix