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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: L.Goldberg,Berkeley,CA,USA 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 ©SpringerInternationalPublishingSwitzerland2016 This work is subject to copyright. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland 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 implementation 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