Cryptography Arithmetic: Algorithms and Hardware Architectures PDF
Preview Cryptography Arithmetic: Algorithms and Hardware Architectures
Advances in Information Security 77 Amos R. Omondi Cryptography Arithmetic Algorithms and Hardware Architectures Advances in Information Security Volume 77 Serieseditor SushilJajodia,GeorgeMasonUniversity,Fairfax,VA,USA Moreinformationaboutthisseriesathttp://www.springer.com/series/5576 Amos R. Omondi Cryptography Arithmetic Algorithms and Hardware Architectures AmosR.Omondi StateUniversityofNewYork–Korea Songdo,SouthKorea ISSN1568-2633 ISSN2512-2193 (electronic) AdvancesinInformationSecurity ISBN978-3-030-34141-1 ISBN978-3-030-34142-8 (eBook) https://doi.org/10.1007/978-3-030-34142-8 ©SpringerNatureSwitzerlandAG2020 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG. Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland ToYokoyamaMasami Preface This book has been developed from notes that I wrote for a course on algorithms and hardware architectures for computer arithmetic. The course was a broad one—covering the usual areas of fixed-point arithmetic, floating-point arithmetic, elementaryfunctions,andsoforth—witha“tailend”onapplications,oneofwhich wascryptography.PartsIIandIIIofthebookarefromthattailend.AddingPartI, on basic integer arithmetic—the part of computer arithmetic that is relevant for cryptography—makesforaself-containedbookonthemainsubject. Thestudentswhotooktheaforementionedcoursewerefinal-yearundergraduate and first-year graduate students in computer science and computer engineering. The book is intended to serve as an introduction to such students and others with an interest in the subject. The required background consists of an understanding of digital logic, at the level of a good first course, and the ability to follow basic mathematical reasoning. No knowledge of cryptography is necessary; brief discussionsofsomehelpfulbasicsareincludedinthebook. Part I is on algorithms and hardware architectures for the basic arithmetic operations:addition,subtraction,multiplication,anddivision. Muchofthearithmeticofmoderncryptographyisthearithmeticoffinitefields andoftwotypesoffieldinparticular:primefields(forwhichthearithmeticisjust modulararithmeticwithprimemoduli)andbinaryfields.PartIIcoverstheformer andPartIIIthelatter.Eachpartincludesachapteronmathematicalfundamentals, a short chapter on well-known cryptosystems (to provide context), and two or more chapters on the arithmetic. Binary-field arithmetic is used in elliptic-curve cryptography (which also uses prime-field arithmetic); an introductory chapter is includedonsuchcryptography. Cryptography involves numbers of high precision, but there is no more under- standing to be gained with examples of such precision, in binary or hexadecimal, thanwiththoseoflowprecisionindecimal.Therefore,forthereader’s“visualease” andtoensurethatheorshecaneasilyworkthroughtheexamples,allexamplesare of small numbers, with most in decimal. It is, however, to be understood that in practice the numbers will be large and the radix will almost always be a power of two;thehardware-relateddiscussionsareforpowersoftwo. vii viii Preface Anoteonwritingstyle:Forbrevity,andprovidednoconfusionispossible,Ihave insomeplacesbeen“sloppy”withthelanguage.Asanexample,writing“number” insteadofthemoreprecise“representationof...number.”Anotherexampleisthe useof“speed”and“cost”inthediscussionofanarchitecture;thetermsrefertothe realizationofthearchitecture. Anoteonnotation:Ihopemeaningwillbeclearfromusage,butthefollowing examples should be noted. x denotes a number; x denotes bit or digit i in the i representation of x; x denotes the number represented by several bits or digits h intherepresentationofx;X denotesthevalueofXiniterationi;andX denotes i i,j bitordigitiintherepresentationofX ;inalgorithms,“=”denotesassignmentand i x denotesthex-coordinateofapointP. P Afinalnoteisontherepetitionofsometext(afewalgorithms).Thisisanaspect I have retained from the original lecture notes, as I imagine the reader will find it convenienttonothavetogobackovernumerouspagesforaparticularalgorithm. This work was supported by the Ministry of Science and ICT (MSIT), Korea, undertheICTConsilienceCreative(IITP-2019-H8601-15-1011),supervisedbythe Institute for Information & Communications Technology Planning & Evaluation (IITP). Songdo,SouthKorea AmosR.Omondi July2019 Acknowledgements AllThanksandPraisetoAlmightyGod ix Contents PartI 1 BasicComputerArithmetic ............................................... 3 1.1 Addition .............................................................. 4 1.1.1 Serial ........................................................ 5 1.1.2 Carry-Ripple................................................ 8 1.1.3 Parallel-Prefix............................................... 9 1.1.4 Carry-Select................................................. 17 1.1.5 HighPrecision.............................................. 20 1.1.6 SignedNumbersandSubtraction.......................... 21 1.2 Multiplication ........................................................ 29 1.2.1 Sequential................................................... 31 1.2.2 HighRadix.................................................. 36 1.2.3 ParallelandSequential-Parallel............................ 41 1.2.4 HighPrecision.............................................. 43 1.2.5 SignedNumbers............................................ 45 1.2.6 Squaring..................................................... 45 1.3 Division............................................................... 47 References.................................................................... 67 PartII 2 MathematicalFundamentalsI:NumberTheory ....................... 71 2.1 Congruences.......................................................... 71 2.2 Modular-ArithmeticOperations..................................... 73 2.2.1 Addition,Subtraction,andMultiplication................. 73 2.2.2 Division ..................................................... 74 2.3 GeneratorsandPrimitiveRoots ..................................... 75 2.4 QuadraticResiduesandSquareRoots............................... 78 2.5 TheChineseRemainderTheorem................................... 82 2.6 ResidueNumberSystems............................................ 85 References.................................................................... 87 xi
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