Cryptography Arithmetic: Algorithms and Hardware Architectures

338 Pages·2020·4.564 MB·English

by  Amos R. Omondi

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
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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