Cryptography in C & C++ 2nd Edition

481 Pages·2005·3.651 MB·English

by  Michael Welschenbach

Cryptography
in C and C++
MICHAELWELSCHENBACH
TranslatedbyDAVIDKRAMER
press
A TM
CryptographyinCandC++
Copyright(cid:1)c 2005byMichaelWelschenbach
TranslatorandCompositor:DavidKramer
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LibraryofCongressCataloging-in-PublicationData
Welschenbach,Michael.
[KryptographieinCundC++.English]
CryptographyinCandC++/MichaelWelschenbach;translatedbyDavidKramer.–
2ndAmericaned.,rev.andenl.
p.cm.
ThefirstAmericaneditionisatranslationofthesecondGermanedition,whichhas
beenrevisedandexpandedfromthefirstGermanedition.
Includesbibliographicalreferencesandindex.
ISBN1-59059-502-5
1.Computersecurity. 2.Cryptography. 3.C(Computerprogram
language) 4.C++(Computerprogramlanguage) I.Title.
QA76.9.A25W43132005
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Contents
Foreword xiii
AbouttheAuthor xv
AbouttheTranslator xvi
PrefacetotheSecondAmericanEdition xvii
PrefacetotheFirstAmericanEdition xix
PrefacetotheFirstGermanEdition xxiii
I ArithmeticandNumberTheoryinC 1
1 Introduction 3
2 NumberFormats:TheRepresentationofLargeNumbersinC 13
3 InterfaceSemantics 19
4 TheFundamentalOperations 23
4.1 AdditionandSubtraction . . . . . . . . . . . . . . . . . . . . . . 24
4.2 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2.1 TheGradeSchoolMethod . . . . . . . . . . . . . . . . . 34
4.2.2 SquaringIsFaster . . . . . . . . . . . . . . . . . . . . . . 40
4.2.3 DoThingsGoBetterwithKaratsuba? . . . . . . . . . . . 45
4.3 DivisionwithRemainder . . . . . . . . . . . . . . . . . . . . . . 50
5 ModularArithmetic:CalculatingwithResidueClasses 67
6 WhereAllRoadsMeet:ModularExponentiation 81
6.1 FirstApproaches . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.2 M-aryExponentiation . . . . . . . . . . . . . . . . . . . . . . . 86
6.3 AdditionChainsandWindows . . . . . . . . . . . . . . . . . . . 101
6.4 MontgomeryReductionandExponentiation . . . . . . . . . . . 106
6.5 CryptographicApplicationofExponentiation . . . . . . . . . . . 118
v
Contents
7 BitwiseandLogicalFunctions 125
7.1 ShiftOperations . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.2 AllorNothing:BitwiseRelations . . . . . . . . . . . . . . . . . . 131
7.3 DirectAccesstoIndividualBinaryDigits. . . . . . . . . . . . . . 137
7.4 ComparisonOperators . . . . . . . . . . . . . . . . . . . . . . . 140
8 Input,Output,Assignment,Conversion 145
9 DynamicRegisters 157
10 BasicNumber-TheoreticFunctions 167
10.1 GreatestCommonDivisor . . . . . . . . . . . . . . . . . . . . . 168
10.2 MultiplicativeInverseinResidueClassRings . . . . . . . . . . . 175
10.3 RootsandLogarithms . . . . . . . . . . . . . . . . . . . . . . . 183
10.4 SquareRootsinResidueClassRings . . . . . . . . . . . . . . . . 191
10.4.1 TheJacobiSymbol . . . . . . . . . . . . . . . . . . . . . 192
10.4.2 SquareRootsModulopk . . . . . . . . . . . . . . . . . . 198
10.4.3 SquareRootsModulon . . . . . . . . . . . . . . . . . . . 203
10.4.4 CryptographywithQuadraticResidues . . . . . . . . . . 211
10.5 APrimalityTest . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
11 Rijndael:ASuccessortotheDataEncryptionStandard 237
11.1 ArithmeticwithPolynomials . . . . . . . . . . . . . . . . . . . . 239
11.2 TheRijndaelAlgorithm . . . . . . . . . . . . . . . . . . . . . . . 244
11.3 CalculatingtheRoundKey . . . . . . . . . . . . . . . . . . . . . 247
11.4 TheS-Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
11.5 TheShiftRowsTransformation . . . . . . . . . . . . . . . . . . . 249
11.6 TheMixColumnsTransformation . . . . . . . . . . . . . . . . . . 250
11.7 TheAddRoundKeyStep . . . . . . . . . . . . . . . . . . . . . . . . 252
11.8 EncryptionasaCompleteProcess . . . . . . . . . . . . . . . . . 253
11.9 Decryption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
11.10 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
11.11 ModesofOperation . . . . . . . . . . . . . . . . . . . . . . . . 260
12 LargeRandomNumbers 261
12.1 ASimpleRandomNumberGenerator . . . . . . . . . . . . . . . 265
12.2 CryptographicRandomNumberGenerators . . . . . . . . . . . 268
12.2.1 TheGenerationofStartValues . . . . . . . . . . . . . . . 269
12.2.2 TheBBSRandomNumberGenerator . . . . . . . . . . . 273
12.2.3 TheAESGenerator . . . . . . . . . . . . . . . . . . . . . 279
12.2.4 TheRMDSHA-1Generator . . . . . . . . . . . . . . . . . 283
vi
Contents
12.3 QualityTesting . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
12.3.1 Chi-SquaredTest . . . . . . . . . . . . . . . . . . . . . . 287
12.3.2 MonobitTest . . . . . . . . . . . . . . . . . . . . . . . . 289
12.3.3 PokerTest . . . . . . . . . . . . . . . . . . . . . . . . . . 289
12.3.4 RunsTest . . . . . . . . . . . . . . . . . . . . . . . . . . 289
12.3.5 LongrunsTest . . . . . . . . . . . . . . . . . . . . . . . . 289
12.3.6 AutocorrelationTest . . . . . . . . . . . . . . . . . . . . 290
12.3.7 QualityoftheFLINT/CRandomNumberGenerators . . . 290
12.4 MoreComplexFunctions . . . . . . . . . . . . . . . . . . . . . . 291
13 StrategiesforTestingLINT 305
13.1 StaticAnalysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
13.2 Run-TimeTests . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
II ArithmeticinC++withtheClassLINT 317
14 LetC++SimplifyYourLife 319
14.1 NotaPublicAffair:TheRepresentationofNumbersinLINT . . . 324
14.2 Constructors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
14.3 OverloadedOperators . . . . . . . . . . . . . . . . . . . . . . . 329
15 TheLINTPublicInterface:MembersandFriends 337
15.1 Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
15.2 NumberTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
15.3 StreamI/OofLINTObjects . . . . . . . . . . . . . . . . . . . . . 352
15.3.1 FormattedOutputofLINTObjects . . . . . . . . . . . . . 353
15.3.2 Manipulators . . . . . . . . . . . . . . . . . . . . . . . . 360
15.3.3 FileI/OforLINTObjects . . . . . . . . . . . . . . . . . . . 362
16 ErrorHandling 367
16.1 (Don’t)Panic... . . . . . . . . . . . . . . . . . . . . . . . . . . 367
16.2 User-DefinedErrorHandling. . . . . . . . . . . . . . . . . . . . 369
16.3 LINTExceptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
17 AnApplicationExample:TheRSACryptosystem 377
17.1 AsymmetricCryptosystems . . . . . . . . . . . . . . . . . . . . 378
17.2 TheRSAAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . 380
17.3 DigitalRSASignatures . . . . . . . . . . . . . . . . . . . . . . . 395
17.4 RSAClassesinC++ . . . . . . . . . . . . . . . . . . . . . . . . . 403
18 DoItYourself:TestLINT 413
vii
Contents
19 ApproachesforFurtherExtensions 417
III Appendices 419
A DirectoryofCFunctions 421
A.1 Input/Output,Assignment,Conversions,Comparisons . . . . . . 421
A.2 BasicCalculations . . . . . . . . . . . . . . . . . . . . . . . . . 422
A.3 ModularArithmetic. . . . . . . . . . . . . . . . . . . . . . . . . 423
A.4 BitwiseOperations . . . . . . . . . . . . . . . . . . . . . . . . . 425
A.5 Number-TheoreticFunctions . . . . . . . . . . . . . . . . . . . 426
A.6 GenerationofPseudorandomNumbers . . . . . . . . . . . . . . 427
A.7 RegisterManagement . . . . . . . . . . . . . . . . . . . . . . . 431
B DirectoryofC++Functions 433
B.1 Input/Output,Conversion,Comparison:MemberFunctions . . . 433
B.2 Input/Output,Conversion,Comparison:FriendFunctions . . . . 436
B.3 BasicOperations:MemberFunctions . . . . . . . . . . . . . . . 438
B.4 BasicOperations:FriendFunctions . . . . . . . . . . . . . . . . 439
B.5 ModularArithmetic:MemberFunctions . . . . . . . . . . . . . . 440
B.6 ModularArithmetic:FriendFunctions . . . . . . . . . . . . . . . 442
B.7 BitwiseOperations:MemberFunctions . . . . . . . . . . . . . . 443
B.8 BitwiseOperations:FriendFunctions . . . . . . . . . . . . . . . 444
B.9 Number-TheoreticMemberFunctions . . . . . . . . . . . . . . 445
B.10 Number-TheoreticFriendFunctions . . . . . . . . . . . . . . . 446
B.11 GenerationofPseudorandomNumbers . . . . . . . . . . . . . . 450
B.12 MiscellaneousFunctions . . . . . . . . . . . . . . . . . . . . . . 450
C Macros 451
C.1 ErrorCodesandStatusValues . . . . . . . . . . . . . . . . . . . 451
C.2 AdditionalConstants . . . . . . . . . . . . . . . . . . . . . . . . 451
C.3 MacroswithParameters . . . . . . . . . . . . . . . . . . . . . . 453
D CalculationTimes 459
E Notation 461
F ArithmeticandNumber-TheoreticPackages 463
References 465
Index 473
viii
List of Figures
4-1 Calculationsformultiplication . . . . . . . . . . . . . . . . . . 35
4-2 Calculationsforsquaring . . . . . . . . . . . . . . . . . . . . . 41
4-3 CPUtimeforKaratsubamultiplication . . . . . . . . . . . . . . 49
4-4 Calculationalschemafordivision . . . . . . . . . . . . . . . . . 51
11-1 LayeringoftransformationsintheRijndaelrounds . . . . . . . . 246
11-2 DiagramforroundkeysforLk =4 . . . . . . . . . . . . . . . . 248
12-1 Periodicbehaviorofapseudorandomsequence . . . . . . . . . 263
17-1 Exampleoftheconstructionofacertificate . . . . . . . . . . . . 401
17-2 Certificationofadigitalsignature . . . . . . . . . . . . . . . . . 402
ix
List of Tables
1-1 ArithmeticandnumbertheoryinCindirectoryflint/src . . . . . 7
1-2 Arithmeticmodulesin80x86assembler(seeChapter19)indirec-
toryflint/src/asm . . . . . . . . . . . . . . . . . . . . . . . . . 7
1-3 Tests(seeSection13.2andChapter18)indirectoriesflint/testand
flint/test/testvals . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1-4 Librariesin80x86assembler(seeChapter19)indirectoriesflint/lib
andflint/lib/dll . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1-5 RSAimplementation(seeChapter17)indirectoryflint/rsa . . . 8
3-1 FLINT/Cerrorcodes . . . . . . . . . . . . . . . . . . . . . . . . 21
5-1 Compositiontableforadditionmodulo5 . . . . . . . . . . . . . 71
5-2 Compositiontableformultiplicationmodulo5 . . . . . . . . . . 71
6-1 Requirementsforexponentiation . . . . . . . . . . . . . . . . . 88
6-2 Numbers of multiplications for typical sizes of exponents and
variousbases2k . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6-3 Valuesforthefactorizationoftheexponentdigitsintoproductsof
apowerof2andanoddfactor . . . . . . . . . . . . . . . . . . . 90
6-4 Numbers of multiplications for typical sizes of exponents and
variousbases2k . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6-5 ExponentiationfunctionsinFLINT/C . . . . . . . . . . . . . . . 117
7-1 ValuesofaBooleanfunction . . . . . . . . . . . . . . . . . . . . 132
7-2 ValuesoftheCLINTfunctionand_l() . . . . . . . . . . . . . . . . 132
7-3 ValuesoftheCLINTfunctionor_l() . . . . . . . . . . . . . . . . 133
7-4 ValuesoftheCLINTfunctionxor_l() . . . . . . . . . . . . . . . . 133
8-1 Diagnosticvaluesofthefunctionvcheck_l() . . . . . . . . . . . 154
10-1 Thetenlargestknownprimes(asofDecember2004) . . . . . . . 215
10-2 Thenumberofprimesuptovariouslimitsx . . . . . . . . . . . 220
10-3 ThenumberkofpassesthroughtheMiller–Rabintesttoachieve
probabilitiesoferrorlessthan2−80and2−100asafunctionofthe
numberlofbinarydigits(after[DaLP]). . . . . . . . . . . . . . . 228
10-4 ApproximatecalculationtimesfortheAKStest,after[CrPa] . . . 234
F
11-1 Elementsof 23 . . . . . . . . . . . . . . . . . . . . . . . . . . 240
xi