Cryptography in C and C++

434 Pages·2001·12.66 MB·English

by  Michael Welschenbach (auth.)

Cryptography
in and
C C++
MICHAEL WELSCHENBACH
Translated by DAVID KRAMER
APress Media, LLC
Cryptography in C and C++
Copyright © 2001 by Michael Welschenbach
Originally published by Apress in 2001
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Contents
list of Figures ix
list of Tables xi
Preface to the Translation of the Second Edition xiii
Preface to the First Edition xvii
I Arithmetic and Number Theory in C 1
1 Introduction 3
2 Number Formats: The Representation of Large Numbers inC 11
3 Interface Semantics 17
4 The Fundamental Operations 21
4.1 Addition and Subtraction. 22
4.2 Multiplication . . . . . . . 32
4.2.1 The Grade School Method 33
4.2.2 Squaring Is Faster . . . . . 39
4.2.3 Do Things Go Better with Karatsuba? 44
4.3 Division with Remainder . . . . . . . . . . . 49
5 Modular Arithmetic: Calculating with Residue Classes 65
6 Where All Roads Meet: Modular Exponentiation 79
6.1 FirstApproaches ....... . 79
6.2 M-ary Exponentiation . . . . . . . . . . . 85
6.3 Addition Chains and Windows . . . . . . . 98
6.4 Montgomery Reduction and Exponentiation 103
6.5 Cryptographic Application of Exponentiation . 114
7 Bitwise and Logical Functions 123
7.1 Shift Operations ............ . 123
7.2 All or Nothing: Bitwise Relations .... . 129
7.3 Direct Access to Individual Binary Digits . 135
7.4 Comparison Operators . . . . . . . . . . 138
Contents
8 Input, Output, Assignment, Conversion 143
9 Dynamic Registers 155
10 Basic Number-Theoretic Functions 165
1001 Greatest Common Divisor 0 0 0 0 0 0 0 0 0 0 166
1002 Multiplicative Inverse in Residue Class Rings 173
10o3 Roots and Logarithms 0 0 0 0 0 0 0 181
10o4 Square Roots in Residue Class Rings 187
10o4o1 The Jacobi Symbol o 0 o 188
100402 Square Roots Modulo pk 0 0 194
10o4o3 Square Roots Modulo n 0 o o 199
10.4.4 Cryptography with Quadratic Residues 207
10o5 APrimalityTest 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 210
11 Large Random Numbers 229
12 Strategies for Testing UNT 243
1201 StaticAnalysis 0 0 245
12o2 Run-TimeTests 0 0 0 247
II Arithmetic in C++ with the Class LINT 255
13 Let C++ Simplify Your life 257
13o1 Not a Public Affair: The Representation of Numbers in LINT 262
13o2 Constructors o 0 0 0 o 263
13o3 Overloaded Operators 0 0 0 0 0 0 0 0 0 0 0 267
14 The LINT Public Interface: Members and Friends 275
14o1 Arithmetic 0 0 0 0 0 0 0 0 275
14o2 NumberTheory 0 0 0 0 0 0 0 0 0 0 0 0 0 0 285
14o3 Stream 110 ofLINT Objects 0 0 0 0 0 0 0 0 290
14o3o1 Formatted Output of LINT Objects 292
14o3o2 Manipulators 0 0 0 0 0 0 299
14o3o3 File 1/0 for LINT Objects 302
15 Error Handling 307
1501 (Don't) Panic 00 0 307
15o2 User-Defined Error Handling 0 309
15o3 LINT Exceptions 0 0 0 0 0 0 0 0 311
16 An Application Example: The RSA Cryptosystem 317
16o1 Asymmetric Cryptosystems 318
16o2 The RSA Algorithm 0 0 0 0 0 0 0 0 0 0 0 0 0 320
vi
Contents
16.3 Digital RSA Signatures 334
16.4 RSA Classes in C++ 341
17 Do It Yourself: Test LINT 351
18 Approaches for Further Extensions 355
19 Rijndael: A Successor to the Data Encryption Standard 357
19.1 Arithmetic with Polynomials . 359
19.2 The RijndaelAlgorithm .. 363
19.3 Calculating the Round Key . . 365
19.4 The S-Box .......... . 367
19.5 The Shi ftRow Transformation 368
19.6 The MixColumn Transformation . 370
19.7 The AddRoundKey Step ..... . 371
19.8 Encryption of a Block as a Complete Process 371
19.9 Decryption ................. . 375
III Appendices 379
A Directory of C Functions 381
A.1 Input/Output, Assignment, Conversions, Comparisons . 381
A.2 Basic Calculations . 382
A.3 Modular Arithmetic . . . . . . 383
A.4 Bitwise Operations . . . . . . 386
A.5 Number-Theoretic Functions 387
A.6 Generation of Pseudorandom Numbers 388
A.7 Register Management 389
B Directory of C++ Functions 391
B.1 Input/Output, Conversion, Comparison: Member Functions . 391
B.2 Input/Output, Conversion, Comparison: Friend Functions . 394
B.3 Basic Operations: Member Functions . . 396
B.4 Basic Operations: Friend Functions . . . 397
B.5 Modular Arithmetic: Member Functions . 398
B.6 Modular Arithmetic: Friend Functions . 399
B. 7 Bitwise Operations: Member Functions 401
B.8 Bitwise Operations: Friend Functions . 402
B.9 Number-Theoretic Member Functions 402
B.10 Number-Theoretic Friend Functions . 404
B.11 Generation of Pseudorandom Numbers 407
B.l2 Miscellaneous Functions ....... . 408
vii
Contents
c Macros 409
C.1 Error Codes and Status Values 409
C.2 Additional Constants . . 409
C.3 Macros with Parameters 411
D Calculation Times 415
E Notation 417
F Arithmetic and Number-Theoretic Packages 419
References 421
Index 427
viii
List of Figures
4.1 Calculations for multiplication .... 33
4.2 Calculations for squaring . . . . . . . 39
4.3 CPU time for Karatsuba multiplication 48
4.4 Calculational schema for division . . . 50
11.1 Periodic behavior of a pseudorandom sequence 231
16.1 Example of the construction of a certificate . 340
16.2 Certification of a digital signature . . . . . . 340
19.1 Layering of transformations in the Rijndael rounds . 364
19.2 Diagram for round keys for Lk = 4 . . . . . . . . . 367
List of Tables
1.1 Arithmetic and Number Theory in C in Directory flint/ src . . . . 6
1.2 Arithmetic and Number Theory in C++ in Directory flint/ src . . . 7
1.3 Arithmetic Module in 80x86 Assembler (see Chapter 18) in Direc-
toryflint/src/asm . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Arithmetic libraries in 80x86 Assembler (see Chapter 18) in Direc-
tory flint/lib . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Tests (see Section 12.2 and Chapter 17) in Directory flint/test 7
1.6 RSA Implementation (see Chapter 16) in Directory flint/ rsa 7
3.1 FUNT/Cerrorcodes. . . . . . . . . . . . 19
5.1 Composition table for addition modulo 5 . 69
5.2 Composition table for multiplication modulo 5 . 69
6.1 Requirements for exponentiation . . . . . . . . 86
6.2 Numbers of multiplications for typical sizes of exponents and
various bases 2k . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.3 Values for the factorization of the exponent digits into products of
a power of 2 and an odd factor . . . . . . . . . . . . . . . . . . . 88
6.4 Numbers of multiplications for typical sizes of exponents and
various bases 2k . . . . . . . . . . . . 89
6.5 Exponentiation functions in FUNT/C . 115
7.1 Values of a Boolean function . . . . . 130
7.2 Values of the CLINT function and_l() 130
7.3 Values of the CLINT function or_l() 130
7.4 Values of the CLINT function xor _l () 131
8.1 Diagnostic values of the function vcheck _l () 152
10.1 The largest known primes (as of August 2000) . 212
10.2 The number of primes up to various limits x 217
12.1 Group law for the integers to help in testing. 252
12.2 PUNT I C test functions . 253
13.1 LINT constructors .... 265
13.2 LINT arithmetic operators 268