Cryptography in C and C (Cryptography in C and C) Mikhail Velshenbah

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Uploaded: 16.10.2007
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The book contains the cryptographic algorithms realizovanna C
Contents

I Arithmetic and Number Theory in C 1
1 Introduction 3
2 Number Formats: The Representation of Large Numbers in C 13
3 Interface Semantics 19
4 The Fundamental Operations 23
4.1 Addition and Subtraction. . . . . . . . . . . . . . . . . . . . . . 24
4.2 Multiplication. . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2.1 The Grade SchoolMethod. . . . . . . . . . . . . . . . . 34
4.2.2 Squaring Is Faster. . . . . . . . . . . . . . . . . . . . . . 40
4.2.3 Do Things Go Better with Karatsuba? . . . . . . . . . . . 45
4.3 Division with Remainder. . . . . . . . . . . . . . . . . . . . . . 50
5 Modular Arithmetic: Calculating with Residue Classes 67
6 Where All RoadsMeet: Modular Exponentiation 81
6.1 First Approaches. . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.2 M-ary Exponentiation. . . . . . . . . . . . . . . . . . . . . . . 86
6.3 Addition Chains andWindows. . . . . . . . . . . . . . . . . . . 101
6.4 Montgomery Reduction and Exponentiation. . . . . . . . . . . 106
6.5 Cryptographic Application of Exponentiation. . . . . . . . . . . 118
v
Contents
7 Bitwise and Logical Functions 125
7.1 Shift Operations. . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.2 All or Nothing: Bitwise Relations. . . . . . . . . . . . . . . . . . 131
7.3 Direct Access to Individual Binary Digits. . . . . . . . . . . . . . 137
7.4 Comparison Operators. . . . . . . . . . . . . . . . . . . . . . . 140
8 Input, Output, Assignment, Conversion 145
9 Dynamic Registers 157
10 Basic Number-Theoretic Functions 167
10.1 Greatest Common Divisor. . . . . . . . . . . . . . . . . . . . . 168
10.2 Multiplicative Inverse in Residue Class Rings. . . . . . . . . . . 175
10.3 Roots and Logarithms. . . . . . . . . . . . . . . . . . . . . . . 183
10.4 Square Roots in Residue Class Rings. . . . . . . . . . . . . . . . 191
10.4.1 The Jacobi Symbol. . . . . . . . . . . . . . . . . . . . . 192
10.4.2 Square RootsModulo pk. . . . . . . . . . . . . . . . . . 198
10.4.3 Square RootsModulo n. . . . . . . . . . . . . . . . . . . 203
10.4.4 Cryptography with Quadratic Residues. . . . . . . . . . 211
10.5 A Primality Test. . . . . . . . . . . . . . . . . . . . . . . . . . . 214
11 Rijndael: A Successor to the Data Encryption Standard 237
11.1 Arithmetic with Polynomials. . . . . . . . . . . . . . . . . . . . 239
11.2 The Rijndael Algorithm. . . . . . . . . . . . . . . . . . . . . . . 244
11.3 Calculating the Round Key. . . . . . . . . . . . . . . . . . . . . 247
11.4 The S-Box. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
11.5 The ShiftRowsTransformation. . . . . . . . . . . . . . . . . . . 249
11.6 The MixColumnsTransformation. . . . . . . . . . . . . . . . . . 250
11.7 The AddRoundKeyStep. . . . . . . . . . . . . . . . . . . . . . . . 252
11.8 Encryption as a Complete Process. . . . . . . . . . . . . . . . . 253
11.9 Decryption. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
11.10 Performance. . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
11.11 Modes of Operation. . . . . . . . . . . . . . . . . . . . . . . . 260
12 Large Random Numbers 261
12.1 A Simple Random Number Generator. . . . . . . . . . . . . . . 265
12.2 Cryptographic Random Number Generators. . . . . . . . . . . 268
12.2.1 The Generation of Start Values. . . . . . . . . . . . . . . 269
12.2.2 The BBS Random Number Generator. . . . . . . . . . . 273
12.2.3 The AES Generator. . . . . . . . . . . . . . . . . . . . . 279
12.2.4 The RMDSHA-1 Generator. . . . . . . . . . . . . . . . . 283

ETC
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