Криптография на С и С++ (Cryptography in C and C++) Михаил Вельшенбах
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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
И ТД
.......
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
И ТД
.......