Contents
7 Bitwise and Logical Functions 125
7.1 ShiftOperations .......................... 125
7.2 All or Nothing: Bitwise Relations . . . . . . . . . . . . . . . . . . 131
7.3 Direct Access to Individual Binary Digits . . . . . . . . . . . . . . 137
7.4 ComparisonOperators....................... 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
10.4.1TheJacobiSymbol ..................... 192
10.4.2 Square Roots Modulo
p
k
.................. 198
10.4.3 Square Roots Modulo
n
................... 203
10.4.4 Cryptography with Quadratic Residues . . . . . . . . . . 211
10.5 APrimalityTest........................... 214
11 Rijndael: A Successor to the Data Encryption Standard 237
11.1 ArithmeticwithPolynomials.................... 239
11.2 TheRijndaelAlgorithm....................... 244
11.3 CalculatingtheRoundKey..................... 247
11.4 TheS-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
11.10Performance ............................ 259
11.11ModesofOperation ........................ 260
12 Large Random Numbers 261
12.1 ASimpleRandomNumberGenerator............... 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
12.2.3TheAESGenerator ..................... 279
12.2.4 The RMDSHA-1 Generator . . . . . . . . . . . . . . . . . 283
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