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Index

ABC conjecture, 417, 434 abelian group, 322, 376, 377

Acar, T. (with Ko¸c et al.), 449, 453 additive number theory, 18, 19, 46,

538

Adleman, L., 193, 194, 199, 200, 209, 211, 289, 306, 371, 389

Adleman, L. (with Rivest et al.), 389

Adleman, Pomerance, Rumely (APR) test, 194

a ne

coordinates, 324, 329 solutions, 320

Agarwal, R., 483

Agrawal, M., 200, 201, 205, 207, 212, 217

Aho, A., 466, 509, 513 Alford, W., 134, 135, 141, 276

algebraic number theory, 284, 381 Algorithm D (of Knuth), 447, 453 Alt, H., 522

ambiguous form, 248, 249 Amdahl Six, 24 ammonia molecule, 421

analytic number theory, 7, 20, 33, 39, 43, 46, 67, 193, 371, 438

Ankeny, N., 42

Apostol, T., 434

Araki, K., 392

Arazi, B., 446, 453, 520 Arch, S., x

Archibald, R., 6 Ares, S., 428

arithmetic–geometric mean (AGM), 115

arithmetic-geometric mean (AGM), 358

Armengaud, J., 23

Artin conjecture, 207, 223 Artin constant, 80 Artjuhov, M., 136 Ashworth, M., 483 asymptotics (rules for), 8

Atkin, A., 4, 169, 170, 356, 358– 361, 371, 372, 382

Atkin–Bernstein theorem, 170 Atkins, D., 3

Australian National University, 7 Avanzi, R., 214, 215

Avogadro number, 5, 60

B¨urgisser, P., 466

baby-steps giant-steps method, 250 baby-steps, giant-steps method, 235,

236, 248, 250, 347, 349, 351, 359

Bach, E., x, 42, 43, 65, 69, 80, 113, 141

Backstrom, R., 5

Bailey, D., x, 31, 52, 58, 483, 485, 530

Baillie, R., 149, 165 Baker, R., 37 Balasubramanian, R., 142 Balazard, M., 440 Ballinger, R., 14

Balog, A., x, 75 Barnick, M., x

Barrett method (for div/mod), 452– 454

Barrett, P., 452, 453, 513 Bateman, P., x, 25

Bays, C., 59 Beal prize, 417 Beeger, N., 165 Bender, E., 251 Bennett, M., 417

578

Berlekamp algorithm, 104 Berlekamp–Massey algorithm, 269 Bernoulli numbers, 492, 493, 533 Bernstein, D., ix, x, 74, 128–131,

169, 170, 214, 215, 217, 441, 470, 473, 476, 507, 534

Berrizbeitia, P., 214, 215 Berry, M., 427

Berson, T. (with Gong et al.), 402, 438

Berta, I., 433 Bertrand postulate, 55

Beukers, F., x, 417, 441 big-O notation, 8 binary divide, 447

binary quadratic forms, 239 binary segmentation, 510 binomial theorem, 105

birthday paradox, 229, 238, 271, 341

bit complexity, 9, 473 Blackburn, S., 236 Bleichenbacher, D., x, 147, 169 Bluestein trick, 514, 532

Blum integers, 110, 397, 433 Blum, L., 401

Blum, M., 401 Bombieri, E., 45, 441 Boneh, D., 432 Bonfim, O., x

Borwein, J., x, 115, 381, 382 Borwein, P., 52, 67, 68, 80, 115,

160, 171, 381, 382, 427, 428, 440, 441

Bosma, W., 200

Bosselaers, A., 449, 452, 453 Bouniakowski, V., 17 Boyle, P., 413

Bradley, D., x

Bradley, D. (with Borwein et al.), 67, 68, 80, 160, 171, 427, 428, 440, 441

Bragdon, N., x

Bragdon, P., x

Bratley, P., 414

INDEX

Bredihin, B., 74

Brent parameterization (for ECM), 385, 386

Brent, R., x, 25, 28, 38, 66, 67, 231, 251, 256, 321, 340– 343, 346, 347, 385, 431, 438, 456, 496, 528

Bressoud, D., x, 81, 107, 389, 396, 433

Brigham Young University, 81 Brillhart, J., 28, 175, 186, 261, 280,

299, 307, 310 Brillhart–Morrison method, 261,

307

Broadhurst, D., x, 223, 436 Broadie, M. (with Boyle et al.), 413 Bruin, N., x, 417, 441

Brun

constant, 16, 17, 80 method, 18–20, 61, 63 theorem, 16, 64

Brun, V., 16, 17, 43 Brun–Titchmarsh inequality, 43 Buchmann, J., 236

Buell, D., 29 Bugeaud, Y., x

Buhler, J., x, 289, 290, 292, 297, 468, 493, 532

Buhler, L., x Burnikel, C., 454 Burthe, R., 138, 142

C language, viii, 543 Caldwell, C., 14, 24, 50, 425 Cameron, M., 23 Campbell, G., x

Campbell, M., x

Canfield, E., 49 Cao, D., x

Carmichael numbers, 64, 133–135, 164–166

Carmichael, R., 133 Carmody, P., x, 14 Carvalho, J., 29 Cassels, J., 322 Castro, M., 428

INDEX

Catalan problem, ix, 415, 416 Catmull, E., x Cauchy–Schwarz inequality, 63 cellular automata (CA), 402 certificate

of primality, 370, 371, 373 succinct, 179

Cesari, G., 466, 467, 473, 524 chaotic behavior, 161 character, 39

Chebotarev density theorem, 295 Chebyshev theorem, 10, 18, 55, 58,

62

Chebyshev, P., 10, 55, 206 Chein, E., 25

Chen, J.-r., 16, 19 Cheng, Q., 214 Cheng, Y., 61

Chinese remainder theorem (CRT), 87–89, 104, 109, 112, 113, 139, 197, 219, 221, 251, 261, 262, 275, 292, 304, 351, 354, 356, 383, 491, 499, 500, 508, 509, 526– 528, 531–533, 539

circle

group, 258 method, 46 problem, 170

Clarkson, R., 23

Clausen, M. (with B¨urgisser et al.), 466

Clearwater, S., 420, 423 Clement, P., 65 Cochrane, T., 114

Cohen, G. (with Brent et al.), 25 Cohen, H., x, 39, 85, 104–107, 138,

240, 248, 280, 293, 307, 322, 333, 349, 358, 360, 361, 460, 523

coin

-flip protocol, 396, 397, 433 flip (random), 37, 114, 334 collateralized mortgage obligation

(CMO), 413 Colquitt, W., 23

579

commutative ring, 89

complex multiplication (CM) curves, 359, 362, 366, 380

complexity (computational), 8, 83, 109, 371, 377, 509, 514

bit, 9, 85, 101, 103, 108, 109, 371, 378, 466, 507, 508, 510, 522, 532, 533, 536

operation, 9, 84, 91, 102, 254, 256, 378, 508

polynomial-time, 9 computational finance, 413 computational number theory, 22,

42, 77, 83, 239, 319, 473, 530

Contini, S., 276 convolution, 46, 72, 488, 491

acyclic, 475, 476, 488–491, 503, 510, 524

cyclic, 483, 488, 489, 494, 496, 499, 503–505, 525, 527, 528, 530

DFT-based, 505

DGT-based, 530 half-cyclic, 512 Mersenne, 539

negacyclic, 316, 488, 489, 492, 494, 502–506, 510, 524– 528, 530, 532

Nussbaumer, 493, 495, 503, 505, 507, 527, 532

right-angle, 488, 494 weighted, 494, 497

Cooklev, T. (with Dimitrov et al.), 530

Cooley, J., 479, 483 Copeland, A., 57 Copeland, D., x, 378 Coppersmith algorithm

lattice basis reduction, 189 NFS variant, 300

Coppersmith, D., x, 189, 221, 300, 301, 432

Cornacchia–Smith algorithm, 106, 359

Cosgrave, J., x, 24, 30

580

Couveignes, J., 292, 317, 356 Cox, D., 359, 361 Craig-Wood, N., 539 Cram´er, H., 38

Crandall, R., 5, 28–31, 33, 52, 58, 60, 68, 82, 160, 170, 200, 217–219, 235, 238, 254, 256, 257, 259, 315, 333, 343, 345, 346, 381, 396, 414, 427, 437, 456, 458, 473, 476, 478, 483, 485, 488, 491, 494, 496–498, 500, 501, 506, 525, 528– 532, 534, 536, 538

Crandall, R. (with Borwein et al.), 67, 68, 80, 160, 171, 427, 428, 440, 441

Crandall, R. (with Brent et al.), 28, 321, 340–342, 347, 385, 456, 496, 528

Creutzburg, R., 501 cryptography, 3, 93, 110, 319, 321,

323, 324, 334, 347, 358, 363, 374, 387, 393, 404, 460, 463, 535

DES, 388, 394 Di e–Hellman, 387, 388 DL, 235

DSA, 394 ECDSA, 393, 394

El Gamal, 394, 395

elliptic curve (ECC), 391, 392, 394, 395, 437

elliptic embedding, 394 elliptic signature, 393 protocols, 396 public-key, 113

RSA, 389

RSA signature, 390 SSSA attack, 392

Cullen numbers, 76, 77, 521 Cunningham numbers, 4, 299 Curry, C., 345, 346

Damg˚ard, I., 138, 164

INDEX

Danielson–Lanczos identity, 478, 480, 483

Darmon, H., x, 417

Davenport, H., 36, 42, 43, 48, 66, 114, 125, 247

Davis, M., 273, 418 Day, T., x

de la Vall´ee Poussin, C., 10 De Win, E., 381, 453, 458, 536 Del´eglise, M., 11, 68, 152, 158 Delescaille, J.-P., 234

Denny, T. (with Schirokauer et al.), 306, 387

Deshouillers, J., 19 Deuring theorem, 333, 334 Deuring, M., 333, 334 Deutsch, D., 420

Dewaghe, L. (with Couveignes et al.), 356

Dickman

function, 49, 155 theorem, 48, 49

Dickson, L., 17

Di e, W., 387

Di e–Hellman key exchange, 387– 389, 391

Dilcher, K., x, 31, 316

Dilcher, K. (with Brent et al.), 28, 321, 340–342, 347, 385, 456, 496, 528

Dilcher, K. (with Crandall et al.), 31, 33, 82, 200, 381, 534

Dimitrov, V., 530

Ding, C., 87, 113

Diophantine analysis, 106, 107, 223, 415, 417, 418, 426, 434, 435, 492

Dirichlet, 35, 41

L functions, 39, 41, 42 characters, 39–42, 194, 211 class number formula, 247, 248,

255

theorem, 12, 17, 18, 35, 41, 57, 289

discrepancy theory, 404

INDEX

discrete arithmetic–geometric mean (DAGM), 115, 382

discrete cosine transform (DCT), 478

discrete Fourier transform (DFT), 99, 342, 425, 429, 455, 476–479, 483–485, 489– 494, 498–502, 505–507, 515, 525, 529, 532, 538

finite-field, 477 integer-ring, 477 Shokrollahi, 532

discrete Galois transform (DGT), 500, 502, 530–532, 539

discrete logarithm (DL), ix, 100, 235, 252, 253, 257, 303, 306, 315, 387, 388, 392, 402, 423, 424, 441, 442

elliptic (EDL), ix, 234, 391, 392, 394

finite fields, 303

hash method, 252, 253 index-calculus, 302, 303, 305,

314

kangaroo method, 233 lambda method, 233, 441 number field sieve, 235, 306 rho method, 232, 235, 257, 441

discrete weighted transform (DWT), 493–498, 500, 502–504, 506, 528, 532

discriminant, 106, 113, 144, 145, 151, 152, 236, 240–242, 245–249, 254, 296, 333, 336, 359–363, 365, 367, 373, 375, 383, 384

Disney, 5

distinguished points, 234 div and mod, 445 division polynomials, 352 Dixon, J., 302

DNA computing, 424 Dodson, B., 5 Doenias, J., x

Doenias, J. (with Crandall et al.), 29, 456, 496, 528

581

Donevsky, B. (with Dimitrov et al.), 530

Doxiadis, A., 62 Dress, F., 53

Dubner, H., 30, 31, 78, 79, 528 Dudon, J., 430

Dutt, A., 485, 487, 530 Dyson, F., 426

Edwards, H., 36, 69 E nger, G., x Einstein, A., 425 Ekert, A., 420

El Gamal, T., 395 Elkenbracht-Huizing, M., 299, 301 Elkies, N., x, 356, 357

Ellenberg, J., 417

elliptic curve, 319, 321, 323, 329, 351, 357–359, 378, 383, 392, 402

cryptography (ECC), 392 factoring method (ECM), 3,

28, 225, 238, 256, 258, 261, 300–302, 304, 305, 323, 325, 335–347, 376, 377, 382, 383, 385, 386, 441, 460, 467

fastECPP, 373

operations, 323, 325, 327, 328, 336, 342, 402, 460, 462

point counting, 347, 350 primality testing (ECPP), 4,

215, 323, 334, 368, 372, 380, 383, 384

Ellison, F., 48, 58, 69, 75, 538 Ellison, W., 48, 58, 69, 75, 538 Enge, A., 357

Engelsma, T., x, 81 epact, 231

equidistribution, 44, 57, 58, 70, 406, 411

Eratosthenes

fancy sieve, 126 pseudocode, 122

sieve, 43, 50, 62, 121–123, 125, 126, 153, 154, 164, 169

582

Erd˝os, P., 13, 14, 38, 52, 55, 57, 77, 132, 134, 149, 165

Erd˝os, P. (with Canfield et al.), 49 Erd˝os–Kac theorem, 82 Erd˝os–Tur´an conjecture, 13

error function, 248 Escott, A., 234 Essick, J., x Estermann, T., 48

Euclid, 6, 7, 25, 50, 89, 426 –Euler theorem, 24

algorithm (for gcd), 84, 85, 108, 190, 463, 464

theorems, 2, 6, 50 Euler

constant, 26, 80, 163 criterion, 22, 166, 174, 184 factors, 34

polynomials, 52 product, 71, 72, 158 pseudoprimes, 166, 167 test, 97, 100

theorems, 22, 28, 34, 40, 85, 173, 193

totient function, 13, 93, 127, 136, 155, 173, 389

Euler, L., 10, 13, 18, 19, 21, 25, 27, 28, 33, 34, 288, 404

exponential sums, 43–47, 58, 59, 61, 69, 71, 72, 98, 111, 526, 536

extended Riemann hypothesis (ERH), 20, 42, 43, 46, 101, 137, 141, 142, 200, 207, 209, 215, 247, 250, 251, 302

factoring, 276, 323 (p + 1), 258

continued fraction, 28, 261, 307, 309, 435

double large-prime variations, 272

Fermat, 225, 226, 228, 255, 262, 264

hyperspherical group, 258 large-prime variations, 270

INDEX

Lehman, 227, 228 Lenstra, 251

Lenstra ECM, 3, 28, 128, 225, 238, 256, 258, 261, 300– 302, 304, 305, 323, 325, 335–347, 376, 377, 382, 383, 385, 386, 441, 460, 467

multiple polynomials, 273, 278 number field sieve (NFS), 3, 28, 131, 261, 268, 269, 278–280, 282, 283, 285–

288, 291–301, 306–308, 335, 339, 424, 441

Pollard (p − 1), 236–238, 337, 339, 343

Pollard rho, 28, 229–231, 233, 256–258, 272, 304, 441

Pollard rho (parallel), 235 Pollard–Strassen, 239, 259, 301,

441

quadratic forms, 242 quadratic sieve (QS), 131, 261,

262, 266, 268, 269, 271, 273, 274, 277–279, 285– 288, 291, 295, 300–302, 307, 310, 335, 339, 424, 441

rho method, 128 rigorous, 301

special number field sieve (SNFS), 3, 288, 298

subexponential, 261 tables, 123, 124

three-large-primes variations, 273

Fagin, B., 345, 491, 496–498, 531 fast Fourier transform (FFT), 9,

73, 113, 128, 160, 344, 345, 419, 420, 422–424, 428, 435, 436, 441, 460, 473, 478–485, 490, 491, 493, 494, 496, 498, 500– 505, 507, 508, 529, 531, 536, 538, 539

3-dimensional, 419

INDEX

complex-field, 477 Cooley–Tukey, 479–482 decimation-in-frequency (DIF),

485, 500 decimation-in-time (DIT), 479,

485, 500

floating-point, 497, 498, 501, 507

for elliptic algebra, 340 Gentleman–Sande, 480–482 Mayer, 485

Montgomery extension, 340, 343

nonuniform, 68, 485, 486 parallel, 483, 484 ping-pong, 481

Sch¨onhage–Strassen, 506, 507 Sorenson, 529

split-radix, 483, 501 Stockham, 481

fastECPP, 373, 374 Faure sequence, 414 Faure, H., 408, 414 Fermat

congruence, 132 conjecture, 28

last theorem (FLT), 31, 32, 415, 492, 532

little theorem, 31, 131, 132, 135, 140, 144, 162, 173, 196, 236

method, 226

number transform, 502, 523, 530

numbers, 3, 22, 27–31, 64, 82, 135, 164, 174, 181, 183, 218–220, 223, 231, 256, 259, 278, 295, 299, 307– 309, 320, 346, 347, 380, 382, 431, 454–457, 494– 496, 503, 528

primes, 27, 31, 64, 76, 380, 431 pseudoprimes, 131–133, 135,

164, 167 quotient, 31, 33 tests, 133, 148, 164

583

Fermat, P., 27, 31, 174, 242 Fermat–Catalan conjecture, ix, 416,

417, 434 Fessler, J., 487 fessler, J., x Feynman, R., 420 Fibonacci

generator, 438

numbers, 32, 108, 142, 146, 149, 189, 223, 430

pseudoprimes, 142, 143, 149, 166, 168

Filaseta, M. (with Brillhart et al.), 280, 310

Findley, J., 23 finite fields, 91 Fix, J., x Flajolet, P., 255

Flannery, B. (with Press et al.), 399, 402, 403

Floyd cycle-finding method, 230– 232

Forbes, T., 30, 77, 78

Forbes, T. (with Dubner et al.), 78, 79

Ford, K., 45

Fourier analysis, 44, 62, 160, 427, 439

Fouvry, E., 207 Fox, B., 414

fractal random walk, 428 Franke, J., 4, 374 Fraser, D., 485, 506 Friedlander, J., 18, 401 Frind, M., 78

Frobenius

automorphism, 144, 145, 152, 191, 211

endomorphism, 352 pseudoprimes, 145, 146, 148–

152, 166–168 relation, 379

test, 145, 146, 148, 150, 152 fundamental problem (of arithmetic),

1

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