C. Acknowledgements for the Third Edition.

Back

We wish to thank the following people who have contributed to the Third

Edition:

New factors of Fermat numbers and numbers in the main tables were discovered

by K. Aardal, W. Alford, G. Axelsson, R. Baillie, R. Ballinger, B. Beesley,

D. Bernstein, M. Bodschwinna, H. Boender, A. Bot, R. Brent, A. Brown, J. Buhler,

S. Cavallar, S. Contini, J. Cosgrave, R. Crandall, C. Curry, N. Daminelli, F. Damm,

V. Danilov, J. Davis, P. Demichel, K. Dilcher, B. Dodson, D. Doligez, H. Dubner,

L. Durman, S. Edick, R. Edwards, M. Elkenbracht-Huizing, A. Erdmann, W. Florek,

T. Forbes, J. Fougeron, J. Franke, Y. Gallot, P. Gaudry, J. Gilchrist, G. Gostin,

W. Grabysz, M. Gra_, T. Granlund, P. Grobstich, G. Gusev, R. Harley, F. Heider,

D. Holdridge, R. Horn, S. Huddleston, M. Hurter, C. Kerchner, Y. Kida, T. Kleinjung,

J. Klos, K. Koyama, A. Kruppa, H. Kuwakado, D. Leclair, J. Leherbauer,

A. Lenstra, R. Lercier, R. Lewis, P. Leyland, W. Lioen, S. Lodin, J. Loho, A. Lynch,

M. Mambo, M. Manasse, J. Marchand, E. Mayer, R. McIntosh, P. McLaughlin, Jr.,

J.-C. Meyrignac, N. Melo, D. Miller, P. Montgomery, F. Morain, D. Morenus,

A. Mu_ett, B. Murphy, P. Nicholson, T. Nohara, T. Nokelby, E. Okamoto, R. Peralta,

C. Pomerance, R. Prethaler, C. Putnam, M. Quercia, J. Rathert, J. Renze,

R. Robson, R. Ruby, D. Rusin, P. Samidoost, G. Sassoon, A. Schmidt, R. Silverman,

N. Smart, V. Stevens, H. Suyama, T. Szep, D. Takahashi, C. Tardif, T. Taura,

H. te Riele, Y. Torii, V. Trevisan, C. van Halewyn, R. Wackerbarth, G. Wambach,

S. Whitaker, M. Wiener, D. Winter, D. Wolf, G. Woltman, J. Young, J. Zayer,

P. Zimmermann, SSW, many volunteers who keep factoring programs running on

their workstations, and an anonymous factorer who calls himself \M. Mersenne".

The new results of the third edition required millions of hours of computer time.

We are grateful to the directors and sta_s of the following computer centers which

xcvii V C ACKNOWLEDGEMENTS FOR THE THIRD EDITION

provided this time: Convex Computer Corporation, NeXT Computer, Inc., the

National Center for Supercomputing Applications, JAIST, the Center for Cryptography,

Computer and Network Security at the University of Wisconsin-Milwaukee,

Microsoft Research, Unisys (formerly System Development Corporation), UCLA

Department of Mathematics, Oregon State University Department of Mathematics,

Centrum voor Wiskunde en Informatica the Dutch National Aerospace Laboratory

in The Netherlands, the Universitat des Sarlandes, Germany, INRIA (Institut National

de Recherche en Informatique et Automatique), the Poznan Supercomputer

and Networking Centre, Medicis Polytechique, the High Performance Computer

Centre North, the Swedish Institute for Computer Science, the Maths Department

of the Stockholm University, Swox, Circus Ulfberg, the Astonomy Department of

Uppsala University, and Purdue University.

M. Senn and R. M. Jegadeesan wrote programs which formatted the tables in

this book.

SSW gratefully acknowledges the support of the Lilly Foundation and the

Center for Education and Research in Information Assurance and Security in the

preparation of this edition and of the annual updates.

V D REFERENCES FOR THE THIRD EDITION xcviii

References

The _rst edition had references 1-118; they appear in III E. The second edition

had references 201-263; they appear in IV D.

301. L. M. Adleman, Factoring numbers using singular integers, Proceedings 23nd Annual ACM

Symposium on Theory of Computing (STOC) (1991), 64{71.

302. W. R. Alford and C. Pomerance, Implementing the self-initializing quadratic sieve on a

distributed network, Number Theoretic and Algebraic Methods in Computer Science, A. van

der Poorten, I. Shparlinski and H. G. Zimmer, editors, Moscow, 1993, pp. 163{174.

303. A. O. L. Atkin and F. Morain, Finding suitable curves for the elliptic curve method of

factorization, Math. Comp. 60 (1993), 399{405, MR 93k:11115.

304. A. O. L. Atkin and F. Morain, Elliptic curves and primality proving, Math.Comp. 61 (1993),

29{68, MR 93m:11136.

305. H. Boender and H. J. J. te Riele, Factoring integers with large-prime variations of the

quadratic sieve, Experimental Math. 5 (1996), 257{273, MR 97m:11155.

306. W. Bosma, Explicit primality criteria for h _ 2k _ 1, Math. Comp. 61 (1993), 97{109, MR

94c:11005.

307. W. Bosma and M. P. van der Hulst, Primality Proving with Cyclotomy, Proefschrift, University

of Amsterdam, Amsterdam, 1990.

308. R. P. Brent, On computing factors of cyclotomic polynomials, Math. Comp. 61 (1993),

131{149, MR 93m:11131.

309. R. P. Brent, Factorization of the tenth Fermat number, Math. Comp. 68 (1999), 429{451,

MR 99e:11154.

310. R. P. Brent, R. E. Crandall, K. Dilcher and C. van Halewyn, Three new factors of Fermat

numbers, Math. Comp. 69 (2000), 1297{1304, MR 2000j:11194.

311. D. M. Bressoud, Factorization and Primality Testing, Springer-Verlag Undergraduate Texts

in Mathematics, Berlin, New York, 1989.

312. W. N. Colquitt and L. Welsh, Jr., A new Mersenne prime, Math. Comp. 56 (1991), 867{870,

MR 91h:11006.

313. Don Coppersmith, Solving linear equations over GF(2): Block Lanczos algorithm, Lin. Alg

and its Apps. 192, 33{60, MR 94c:11124.

314. Don Coppersmith, Solving homogeneous linear equations over GF(2) via block Wiedemann

algorithm, Math. Comp. 62, 333{350.

315. R. Crandall, J. Doenias, C. Norrie and J. Young, The twenty-second Fermat number is

composite, Math. Comp. 64 (1995), 863{868, MR 95f:11104.

316. B. Dixon and A. K. Lenstra, Massively parallel elliptic curve factoring, Advances in Cryptology,

Proceedings of Eurocrypt '92, Lecture Notes in Computer Science 658, Springer-Verlag,

Berlin, New York, 1993, pp. 183{193.

317. Harvey Dubner, Generalized repunit primes, Math. Comp. 61 (1993), 927{930, MR 94a:-

11009.

318. Harvey Dubner and Wilfrid Keller, Factors of generalized Fermat numbers, Math. Comp.

64 (1995), 397{405, MR 95c:11010.

319. R.-M. Elkenbracht-Huizing, An implementation of the number _eld sieve, Experimental

Math. 5 (1996), 231{253, MR 98a:11182.

320. R.-M. Elkenbracht-Huizing, P. L. Montgomery, R. D. Silverman, R. K. Wackerbarth and S.

S. Wagsta_, Jr., The number _eld sieve on many computers, CRM Proceedings and Lecture

Notes 19 (1999), 81{85, MR 2000e:11157.

321. Roger A. Golliver, Arjen K. Lenstra and Kevin S. McCurley, Lattice sieving and trial division,

Algorithmic Number Theory (Ithaca, NY 1994), Lecture Notes in Computer Science

877, Springer-Verlag, Berlin, New York, 1994, pp. 18{27.

322. Gary B. Gostin, New factors of Fermat numbers, Math. Comp. 64 (1995), 393{395, MR

95c:11151.

323. B. A. LaMacchia and A. M. Odlyzko, Solving large sparse linear systems over _nite _elds, Advances

in Cryptology{Crypto '90, Lecture Notes in Computer Science 537, Springer-Verlag,

Berlin, New York, 1991, pp. 109{133.

324. A. K. Lenstra, Massively parallel computing and factoring, Proceedings of LATIN '92, Lecture

Notes in Computer Science 583, Springer-Verlag, Berlin, New York, 1992, pp. 344{355.

325. A. K. Lenstra and H. W. Lenstra, Jr., The development of the number _eld sieve, Lecture

Notes in Mathematics 1554, Springer-Verlag, Berlin, New York, 1993, MR 96m:11116.

xcix V D REFERENCES FOR THE THIRD EDITION

326. A. K. Lenstra, H. W. Lenstra, Jr., M. S. Manasse and J. M. Pollard, The number _eld sieve,

Proceedings 22nd Annual ACM Symposium on Theory of Computing (STOC), Baltimore,

1990, pp. 564{572.

327. A. K. Lenstra, H. W. Lenstra, Jr., M. S. Manasse and J. M. Pollard, The factorization of

the ninth Fermat number, Math. Comp. 61 (1993), 319{349, MR 93k:11116.

328. A. K. Lenstra and M. S. Manasse, Factoring by electronic mail, Advances in Cryptology {

Proceedings of Eurocrypt '89, Springer-Verlag Lecture Notes in Computer Science, Berlin,

New York, 1990, pp. 355{371.

329. A. K. Lenstra and M. S. Manasse, Factoring with two large primes, Math. Comp. 63 (1994),

785{798, MR 95a:11107.

330. Peter L. Montgomery, An FFT Extension of the Elliptic Curve Method of Factorization,

Ph. D. thesis at the University of California, Los Angeles, 1992.

331. Peter L. Montgomery, Square roots of products of algebraic numbers, Mathematics of Computation

1943{1993: a Half-Century of Computational Mathematics, Walter Gautschi, editor.

Proceedings of Symposia in Applied Mathematics, Amer. Math. Soc., Providence, 1994,

pp. 567{571, MR 96a:11148.

332. Peter L. Montgomery, A survey of modern integer factorization algorithms, CWI Quarterly

7 (4) (1994), 337{366, MR 96b:11161.

333. Peter L. Montgomery and Robert D. Silverman, An FFT extension to the P 􀀀 1 factoring

algorithm, Math. Comp. 54 (1990), 839{854, MR 90j:11142.

334. Fran_cois Morain, Atkin's test: News from the front, Advances in Cryptology { Proceedings

of Eurocrypt '89, Springer-Verlag Lecture Notes in Computer Science, Berlin, New York,

1990, pp. 626{635, MR 91m:11111.

335. Fran_cois Morain, Courbes elliptiques et tests de primalit_e, Ph. D. thesis at Universit_e Claude

Bernard{Lyon I, 1990, MR 95i:11149.

336. Mitsuo Morimoto and Y^uji Kida, Factorization of Cyclotomic Numbers, Department of

Mathematics, Sophia University, Tokyo, 1987 (in Japanese).

337. Mitsuo Morimoto, Y^uji Kida and Michiyo Sait^o, Factorization of Cyclotomic Numbers, II,

Department of Mathematics, Sophia University, Tokyo, 1989 (in Japanese).

338. R. Peralta, A quadratic sieve on the n-dimensional hypercube, Proceedings of Crypto '92,

vol. 740, Springer-Verlag, pp. 324{332, MR 95f:11108.

339. Carl Pomerance, The number _eld sieve, Mathematics of Computation 1943{1993: a Half-

Century of Computational Mathematics, Walter Gautschi, editor. Proceedings of Symposia

in Applied Mathematics, Amer. Math. Soc., Providence, 1994, pp. 465{480.

340. Hans Riesel, Prime Numbers and Computer Methods for Factorization, second edition,

Birkhauser, Boston, 1994, MR 95h:11142.

341. Je_rey Shallit, Hugh C.Williams and Fran_cois Morain, Discovery of a lost factoring machine,

Math. Intel. 17 (1995), 41{47, MR 96f:01029.

342. Robert D. Silverman and Samuel S. Wagsta_, Jr., A practical analysis of the elliptic curve

factoring algorithm, Math. Comp. 61 (1993), 445{462, MR 93k:11117.

343. N. M. Stephens, Lenstra's factorisation method based on elliptic curves, Advances in Cryptology,

Proceedings of Crypto '85, H. C. Williams, ed., Springer-Verlag, Berlin, New York,

1986, pp. 409{416, MR 87k:11140.

344. Peter Stevenhagen, On Aurifeuillian factorizations, Nederl. Akad. Wetensch. Indag. Math.

49 (1987), 451{468, MR 89a:11015.

345. Hideo Wada, Computers and Prime Factorization, Nebula, Tokyo, 1987 (in Japanese).

346. Samuel S. Wagsta_, Jr., Aurifeuillian factorizations and the period of the Bell numbers

modulo a prime, Math. Comp. 65 (1996), 383{391, MR 96f:11033.

347. H. C. Williams and J. O. Shallit, Factoring integers before computers, Mathematics of Computation

1943{1993: a Half-Century of Computational Mathematics, Walter Gautschi, editor.

Proceedings of Symposia in Applied Mathematics, Amer. Math. Soc., Providence, 1994,

pp. 481{531, MR 95m:11143.

348. J. Young, Large primes and Fermat factors, Math. Comp. 67 (1998), 1735{1738, MR

99a:11010.

Навигация