Abstract
Public key cryptosystems require the use of large prime numbers, numbers with at least 256 bits (80 decimal digits), see for example [12]. One needs to generate these numbers as fast as possible. One way of dealing with this problem is the use of special primes built up using the converse of Fermat’s theorem [35, 14, 17, 29]. Another is to use sophisticated primality proving algorithms, that are fast but need a. careful implementation [13, 9].
Research partially supported by the Programme de Recherches Coordonnées (PRC) Maths-Info.
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References
W. ADAMS. Splitting of quartic polynomials. Math. Comp. 43,167 (July 1984), 329343.
W. ADAMS. Characterizing pseudoprimes for third order linear recurrences. Math. Comp. 48, 177 (Jan. 1987), 1–15.
W. ADAMS AND D. SHANKS. Strong primality tests that are not sufficient. Math. Comp. 39, 159 (July 1982), 255–300.
W. R. ALFORD, A. GRANVILLE, AND C. POMERANCE. There are infinitely many Carmichael numbers. Preprint, July 13th 1992.
F. ARNAULT. Le test de primauté de Rabin—Miller: un nombre composé qui le “passe”. Report 61, Université de Poitiers — Département (le Mathématiques, Nov. 1991.
F. ARNAULT. Carmichaels fortement pseudo-premiers. Manuscript, 1992.
S. ARNO. A note ou Perrin pseudoprimes. Math. Comp. 56, 193 (Jan. 1991), 371–376.
A. O. L. ATKIN. Probabilistic primality testing. In Analysis of Algorithms Seminar I (1992), P. Flajolet and P. Zimmermann, Eds., INRIA Research Report XXX. Summary by F. Morain.
A. O. L. ATKIN AND F. MORAIN. Elliptic curves and primality proving. Research Report 1256, INR.IA, Juin 1990. Submitted to Math. Comp.
R. SAILLIE AND S. S. WAGSTAFF, JR. Lucas pseudoprimes. Math. Comp. 35, 152 (Oct. 1980), 1391–1417.
R. BALASUBRAMANIAN AND M. R. MURTY. Elliptic pseudoprimes, II: Submitted for publication.
G. BRASSARD. Modern Cryptology, vol. 325 of Lect. Notes in Computer Science. Springer- Verlagq 1988.
H. COHEN AND A. K. LENSTRA. Implementation of a new primality test. Math. Comp. 48, 177 (1987), 103–121.
C. COUVREUR AND J. QUISQUATER. An introduction to fast generation of large prime numbers. Philips J. Research 37 (1982), 231–264.
I. DAMGÂRD AND P. LANDROCK. Improved bounds for the Rabin primality test. In Proc. 3rd IMA conference on Coding and Cryptography (1991), M. Ganley, Ed., Oxford University Press.
J. H. DAVENPORT. Primality testing revisited. In ISSAC’92 (New York, 1992), P. S. Wang, Ed., ACM Press, pp. 123–129. Proceedings, July 27–29, Berkeley.
D. GORDON. Strong primes are easy to find. In Advances in Cryptology (1985), T. Beth, N. Cot, and I. Ingemarsson, Eds., vol. 209 of Lect. Notes in Computer Science, Springer-Verlag, pp. 216–223. Proceedings Eurocrypt ‘84, Paris ( France ), April 9–11, 1984.
D. M. GORDON. On the number of elliptic pseudoprimes. Math. Comp. 52, 185 (Jan. 1989), 231–245.
D. M. GORDON AND C. POMERANCE. The distribution of Lucas and elliptic pseudo-primes. Math. Comp. 57, 196 (Oct. 1991), 825–838.
D. GUILLAUME AND F. MORAIN. Building Carmichael numbers with a large number of prime factors and generalization to other numbers. Research Report 1741, INRIA, Aug. 1992.
S. GuRAK. Pseudoprimes for higher-order linear recurrence sequences. Math. Comp. 55, 192 (Oct. 1990), 783–813.
G. H. HARDY AND E. M. WRIGHT. An introduction to the theory of numbers, 5th ed. Clarendon Press, Oxford, 1985.
D. HUSEMÖLLER. Elliptic curves, vol. 111 of Graduate Texts in Mathematics. Springer, 1987.
G. JAESCHKE. The Carmichael numbers to 1012. Math. Comp. 55, 191 (July 1990), 383–389.
W. KELLER. The Carmichael numbers to 1013. AMS Abstracts 9 (1988), 328–329. Abstract 88T-11–150.
G. C. KURTZ, D. SHANKS, AND H. C. WILLIAMS. Fast primality tests for numbers less than 50. 109. Math. Comp. 0, 174 (Apr. 1986), 691–701.
G. LÖH. Carmichael numbers with a large number of prime factors. AMS Abstracts 9 (1988), 329. Abstract 88T-11–151
G. LÖH AND W. NIEBUHR. Carmichael numbers with a large number of prime factors, II. AMS Abstracts 10 (1989), 305. Abstract 89T-11–131
U. M. MAURER. Fast generation of secure RSA-products with almost maximal diversity. In Advances in Cryptology (1990), J.-J. Quisquater, Ed., vol. 434 of Lect. Notes in Computer Science, Springer-Verlag, pp. 636–647. Proc. Eurocrypt ‘89, Houthalen, April 10–13.
R. MESHULAM. An uncertainty inequality and zero subsums. Discrete Mathematics 8.4 (1990), 197–200.
I. MIYAMOTO AND M. R. MURTY. Elliptic psendoprimes. Math. Comp. 53, 187 (July 1989), 415–430.
L. MONIER. Evaluation and comparison of two efficient probabilistic primality testing algorithms. Theoretical Computer Science 12 (1980), 97–108.
R. PINCH. The Carmichael numbers to 1016. In preparation, 1992.
R. PINCH. The pseudoprimes up to 1012. In preparation, Sept. 1992.
D. A. PLAISTED. Fast verification, testing and generation of large primes. Theoretical Computer Science 9 (1979), 1–16.
C. POMERANCE. Carmichael numbers. To appear in Nieuw Arch. Wisk., 1992.
C. POMERANCE, J. L. SELFRIDGE, AND S. S. WAGSTAFF, JR. The pseudoprimes to 25.10. Math. Covrip. 35, 151 (1980), 1003–1026.
M. O. R.ABIN. Probabilistic algorithms in finite fields. SIAM J. Comput. 9, 2 (1980), 273–280.
P. R.IBENBOIM. The book of prime number accords, 2nd ed: Springer, 1989.
R. SCHROEPPEL. Richard Pinch’s list of pseudoprimes. E-mail to the NMBRTHRY list, June 1992.
P. VAN EMDE BOAS. A combinatorial problem on finite abelian groups, II. Tech. Rep. ZW-007, Math. Centrum Amsterdam Afd. Zuivere Wisk., 1969. 60 pp.
P. VAN EMDE BOAS AND D. Kau swiiK. A combinatorial problem on finite abelian groups, III. Tech. Rep. ZW-008, Math. Centrum Amsterdam Afd. Zuivere Wisk., 1969.
M. ZIIANG. Searching for large Carmichael numbers. To appear in Sichuan Daxue Xuebao, Dec. 1991.
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Morain, F. (1993). Pseudoprimes: A Survey of Recent Results. In: Camion, P., Charpin, P., Harari, S. (eds) Eurocode ’92. International Centre for Mechanical Sciences, vol 339. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2786-5_18
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DOI: https://doi.org/10.1007/978-3-7091-2786-5_18
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