Abstract
The primality testing problem (PTP) may be described as the following simple decision (i.e., yes/no) problem:
It would be interesting to know, for example, what the situation is with the determination if a number is a prime, and in general how much we can reduce the number of steps from the method of simply trying for finite combinatorial problems.
Kurt GÖdel (1906–1978)
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
L. M. Adleman, “Algorithmic Number Theory–The Complexity Contribution”, Proceedings of the 35thAnnual IEEE Symposium on Foundations of Computer Science, IEEE Press, 1994, 88–113.
L. M. Adleman, C. Pomerance, and R. S. Rumely, “On Distinguishing Prime Numbers from Composite Numbers”, Annals of Mathematics, 117 (1983), 173–206.
L. M. Adleman and M. D. A. Huang, Primality Testing and Abelian Varieties over Finite Fields, Lecture Notes in Mathematics 1512, Springer-Verlag, 1992.
M. Agrawal, N. Kayal and N. Saxena, Primes is in P, Dept of Computer Science & Engineering, Indian Institute of Technology Kanpur, India, 6 August 2002.
W. Alford, G. Granville and C. Pomerance, “There Are Infinitely Many Carmichael Numbers”, Annals of Mathematics, 140 (1994), 703–722.
A. O. L. Atkin and F. Morain, “Elliptic Curves and Primality Proving”, Mathematics of Computation, 61 (1993), 29–68.
E. Bach and J. Shallit, Algorithmic Number Theory I - Efficient Algorithms, MIT Press, 1996.
A. Baker, AConcise Introduction to theTheory of Numbers, Cambridge University Press, 1984.
R. Bhattacharjee and P. Pandey, “Primality Testing”, Dept of Computer Science & Engineering, Indian Institute of Technology Kanpur, India, 2001.
G. Brassard, “A Quantum Jump in Computer Science”, Computer Science Today — Recent Trends and Development, Lecture Notes in Computer Science 1000, Springer-Verlag, 1995, 1–14.
R. P. Brent, “Primality Testing and Integer Factorization”, Proceedings of Australian Academy of Science Annual General Meeting Symposium on the Role of Mathematics in Science, Canberra, 1991, 14–26.
H. Cohen, ACourse in Computational Algebraic Number Theory, Graduate Texts in Mathematics138, Springer-Verlag, 1993.
T. H. Cormen, C. E. Ceiserson and R. L. Rivest, Introduction to Algorithms, MIT Press, 1990.
D. A. Cox, Primes of the Form x 2 + ny2, Wiley, 1989.
J. D. Dixon, “Factorization and Primality tests”, The American Mathematical Monthly, June-July 1984, pp 333–352.
S. Goldwasser and J. Kilian, “Almost All Primes Can be Quickly Certified”, Proceedings of the 18th ACM Symposium on Theory of Computing, Berkeley, 1986, 316–329.
S. Goldwasser and J. Kilian, “Primality Testing Using Elliptic Curves”, Journal of ACM, 46, 4 (1999), 450–472.
J. Kilian, Uses of Randomness in Algorithms and Protocols, MIT Press, 1990.
D. E. Knuth, The Art of Computer Programming II - Seminumerical Algorithms, 3rd Edition, Addison-Wesley, 1998.
E. Kranakis, Primality and Cryptography, John Wiley & Sons, 1986.
G. Miller, “Riemann’s Hypothesis and Tests for Primality”, Journal of Systems and Computer Science, 13 (1976), 300–317.
R. G. E. Pinch, “Some Primality Testing Algorithms”, Notices of the American Mathematical Society, 40, 9 (1993), 1203–1210.
C. Pomerance, “Very Short Primality Proofs”, Mathematics of Computation, 48 (1987), 315–322.
C. Pomerance, J. L. Selfridge and S. S. Wagstaff, Jr., “The Pseudoprimes to 25 • 109i, Mathematics of Computation, 35 (1980), 1003–1026.
V. R. Pratt, “Every Prime Has a Succinct Certificate”, SIAM Journal on Computing, 4 (1975), 214–220.
M. O. Rabin, “Probabilistic Algorithms for Testing Primality”, Journal of Number Theory, 12 (1980), 128–138.
P. Ribenboim, “Selling Primes”, Mathematics Magazine, 68, 3(1995), 175182.
H. Riesel, Prime Numbers and Computer Methods for Factorization, Birkhäuser, Boston, 1990.
R. Solovay and V. Strassen, “A Fast Monte-Carlo Test for Primality”, SIAM Journal on Computing, 6, 1(1977), 84–85. “Erratum: A Fast Monte-Carlo Test for Primality”, SIAM Journal on Computing, 7, 1 (1978), 118.
S. Wagon, “Primality Testing”, The Mathematical Intelligencer, 8, 3 (1986), 58–61.
S. S. Wagstaff, Jr., Cryptanalysis of Number Theoretic Ciphers, Chapman & Hall/CRC Press, 2002.
H. S. Wilf, Algorithms and Complexity, 2nd Edition, A. K.Peters, 2002.
H. C. Williams, Édouard Lucas and Primality Testing, John Wiley Sons, 1998.
S. Y. Yan, “Primality Testing of Large Numbers in Maple”, Computers &Mathematics with Applications, 29, 12 (1995), 1–8.
S. Y. Yan, Number Theory for Computing, 2nd Edition, Springer-Verlag, 2002.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer Science+Business Media New York
About this chapter
Cite this chapter
Yan, S.Y. (2004). Primality Testing and Prime Generation. In: Primality Testing and Integer Factorization in Public-Key Cryptography. Advances in Information Security, vol 11. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3816-2_2
Download citation
DOI: https://doi.org/10.1007/978-1-4757-3816-2_2
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4757-3818-6
Online ISBN: 978-1-4757-3816-2
eBook Packages: Springer Book Archive