Primality Testing and Prime Generation

  • Song Y. Yan
Part of the Advances in Information Security book series (ADIS, volume 11)


The primality testing problem (PTP) may be described as the following simple decision (i.e., yes/no) problem:


Prime Number Elliptic Curve Elliptic Curf Prime Generation Discrete Logarithm Problem 
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  1. 2.
    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.Google Scholar
  2. 3.
    L. M. Adleman, C. Pomerance, and R. S. Rumely, “On Distinguishing Prime Numbers from Composite Numbers”, Annals of Mathematics, 117 (1983), 173–206.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 4.
    L. M. Adleman and M. D. A. Huang, Primality Testing and Abelian Varieties over Finite Fields, Lecture Notes in Mathematics 1512, Springer-Verlag, 1992.Google Scholar
  4. 5.
    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.Google Scholar
  5. 6.
    W. Alford, G. Granville and C. Pomerance, “There Are Infinitely Many Carmichael Numbers”, Annals of Mathematics, 140 (1994), 703–722.MathSciNetCrossRefGoogle Scholar
  6. 10.
    A. O. L. Atkin and F. Morain, “Elliptic Curves and Primality Proving”, Mathematics of Computation, 61 (1993), 29–68.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 12.
    E. Bach and J. Shallit, Algorithmic Number Theory I - Efficient Algorithms, MIT Press, 1996.Google Scholar
  8. 13.
    A. Baker, AConcise Introduction to theTheory of Numbers, Cambridge University Press, 1984.Google Scholar
  9. 24.
    R. Bhattacharjee and P. Pandey, “Primality Testing”, Dept of Computer Science & Engineering, Indian Institute of Technology Kanpur, India, 2001.Google Scholar
  10. 27.
    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.Google Scholar
  11. 31.
    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.Google Scholar
  12. 41.
    H. Cohen, ACourse in Computational Algebraic Number Theory, Graduate Texts in Mathematics138, Springer-Verlag, 1993.Google Scholar
  13. 44.
    T. H. Cormen, C. E. Ceiserson and R. L. Rivest, Introduction to Algorithms, MIT Press, 1990.Google Scholar
  14. 45.
    D. A. Cox, Primes of the Form x 2 + ny2, Wiley, 1989.Google Scholar
  15. 53.
    J. D. Dixon, “Factorization and Primality tests”, The American Mathematical Monthly, June-July 1984, pp 333–352.Google Scholar
  16. 61.
    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.Google Scholar
  17. 62.
    S. Goldwasser and J. Kilian, “Primality Testing Using Elliptic Curves”, Journal of ACM, 46, 4 (1999), 450–472.MathSciNetzbMATHCrossRefGoogle Scholar
  18. 79.
    J. Kilian, Uses of Randomness in Algorithms and Protocols, MIT Press, 1990.Google Scholar
  19. 80.
    D. E. Knuth, The Art of Computer Programming II - Seminumerical Algorithms, 3rd Edition, Addison-Wesley, 1998.Google Scholar
  20. 87.
    E. Kranakis, Primality and Cryptography, John Wiley & Sons, 1986.Google Scholar
  21. 101.
    G. Miller, “Riemann’s Hypothesis and Tests for Primality”, Journal of Systems and Computer Science, 13 (1976), 300–317.zbMATHCrossRefGoogle Scholar
  22. 118.
    R. G. E. Pinch, “Some Primality Testing Algorithms”, Notices of the American Mathematical Society, 40, 9 (1993), 1203–1210.Google Scholar
  23. 123.
    C. Pomerance, “Very Short Primality Proofs”, Mathematics of Computation, 48 (1987), 315–322.MathSciNetzbMATHCrossRefGoogle Scholar
  24. 127.
    C. Pomerance, J. L. Selfridge and S. S. Wagstaff, Jr., “The Pseudoprimes to 25 • 109i, Mathematics of Computation, 35 (1980), 1003–1026.MathSciNetzbMATHGoogle Scholar
  25. 128.
    V. R. Pratt, “Every Prime Has a Succinct Certificate”, SIAM Journal on Computing, 4 (1975), 214–220.MathSciNetzbMATHCrossRefGoogle Scholar
  26. 129.
    M. O. Rabin, “Probabilistic Algorithms for Testing Primality”, Journal of Number Theory, 12 (1980), 128–138.MathSciNetzbMATHCrossRefGoogle Scholar
  27. 132.
    P. Ribenboim, “Selling Primes”, Mathematics Magazine, 68, 3(1995), 175182.Google Scholar
  28. 137.
    H. Riesel, Prime Numbers and Computer Methods for Factorization, Birkhäuser, Boston, 1990.Google Scholar
  29. 164.
    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.MathSciNetCrossRefGoogle Scholar
  30. 170.
    S. Wagon, “Primality Testing”, The Mathematical Intelligencer, 8, 3 (1986), 58–61.MathSciNetzbMATHCrossRefGoogle Scholar
  31. 171.
    S. S. Wagstaff, Jr., Cryptanalysis of Number Theoretic Ciphers, Chapman & Hall/CRC Press, 2002.Google Scholar
  32. 174.
    H. S. Wilf, Algorithms and Complexity, 2nd Edition, A. K.Peters, 2002.Google Scholar
  33. 177.
    H. C. Williams, Édouard Lucas and Primality Testing, John Wiley Sons, 1998.Google Scholar
  34. 179.
    S. Y. Yan, “Primality Testing of Large Numbers in Maple”, Computers &Mathematics with Applications, 29, 12 (1995), 1–8.MathSciNetzbMATHCrossRefGoogle Scholar
  35. 180.
    S. Y. Yan, Number Theory for Computing, 2nd Edition, Springer-Verlag, 2002.Google Scholar

Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Song Y. Yan
    • 1
  1. 1.Coventry UniversityUK

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