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Computational/Algorithmic Number Theory

  • Song Y. Yan
Chapter

Abstract

Computational and algorithmic number theory are two very closely related subjects; they are both concerned with, among many others, computer algorithms, particularly efficient algorithms (including parallel and distributed algorithms, sometimes also including computer architectures), for solving different sorts of problems in number theory and in other areas, including computing and cryptography. Primality testing, integer factorization and discrete logarithms are, amongst many others, the most interesting, difficult and useful problems in number theory, computing and cryptography. In this chapter, we shall study both computational and algorithmic aspects of number theory. More specifically, we shall study various algorithms for primality testing, integer factorization and discrete logarithms that are particularly applicable and useful in computing and cryptography, as well as methods for many other problems in number theory, such as the Goldbach conjecture and the odd perfect number problem.

Keywords

Elliptic Curve Turing Machine Discrete Logarithm Integer Factorization Generalize Riemann Hypothesis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Bibliographic Notes and Further Reading

  1. [16]
    E. Bach and J. Shallit, Algorithmic Number Theory I — Efficient Algorithms, MIT Press, 1996.Google Scholar
  2. [33]
    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
  3. [50]
    H. Cohen, A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics 138, Springer-Verlag, 1993.Google Scholar
  4. [83]
    P. Giblin, Primes and Programming — An Introduction to Number Theory with Computing, Cambridge University Press, 1993.Google Scholar
  5. [81]
    P. Garrett, Making, Breaking Codes: An Introduction to Cryptology, Prentice-Hall, 2001.Google Scholar
  6. [123]
    D. E. Knuth, The Art of Computer Programming II — Seminumerical Algorithms, 3rd Edition, Addison-Wesley, 1998.Google Scholar
  7. [128]
    N. Koblitz, A Course in Number Theory and Cryptography, 2nd Edition, Graduate Texts in Mathematics 114, Springer-Verlag, 1994.Google Scholar
  8. [129]
    N. Koblitz, Algebraic Aspects of Cryptography, Algorithms and Computation in Mathematics 3, Springer-Verlag, 1998.Google Scholar
  9. [133]
    H. Krishna, B. Krishna, K. Y. Lin, and J. D. Sun, Computational Number Theory and Digital Signal Processing, CRC Press, 1994.Google Scholar
  10. [134]
    E. Kranakis, Primality and Cryptography, John Wiley & Sons, 1986.Google Scholar
  11. [207]
    H. Riesel, Prime Numbers and Computer Methods for Factorization, Birkhäuser, Boston, 1990.Google Scholar
  12. [222]
    M. R. Schroeder, Number Theory in Science and Communication, 3rd Edition, Springer Series in Information Sciences 7, Springer-Verlag, 1997.Google Scholar
  13. [200]
    P. Ribenboim, The New Book of Prime Number Records, Springer-Verlag, 1996.Google Scholar
  14. [234]
    R. D. Silverman, “A Perspective on Computational Number Theory”, Notices of the American Mathematical Society, 38 6(1991), 562–568.Google Scholar
  15. [14]
    E. Bach, M. Giesbrecht and J. McInnes, The Complexity of Number Theoretical Algorithms, Technical Report 247/91, Department of Computer Science, University of Toronto, 1991.Google Scholar
  16. [79]
    M. R. Garey and D. S. Johnson, Computers and Intractability — A Guide to the Theory of NP-Completeness, W. H. Freeman and Company, 1979.Google Scholar
  17. [114]
    D. S. Johnson, “A Catalog of Complexity Classes”, Handbook of Theoretical Computer Science, edited by J. van Leeuwen, MIT Press, 1990, 69–161.Google Scholar
  18. [142]
    H. R. Lewis and C. H. Papadimitriou, Elements of the Theory of Computation, 2nd Edition, Prentice-Hall, 1998.Google Scholar
  19. [143]
    P. Linz, An Introduction to Formal Languages and Automata, 2nd Edition, Jones and Bartlett Publishers, 1997.Google Scholar
  20. [170]
    R. Motwani and P. Raghavan, Randomized Algorithms, Cambridge University Press, 1995.Google Scholar
  21. [214]
    G. Rozenberg and A. Salomaa, Cornerstones of Undecidability, Prentice-Hall, 1994.Google Scholar
  22. [261]
    S. Y. Yan, Perfect, Amicable and Sociable Numbers — A Computational Approach, World Scientific, 1996.Google Scholar
  23. [43]
    J. P. Buhler (editor), Algorithmic Number Theory, Third International Symposium, ANTS-III, Proceedings, Lecture Notes in Computer Science 1423, Springer-Verlag, 1998.Google Scholar
  24. [226]
    P. Shor, “Algorithms for Quantum Computation: Discrete Logarithms and Factoring”, Proceedings of 35th Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press, 1994, 124–134.Google Scholar
  25. [23]
    P. Benioff, “The Computer as a Physical System — A Microscopic Quantum Mechanical Hamiltonian Model of Computers as Represented by Turing Machines”, Journal of Statistical Physics, 22 (1980), 563–591.MathSciNetCrossRefGoogle Scholar
  26. [63]
    D. Deutsch, “Quantum Theory, the Church-Turing Principle and the Universal Quantum Computer”, Proceedings of the Royal Society of London, Series A, 400 (1985), 96–117.MathSciNetGoogle Scholar
  27. [74]
    R. P. Feynman, “Simulating Physics with Computers”, International Journal of Theoretical Physics, 21 (1982), 467–488.MathSciNetCrossRefGoogle Scholar
  28. [75]
    R. P. Feynman, Feynman Lectures on Computation, Edited by A. J. G. Hey and R. W. Allen, Addison-Wesley, 1996.Google Scholar
  29. [258]
    C. P. Williams and S. H. Clearwater, Explorations in Quantum Computation, The Electronic Library of Science (TELOS), Springer-Verlag, 1998.Google Scholar
  30. [1]
    U. Vazirani, “Introduction to Special Section on Quantum Computation”, p 1409–1410.Google Scholar
  31. [2]
    E. Bernstein and U. Vazirani, “Quantum Complexity Theory”, pp 1411–1473.Google Scholar
  32. [3]
    D. R. Simon, “On the Power of Quantum Computation”, pp 1474–1483.Google Scholar
  33. [4]
    P. Shor, “Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer”, pp 1484–1509.Google Scholar
  34. [5]
    C. H. Bennett et al., “Strengths and Weakness of Quantum Computing”, pp 1510–1523.Google Scholar
  35. [6]
    L. M. Adleman et al., “Quantum Computability”, pp 1524–1540.Google Scholar
  36. [7]
    A. Barenco et al., “Stabilization of Quantum Computations by Symmetrization”, pp 1541–1557.Google Scholar
  37. [24]
    C. H. Bennett, “Quantum Information and Computation”, Physics Today, October 1995, 24–30.Google Scholar
  38. [115]
    R. Jozsa, “Quantum Factoring, Discrete Logarithms, and the Hidden Subgroup Problem”, Computing in Science and Engineering, March/April 2001, 34–43.Google Scholar
  39. [202]
    E. Rieffel and W. Polak, “An Introduction to Quantum Computing for Non-Physicists”, ACM Computing Surveys, 32 3(2000), 300–335.CrossRefGoogle Scholar
  40. [217]
    V. Scarani, “Quantum Computing”, American Journal of Physics, 66, 11(1998), 956–960.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Song Y. Yan
    • 1
  1. 1.Computer ScienceAston UniversityBirminghamUK

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