Computational Preliminaries

  • Song Y. Yan


Computation has long been the deriving force in the development of both mathematics and cryptography, modern computation theory is however rooted in Turing’s 1936 paper on Computable Numbers, where a universal computing model, now called Turing machine is proposed.


  1. 1.
    L. M. Adleman, J. DeMarrais and M-D. A. Huang, “Quantum Computability”, SIAM Journal on Computing, 26, 5(1996), pp 1524–1540.MathSciNetCrossRefGoogle Scholar
  2. 2.
    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, 5(1980), pp 563–591.MathSciNetCrossRefGoogle Scholar
  3. 3.
    C. H. Bennett, “Strengths and Weakness of Quantum Computing”, SIAM Journal on Computing, 26, 5(1997), pp 1510–1523.MathSciNetCrossRefGoogle Scholar
  4. 4.
    C. H. Bennett and D. P. DiVincenzo, “Quantum Information and Computation”, Nature, 404, 6775(2000), pp 247–255.CrossRefGoogle Scholar
  5. 5.
    E. Bernstein and U. Vazirani, “Quantum Complexity Theory”, SIAM Journal on Computing, 26, 5(1997), pp 1411–1473.MathSciNetCrossRefGoogle Scholar
  6. 6.
    I. L Change, R. Laflamme, P, Shor, and W. H. Zurek, “Quantum Computers, Factoring, and Decoherence, Science, 270, 5242(1995), pp 1633–1635.Google Scholar
  7. 7.
    A. Church, “An Unsolved Problem of Elementary Number Theory” The American Journal of Mathematics, 58, 2(1936), pp 345–363.MathSciNetCrossRefGoogle Scholar
  8. 8.
    A. Church, “Book Review: On Computable Numbers, with an Application to the Entscheidungsproblem by Turing”, Journal of Symbolic Logic, 2, 1(1937), pp 42–43.Google Scholar
  9. 9.
    H. Cohen, A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics 138, Springer, 1993.Google Scholar
  10. 10.
    S. Cook, The Complexity of Theorem-Proving Procedures, Proceedings of the 3rd Annual ACM Symposium on the Theory of Computing, New York, 1971, pp 151–158.Google Scholar
  11. 11.
    S. Cook, “The Importance of the P versus NP Question”, Journal of ACM, 50, 1(2003), pp 27–29.MathSciNetCrossRefGoogle Scholar
  12. 12.
    S. Cook, The P versus NP Problem, In: J. Carlson, A. Jaffe and A. Wiles, Editors, The Millennium Prize Problems, Clay Mathematics Institute and American Mathematical Society, 2006, pp 87–104.Google Scholar
  13. 13.
    T. H. Cormen, C. E. Ceiserson and R. L. Rivest, Introduction to Algorithms, 3rd Edition, MIT Press, 2009.Google Scholar
  14. 14.
    R. Crandall and C. Pomerance, Prime Numbers – A Computational Perspective, 2nd Edition, Springer, 2005.Google Scholar
  15. 15.
    D. Deutsch, “Quantum Theory, the Church–Turing Principle and the Universal Quantum Computer”, Proceedings of the Royal Society of London, Series A, 400, 1818(1985), pp 96–117.MathSciNetCrossRefGoogle Scholar
  16. 16.
    R. P. Feynman, “Simulating Physics with Computers”, International Journal of Theoretical Physics, 21, 6(1982), 467–488.MathSciNetCrossRefGoogle Scholar
  17. 17.
    R. P. Feynman, Feynman Lectures on Computation, Edited by A. J. G. Hey and R. W. Allen, Addison-Wesley, 1996.Google Scholar
  18. 18.
    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
  19. 19.
    O. Goldreich, Foundations of Cryptography: Basic Tools, Cambridge University Press, 2001.Google Scholar
  20. 20.
    O. Goldreich, Foundations of Cryptography: Basic Applications, Cambridge University Press, 2004.Google Scholar
  21. 21.
    O. Goldreich, P, NP, and NP-Completeness, Cambridge University Press, 2010.Google Scholar
  22. 22.
    J. Grustka, Quantum Computing, McGraw-Hill, 1999.Google Scholar
  23. 23.
    M. Hirvensalo, Quantum Computing, 2nd Edition, Springer, 2004.Google Scholar
  24. 24.
    J. Hopcroft, R. Motwani and J. Ullman, Introduction to Automata Theory, Languages, and Computation, 3rd Edition, Addison-Wesley, 2007.Google Scholar
  25. 25.
    R. Karp, “Reducibility among Cominatorial Problems”, Complexity of Computer Computations, Edited by R. E. Miller and J. W. Thatcher, Plenum Press, New York, 1972, pp 85–103.CrossRefGoogle Scholar
  26. 26.
    D. E. Knuth, The Art of Computer Programming II – Seminumerical Algorithms, 3rd Edition, Addison-Wesley, 1998.Google Scholar
  27. 27.
    M. Le Bellac, A Short Introduction to Quantum Information and Quantum Computation, Cambridge University Press, 2005.Google Scholar
  28. 28.
    H. R. Lewis and C. H. Papadimitrou, Elements of the Theory of Computation, Prentice-Hall, 1998.Google Scholar
  29. 29.
    P. Linz, An Introduction to Formal Languages and Automata, 5th Edition, Jones and Bartlett Publishers, 2011.Google Scholar
  30. 30.
    Y. V. Matiyasevich, Hilbert’s Tenth Problem, MIT Press, 1993.Google Scholar
  31. 31.
    N. D. Mermin, Quantum Computer Science, Cambridge University Press, 2007.Google Scholar
  32. 32.
    M. A. Nielson and I. L. Chuang, Quantum Computation and Quantum Information, 10th Anniversary Edition, Cambridge University Press, 2010.Google Scholar
  33. 33.
    C. H. Papadimitrou, Computational Complexity, Addison Wesley, 1994.Google Scholar
  34. 34.
    E. Rieffel and W. Polak, Quantum Computing: A Gentle Introduction, MIT Press, 2011.Google Scholar
  35. 35.
    H. Riesel, Prime Numbers and Computer Methods for Factorization, Birkhäuser, Boston, 1990.zbMATHGoogle Scholar
  36. 36.
    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, pp 124–134.Google Scholar
  37. 37.
    P. Shor, “Polynomial-Tme Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer”, SIAM Journal on Computing, 26, 5(1997), pp 1411–1473.MathSciNetCrossRefGoogle Scholar
  38. 38.
    P. Shor, “Quantum Computing”, Documenta Mathematica, Extra Volume ICM 1998, I, pp 467–486.Google Scholar
  39. 39.
    P. Shor, “Introduction to Quantum Algorithms”, AMS Proceedings of Symposium in Applied Mathematics, 58, 2002, pp 143–159.MathSciNetCrossRefGoogle Scholar
  40. 40.
    P. Shor, “Why Haven’t More Quantum Algorithms Been Found?”, Journal of the ACM, 50, 1(2003), pp 87–90.MathSciNetCrossRefGoogle Scholar
  41. 41.
    D. R. Simon, “On the Power of Quantum Computation”, Proceedings of the 35 Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press, 1994, pp 116–123.Google Scholar
  42. 42.
    D. R. Simon, “On the Power of Quantum Computation”, SIAM Journal on Computing, 25, 5(1997), pp 1474–1483.MathSciNetCrossRefGoogle Scholar
  43. 43.
    M. Sipser, Introduction to the Theory of Computation, 2nd Edition, Thomson, 2006.Google Scholar
  44. 44.
    W. Trappe and L. Washington, Introduction to Cryptography with Coding Theory, 2nd Edition, Prentice-Hall, 2006.Google Scholar
  45. 45.
    A. Turing, “On Computable Numbers, with an Application to the Entscheidungsproblem”, Proceedings of the London Mathematical Society, S2-42, 1(1937), pp 230–265.Google Scholar
  46. 46.
    C. P. Williams and S. H. Clearwater, Explorations in Quantum Computation, The Electronic Library of Science, Springer, 1998.Google Scholar
  47. 47.
    U. V. Vazirani, “On the Power of Quantum Computation”, Philosophical Transactions of the Royal Society London, A356, 1743(1998), pp 1759–1768.Google Scholar
  48. 48.
    U. V. Vazirani, “Fourier Transforms and Quantum Computation”, Proceedings of Theoretical Aspects of Computer Science, Lecture Notes in Computer Science 2292, Springer, 2000, pp 208–220.Google Scholar
  49. 49.
    U. V. Vazirani, “A Survey of Quantum Complexity Theory”, AMS Proceedings of Symposium in Applied Mathematics, 58, 2002, pp 193–220.MathSciNetCrossRefGoogle Scholar
  50. 50.
    J. Watrous, “Quantum Computational Complexity”, . Encyclopedia of Complexity and System Science, Springer, 2009, pp 7174–7201.Google Scholar
  51. 51.
    C. P. Williams, Explorations in Quantum Computation, 2nd Edition, Springer, 2011.Google Scholar
  52. 52.
    N. S. Yanofsky and M. A. Mannucci, Quantum Computing for Computer Scientists, Cambridge University Press, 2008.Google Scholar
  53. 53.
    A. Yao, “Classical Physics and the Church Turing Thesis”, Journal of ACM, 50, 1(2003), pp 100–105.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Song Y. Yan
    • 1
  1. 1.Xingzhi CollegeZhejiang Normal UniversityJinhuaChina

Personalised recommendations