Skip to main content
SpringerLink
Log in

Computational complexity of decision problems in elementary number theory

  • Conference paper
  • First Online:
Model Theory of Algebra and Arithmetic

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 834))

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Adleman, L., A Subexponential Algorithm for the Discrete Logarithm Problem, with Applications to Cryptography. Proc. 20th Annual IEEE Symp. on Foundations of Computer Science, 1979, pp. 55–60.

    Google Scholar 

  2. Adleman, L. and Manders, K., Diophantine Complexity. Proc. 17th Annual Symp. on Foundations of Computer Science (IEEE), 1976, pp. 81–88.

    Google Scholar 

  3. Adleman, L. and Manders, K., Reducibility, Randomness and Intractability. Proc. 9th Annual ACM Symp. on Theory of Computing, 1977, pp. 151–163.

    Google Scholar 

  4. Adleman, L. and Manders, K., Reductions that lie. Proc. 20th Annual IEEE Symp. on Foundations of Computer Science, 1979, pp. 397–410.

    Google Scholar 

  5. Adleman, L., Manders, K., and Miller, G.L., On Taking Roots in Finite Fields. Proc. 18th Annual Symp. on Foundations of Computer Science (IEEE), 1977, pp. 175–178.

    Google Scholar 

  6. Ankeny, N., The Least Quadratic Nonresidue. Ann. of Math. 55 (1952), pp. 65–72.

    Article  MathSciNet  MATH  Google Scholar 

  7. Chandra, A. and Stockmeyer, L., Alternation. Proc. 17th Annual Symp. on Foundations of Computer Science (IEEE), 1976, pp. 98–108.

    Google Scholar 

  8. Cobham, A., The Intrinsic Computational Difficulty of Functions. In Y. Bar-Hillel (ed.) Proc. 1969 Intern. Congress for Logic, Methodology and Philosophy of Science, North-Holland, Amsterdam, pp. 24–30.

    Google Scholar 

  9. Cook, S., The Complexity of Theorem-Proving Procedures. Proc. 3rd Annual ACM Symp. on Theory of Computing, 1971, pp. 151–158.

    Google Scholar 

  10. Edmonds, J., Paths, Trees and Flowers. Canadian J. Math. 17 (1965), pp. 449–467.

    Article  MathSciNet  MATH  Google Scholar 

  11. Garey, M. and Johnson, D., Computers and Intractability. W. H. Freeman, 1979.

    Google Scholar 

  12. Karp, R., Reducibility Among Combinatorial Problems. In: Complexity of Computer Computation, eds. Miller, R. N. and Thatcher, J., Plenum Press, 1972, pp. 85–104.

    Google Scholar 

  13. Kozen, D., On Parallelism in Turing Machines. Proc. 17th Annual Symp. on Foundations of Computer Science (IEEE), 1976, pp. 89–97.

    Google Scholar 

  14. Ladner, R., On the Structure of Polynomial Time Reducibility. J. ACM 22 (1975), p. 155–171.

    Article  MathSciNet  MATH  Google Scholar 

  15. Lagarias, J., Worst-Case Complexity Bounds in the Theory of Integral Quadratic Forms, to appear in J. Algorithms.

    Google Scholar 

  16. Lagarias, J., On the Computational Complexity of Determining the Solvability or Unsolvability of the Equation X2-DY2=−1, to appear in Trans. AMS.

    Google Scholar 

  17. Lagarias, J., Succinet Certificates for the Solvability of Binary Quadratic Diophantine Equations. (Extended Abstract.) Proc. 20th Annual Symp. on Foundations of Computer Science (TEEE), 1979, pp. 47–54.

    Google Scholar 

  18. Lamé, M., Note sur la limite du nombre des divisions..., C. R. Acad. Sci. Paris Ser. A-B 19 (1844), pp. 867–869.

    Google Scholar 

  19. Lehmer, D.H., Computer Technology Applied to the Theory of Numbers. Studies in Number Theory, M.A.A., 1969, pp. 117–151.

    Google Scholar 

  20. Levin, P.A., Universal Sorting Problems (Russian). Problemi Peredaci Informacii IX (1973), pp. 115–116.

    Google Scholar 

  21. Machtey, M. and Young, P., An Introduction to the General Theory of Algorithms. North-Holland, 1978.

    Google Scholar 

  22. Manders, K., Studies in Applied Logic. Ph.D. Thesis, Berkeley, 1978.

    Google Scholar 

  23. Manders, K., and Adleman, L., NP-complete Decision Problems for Binary Quadratics, J. Comp. Sys. Sci. 15 (1978), pp. 168–184. Earlier version: NP-Complete Decision Problems for Quadratic Polynomials. Proc. 8th ACM Conf. on Theory of Computing (1976), pp. 23–29.

    Article  MathSciNet  MATH  Google Scholar 

  24. Miller, G.L., Riemann’s Hypothesis and Tests for Primality, J. Comp. Sys. Sci. 13 (1976), pp. 300–317.

    Article  MATH  Google Scholar 

  25. Montgomery, H., Topics in Multiplicative Number Theory. Springer LNM 227, 1971.

    Google Scholar 

  26. Mordell, L., Diophantine Equations, Pure and Applied Mathematics. Vol. 30, Academic Press, 1969.

    Google Scholar 

  27. Niven, I., and Zuckerman, H., An Introduction to the Theory of Numbers. J. Wiley, 1960, 1966.

    Google Scholar 

  28. Plaisted, D., New NP-Hard and NP-Complete Polynomial and Integer Divisibility Problems. Proc. 18th Annual Symp. on Foundations of Computer Science (IEEE), 1977, pp. 241–253.

    Google Scholar 

  29. Pratt, V., Every Prime has a Succinct Certificate. SIAM J. Comput. 4 (1975), pp. 214–220.

    Article  MathSciNet  MATH  Google Scholar 

  30. Rabin, M., Digitalized Signatures and Public Key Functions as Intractable as Factorization. Memo MIT/LCS/TR-212, 1979.

    Google Scholar 

  31. Rivest, R., Shamir, A. and Adleman, L., A Method for Obtaining Digital Signatures and Public Key Cryptosystems. Comm. ACM 21 (1978), pp. 120–126.

    Article  MathSciNet  MATH  Google Scholar 

  32. Shanks, D., The Infrastructure of a Real Quadratic Field and Its Applications. Proc. 1972 Number Theory Conf., University of Colorado, Boulder (1972), pp. 217–224.

    Google Scholar 

  33. Weinberger, P., Private Communication.

    Google Scholar 

  34. Wilkie, A., this volume.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Philosophy and Department of Mathematics, University of Pittsburgh, 15260, Pittsburgh, PA, USA

    Kenneth L. Manders

Authors
  1. Kenneth L. Manders
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Leszek Pacholski Jedrzej Wierzejewski Alec J. Wilkie

Rights and permissions

Reprints and permissions

Copyright information

© 1980 Springer-Verlag

About this paper

Cite this paper

Manders, K.L. (1980). Computational complexity of decision problems in elementary number theory. In: Pacholski, L., Wierzejewski, J., Wilkie, A.J. (eds) Model Theory of Algebra and Arithmetic. Lecture Notes in Mathematics, vol 834. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0090168

Download citation

  • DOI: https://doi.org/10.1007/BFb0090168

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-10269-4

  • Online ISBN: 978-3-540-38393-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation