Skip to main content
SpringerLink
Log in

Efficient verification of Tunnell’s criterion

  • Published:
Japan Journal of Industrial and Applied Mathematics Aims and scope Submit manuscript

Abstract

An integern is congruent if there is a triangle with rational sides whose area isn. In the 1980s Tunnell gave an algorithm to test congruence which relied on counting integral points on the ellipsoids 2x 2 +y 2 + 8z 2 =n and 2x 2 +y 2 + 32z 2 =n. The correctness of this algorithm is conditional on the conjecture of Birch and Swinnerton-Dyer. The known methods for testing Tunnell’s criterion useO(n) operations. In this paper we give several methods with smaller exponents, including a randomized algorithm using timen 11/2+o(1) and spacen o(1) and a deterministic algorithm using space and timen 1/2+o(1).

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R.P. Brent, An improved Monte Carlo factorization algorithm. BIT,20 (1980), 176–184.

    Article  MATH  MathSciNet  Google Scholar 

  2. J. Cilleruelo and A. Córdoba, Lattice points on ellipses. Duke Math. J.,76 (1994), 741–750.

    Article  MATH  MathSciNet  Google Scholar 

  3. G.E. Collins and R.G.K. Loos, The Jacobi symbol algorithm. SIGSAM Bull.,16 (1982), 12–16.

    Article  MATH  Google Scholar 

  4. L.E. Dickson, History of the Theory of Numbers, Three Volumes. Chelsea Publishing Co., New York, 1966.

    Google Scholar 

  5. J.D. Dixon, Asymptotically fast factorization of integers. Math. Comp.,36 (1981), 255–260.

    Article  MATH  MathSciNet  Google Scholar 

  6. N.D. Elkies, CurvesDy 2 =x 3 −x of odd analytic rank. Algorithmic number theory (Sydney, 2002), Lecture Notes in Comput. Sci.,2369, Springer-Verlag, Berlin, 2002, 244–251.

    Chapter  Google Scholar 

  7. P. Erdős, Remarks on number theory I. Mat. Lapok,12 (1961), 10–17.

    MathSciNet  Google Scholar 

  8. S.A. Fenner, F. Green, S. Homer and R. Pruim, Determining acceptance possibility for a quantum computation is hard for the polynomial hierarchy. Proc. Royal Society, London (A),455 (1999), 3953–3966.

    Article  MATH  MathSciNet  Google Scholar 

  9. L. Fortnow, Counting complexity. Complexity Theory Retrospective II (L. Hemaspaandra and A. Selman, eds.), Lecture Notes in Comput. Sci., Springer-Verlag, Berlin, 1997, 81–107.

    Google Scholar 

  10. C.F. Gauss, Disquisitiones Arithmeticae. Springer-Verlag, New York, 1986.

    MATH  Google Scholar 

  11. L.K. Hua, Introduction to Number Theory. Springer-Verlag, Berlin, 1982.

    MATH  Google Scholar 

  12. K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, second ed. Graduate Texts in Mathematics,84, Springer-Verlag, New York, 1990.

    Google Scholar 

  13. N. Koblitz, Introduction to Elliptic Curves and Modular Forms. Graduate Texts in Mathematics,97, Springer-Verlag, New York, 1984.

    Google Scholar 

  14. M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information. Cambridge University Press, Cambridge, 2000.

    MATH  Google Scholar 

  15. J.M. Pollard, Theorems on factorization and primality testing. Proc. Cambridge Philos. Soc.,76 (1974), 521–528.

    Article  MATH  MathSciNet  Google Scholar 

  16. C. Pomerance, The quadratic sieve factoring algorithm. Advances in cryptology (Paris, 1984), Lecture Notes in Comput. Sci.,209, Springer-Verlag, Berlin, 1985, 169–182.

    Chapter  Google Scholar 

  17. S. Rudich and A. Wigderson (eds.), Computational Complexity Theory. American Mathematical Society, Providence, 2004.

    MATH  Google Scholar 

  18. R.J. Schoof, Quadratic fields and factorization. Computational methods in number theory, Part II, Math. Centre Tracts,155, Math. Centrum, Amsterdam, 1982, 235–286.

    Google Scholar 

  19. D. Shanks, Five number-theoretic algorithms. Proceedings of the Second Manitoba Conference on Numerical Mathematics (Winnipeg), Congressus Numerantium, No. VII, Utilitas Math., 1973, 51–70.

  20. V. Shoup, NTL: A Library for Doing Number Theory. Available from the author’s home page at New York University.

  21. J.H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.

    MATH  Google Scholar 

  22. N.P. Smart, The Algorithmic Resolution of Diophantine Equations. Cambridge U. Press, New York, 1998.

    MATH  Google Scholar 

  23. M. Tompa, Probabilistic Factoring Algorithms can be made errorless. Technical Report 83-09-01, Univ. Washington, Dept. of Computer Science, 1983.

  24. J.B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2. Invent. Math.,72 (1983), 323–334.

    Article  MATH  MathSciNet  Google Scholar 

  25. K. Wagner, The complexity of combinatorial problems with succinct input representation. Acta Inform.,23 (1986), 325–356.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computer Sciences Department, University of Wisconsin, 1210 W. Dayton St., 53706-1685, Madison, WI

    Eric Bach

  2. Department of Mathematics, Bucknell University, 17870, Lewisburg, PA

    Nathan C. Ryan

Authors
  1. Eric Bach
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nathan C. Ryan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eric Bach.

Additional information

The RMMC Summer School in Computational Number Theory and Cryptography was funded by NSF grant DMS-0612103, the Rocky Mountain Mathematics Consortium, The Number Theory Foundation, The Institute for Mathematics and its Applications (IMA), The Fields Institute, The Centre for Information Security and Cryptography (CISaC) and iCORE of Canada. As this paper is an outgrowth of this conference, the authors would like to thank all these sponsors. We would also like to thank the University of Wyoming for hosting the conference. Thanks to Dieter van Melkebeek for quantum advice. Also the authors are indebted to the referees for their comments that made the exposition more clear.

About this article

Cite this article

Bach, E., Ryan, N.C. Efficient verification of Tunnell’s criterion. Japan J. Indust. Appl. Math. 24, 229 (2007). https://doi.org/10.1007/BF03167537

Download citation

  • Received:

  • Revised:

  • DOI: https://doi.org/10.1007/BF03167537

Key words

Search

Navigation