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A New Look at an Old Equation

  • Conference paper
Algorithmic Number Theory (ANTS 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5011))

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Abstract

The general binary quadratic Diophantine equation

$$ ax^2 + bxy + cy^2 + dx + ey + f = 0 $$

was first solved by Lagrange over 200 years ago. Since that time little improvement has been made to Lagrange’s technique. In this paper we show how to reduce this problem to that of determining whether or not an ideal of a certain quadratic order is principal and if so exhibiting a generator of that ideal. In the difficult case of the discriminant \(\ensuremath{\Delta}\) of this order being positive, we develop a Las Vegas algorithm for solving the principal ideal problem that executes in expected time bounded by \(O(\ensuremath{\Delta}^{1/6 + \epsilon})\), whereas the complexity of Lagrange’s (unconditional) technique for solving this problem is \(O(\ensuremath{\Delta}^{1/2 + \epsilon})\).

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References

  1. Booker, A.: Quadratic class numbers and character sums. Math. Comp. 75, 1481–1492 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  2. Buchmann, J., Thiel, C., Williams, H.C.: Short representation of quadratic integers. In: Mathematics and its Applications, vol. 325, pp. 159–185. Kluwer Academic Publishers, Dordrecht (1995)

    Google Scholar 

  3. Buchmann, J., Vollmer, U.: Binary Quadratic Forms: An Algorithmic Approach. Algorithms and Computation in Mathematics, vol. 20. Springer, Berlin (2007)

    Google Scholar 

  4. Cojocaru, A.C., Murty, M.R.: An Introduction to Sieve Methods and their Application. Cambridge University Press, Cambridge (2005)

    Google Scholar 

  5. Cohen, H., Lenstra Jr., H.W.: Heuristics on class groups of number fields. In: Number Theory. Lecture Notes in Math., vol. 1068, pp. 33–62. Springer, New York (1983)

    Google Scholar 

  6. Dickson, L.E.: History of the Theory of Numbers, Carnegie Institution of Washington, Publication No. 256 (1919), vol. 2. Dover Publications, New York (2005)

    Google Scholar 

  7. de Haan, R.: A fast, rigourous technique for verifying the regulator of a real quadratic field. Master’s thesis, University of Amsterdam (2004)

    Google Scholar 

  8. de Haan, R., Jacobson Jr., M.J., Williams, H.C.: A fast, rigorous technique for computing the regulator of a real quadratic field. Math. Comp. 76, 2139–2160 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Jacobson Jr., M.J.: Subexponential Class Group Computation in Quadratic Orders. PhD thesis, Technische Universität Darmstadt, Darmstadt, Germany (1999)

    Google Scholar 

  10. Jacobson Jr., M.J.: Computing discrete logarithms in quadratic orders. Journal of Cryptology 13, 473–492 (2000)

    Article  MATH  Google Scholar 

  11. Jacobson Jr., M.J., Scheidler, R., Williams, H.C.: The efficiency and security of a real quadratic field based key exchange protocol, Walter de Gruyter, Berlin, pp. 89–112 (2001)

    Google Scholar 

  12. Jacobson Jr., M.J., Sawilla, R.E., Williams, H.C.: Efficient ideal reduction in quadratic fields. International Journal of Mathematics and Computer Science 1, 83–116 (2006)

    MATH  MathSciNet  Google Scholar 

  13. Jacobson Jr., M.J., Scheidler, R., Williams, H.C.: An improved real quadratic field based key-exchange procedure. J. Cryptology 19, 211–239 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  14. Jacobson Jr., M.J., van der Poorten, A.J.: Computational aspects of NUCOMP. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 120–133. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  15. Jacobson Jr., M.J., Williams, H.C.: Modular arithmetic on elements of small norm in quadratic fields. Designs, Codes and Cryptography 27, 93–110 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kornhauser, D.M.: On the smallest solution to the general binary quadratic Diophantine equation. Acta Arith. 55, 83–94 (1990)

    MATH  MathSciNet  Google Scholar 

  17. Lagrange, J.L.: Sur la solution des problèmes indéterminés du second degré. In: Oeuvres, Gauthier-Villars, Paris, vol. II, pp. 377–535 (1868)

    Google Scholar 

  18. Lenstra Jr., H.W.: On the calculation of regulators and class numbers of quadratic fields. London Math. Soc. Lecture Notes Series 56, 123–150 (1982)

    MathSciNet  Google Scholar 

  19. Maurer, M.H.: Regulator Approximation and Fundamental Unit Computation for Real-Quadratic Orders. PhD thesis, Technische Universität Darmstadt, Darmstadt, Germany (2000)

    Google Scholar 

  20. Nagell, T.: Introduction to Number Theory, Chelsea, NY (1964)

    Google Scholar 

  21. Nitaj, A.: L’algorithme de Cornacchia. Expositiones Math. 13, 358–365 (1995)

    MATH  MathSciNet  Google Scholar 

  22. van der Poorten, A.J.: A note on NUCOMP. Math. Comp. 72, 1935–1946 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  23. Shanks, D.: The infrastructure of real quadratic fields and its applications. In: Proc. 1972 Number Theory Conf., Boulder, Colorado, pp. 217–224 (1972)

    Google Scholar 

  24. Shanks, D.: On Gauss and composition I, II. NATO ASI, Series C, vol. 265, pp. 163–204. Kluwer, Dordrecht (1989)

    Google Scholar 

  25. Srinivasan, A.: Computations of class numbers of real quadratic fields. Math. Comp. 67, 1285–1308 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  26. Silvester, A.K.: Fast and unconditional principal ideal testing. Master’s thesis, University of Calgary (2006)., http://math.ucalgary.ca/~aksilves/papers/msc-thesis.pdf

  27. Stolt, B.: On the Diophantine equation u 2 − D v 2 = ±4N, Parts I, II, III. Ark. Mat. 2, 1–23, 251–268 (1952); 3, 117–132 (1955)

    Google Scholar 

  28. Williams, H.C., Wunderlich, M.C.: On the parallel generation of the residues for the continued fraction factoring algorithm. Math. Comp. 48, 405–423 (1987)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Defense Research & Development Canada, 3701 Carling Ave., Ottawa, ON, K1A 0Z4, Canada

    R. E. Sawilla

  2. Dept. of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, AB, T2N 1N4, Canada

    A. K. Silvester & H. C. Williams

Authors
  1. R. E. Sawilla
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  2. A. K. Silvester
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  3. H. C. Williams
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Editor information

Alfred J. van der Poorten Andreas Stein

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Sawilla, R.E., Silvester, A.K., Williams, H.C. (2008). A New Look at an Old Equation. In: van der Poorten, A.J., Stein, A. (eds) Algorithmic Number Theory. ANTS 2008. Lecture Notes in Computer Science, vol 5011. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79456-1_2

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  • DOI: https://doi.org/10.1007/978-3-540-79456-1_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-79455-4

  • Online ISBN: 978-3-540-79456-1

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