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Solving Norm Form Equations over Number Fields

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Algebraic Informatics (CAI 2009)

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Abstract

Let K be a number field and L a finite extension of K of degree ℓ. Let ω 1 = 1,ω 2,..., ω be K-linearly independent integers of L and k an integer of K. We denote by N L/K the norm from L to K. In this paper we give an algorithm for the computation of algebraic integers, x 1,..., x  ∈ K satisfying the equation N L/K (ω 1 x 1 + ⋯ + x ω ) = k.

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References

  1. Abel, C.S.: Ein Algorithmus zur Berechnung der Klassenzahl und des Regulators reell quadratischer Ordnungen. Ph.D. Thesis, Universität des Saarlandes, Saarbrücken, Germany (1994)

    Google Scholar 

  2. Arvind, V., Kurur, P.P.: On the complexity of computing units in a number field. In: Buell, D.A. (ed.) ANTS 2004. LNCS, vol. 3076, pp. 72–86. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  3. Assmann, B., Eick, B.: Computing polycyclic presentations for polycyclic rational matrix groups. J. Symbolic Comput. 40(6), 1269–1284 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Barbeau, E.J.: Pell’s Equation. Springer, Heidelberg (2003)

    Book  MATH  Google Scholar 

  5. Bertrand, D.: Dyality on tori and dependence relations. J. Austral. Math. Soc (Series A) 62, 198–216 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Buchmann, J.: On the Computation of Units and Class Numbers by a Generalization of Langrange’s Algorithm. J. Number Theory 26, 8–30 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  7. Buchmann, J., Williams, H.C.: On the Infrastructure of the Principal Ideal Class of an Algebraic Number Field of Unit Rank One. Mathematics of Computation 50(182), 569–579 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, C.-Y., Chang, C.-C., Yang, W.-P.: Fast RSA-type Schemes Based on Pell Equations over \({\mathbb Z}_N\). In: Joint Conference of, International Computer Symposium, Kaohsiung, Taiwan, R.O.C, December 19-21 (1996)

    Google Scholar 

  9. Cohen, H., Frey, G.: Handbook of Elliptic and Hyperelliptic Cryptography. Chapman and Hall/CRC (2006)

    Google Scholar 

  10. Dirichlet, Recherches sur les formes quadratiques à coefficients et à indeterminées complexes, J. Reine Angew. Math. 24, 291-371 = Werke I, 535–618 (1842)

    Google Scholar 

  11. Fjellstedt, L.: On a class of Diophantine equations of the second degree in imaginary quadratic fields. Arkiv. Mat. 2, 435–461 (1952-1954)

    Google Scholar 

  12. 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(260), 2139–2160 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hallgren, S.: Polynomial-Time Quantum Algorithms for Pell’s Equation and the Principal Ideal Problem. Journal of ACM 54(1), Art. 4, 19 (2007) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  14. Huber, K.: On the Period Length of Generalized Inverse Pseudorandom Generators. AAECC 5, 255–260 (1994)

    Article  MATH  Google Scholar 

  15. Lenstra Jr., H.W.: Solving the Pell equation. Notices Amer. Math. Soc. 49, 182–192 (2002)

    MathSciNet  MATH  Google Scholar 

  16. Matthews, K.: The Diophantine Equation x 2 − Dy 2 = N, D > 0. Expos. Math. 18, 323–331 (2000)

    MathSciNet  MATH  Google Scholar 

  17. Mollin, R.A.: Fundamental Number Theory with Applications. CRC Press, Boca Raton (1998)

    MATH  Google Scholar 

  18. Mollin, R.A.: Simple Continued Fraction Solutions for Diophantine Equations. Expos. Math. 19, 55–73 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  19. Murthy, N.R., Swamy, M.N.S.: Cryptographic Applications of Brahmagupta-Bhãskara Equation. IEEE Transactions on Circuits and Systems 53(7), 1565–1571 (2006)

    Article  MathSciNet  Google Scholar 

  20. Niven, I.: Quadratic Diophantine Equations in the Rational and Quadratic Fields. Tras. of Amer. Math. Soc. 52(1), 1–11 (1942)

    Article  MathSciNet  MATH  Google Scholar 

  21. Niven, I.: The Pell equation in quadratic fields. Bull. Amer. Math. Soc. 49, 413–416 (1943)

    Article  MathSciNet  MATH  Google Scholar 

  22. Pohst, M.: Computational Algebraic Number Theory. DMV Seminar Band 21. Birkhauser Verlag, Basel (1993)

    Google Scholar 

  23. Schmid, W.A.: On the set of integral solutions of the Pell equation in number fields. Aequationes Math 71, 109–114 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Shastri, P.: Integral points on the unit circle. J. Number Theory 91(1), 67–70 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  25. Shastri, P.: Integral points on the circle \(X\sp 2+Y\sp 2=c\). Currents trends in number theory (Allahabad, 2000), 175–184. Hindustan Book Agency, New Delhi (2002)

    Google Scholar 

  26. Skolem, T.: A theorem on the equation ζ 2 − δη 2 = 1, where δ, ζ, η are integers in an imaginary quadratic field. Avh. Norske Akad. Oslo 1, 1–13 (1945)

    Google Scholar 

  27. Skolem, T.: A remark on the equation ζ 2 − δη 2 = 1, δ > 0, δ′,δ, ⋯ < 0 where δ,ζ, η belong to a total real number field. Avh. Norske Vid. Akad. Oslo. I(12), 15 (1945)

    MathSciNet  Google Scholar 

  28. Val´fiš, A.Z.: Elementary solution of Pell’s equation (Russian). Akad. Nauk Gruzin. SSR. Trudy Mat. Inst. Razmadze 18, 116–132 (1951)

    MathSciNet  Google Scholar 

  29. Vollmer, U.: Asymptotically fast discrete logarithms in quadratic number fields. In: Bosma, W. (ed.) ANTS 2000. LNCS, vol. 1838, pp. 581–594. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  30. Williams, H.C.: Solving the Pell Equation, Number theory for the millennium, III, Urbana, IL, A K Peters, Natick, MA, pp. 397–435 (2002)

    Google Scholar 

  31. http://www.math.tu-berlin.de/~kant/

  32. http://magma.maths.usyd.edu.au/magma/

  33. http://www.gap-system.org/

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Alvanos, P., Poulakis, D. (2009). Solving Norm Form Equations over Number Fields. In: Bozapalidis, S., Rahonis, G. (eds) Algebraic Informatics. CAI 2009. Lecture Notes in Computer Science, vol 5725. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03564-7_8

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  • DOI: https://doi.org/10.1007/978-3-642-03564-7_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-03563-0

  • Online ISBN: 978-3-642-03564-7

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