Abstract
We present an algorithm to compute a greatest common divisor of two integers in a quadratic number ring that is a unique factorization domain. The algorithm uses \(O(n {\rm log}^{2} n {\rm log log} n + \Delta^ {\raisebox{0.8mm}{\scriptsize 1}{\scriptsize /}\raisebox{-0.5mm}{\scriptsize 2}} +^{\epsilon})\) bit operations in a ring of discriminant Δ. This appears to be the first gcd algorithm of complexity o(n 2) for any fixed non-Euclidean number ring. The main idea behind the algorithm is a well known relationship between quadratic forms and ideals in quadratic rings. We also give a simpler version of the algorithm that has complexity O(n 2) in a fixed ring. It uses a new binary algorithm for reducing quadratic forms that may be of independent interest.
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Agarwal, S., Frandsen, G.: Binary GCD like algorithms in some complex quadratic rings. In: Buell, D.A. (ed.) ANTS 2004. LNCS, vol. 3076, pp. 57–71. Springer, Heidelberg (2004)
Buchmann, J., Biehl, I.: An analysis of the reduction algorithms for binary quadratic forms. In: Voronoi’s impact on Modern Science, pp. 71–98 (1998)
Buchmann, J., Thiel, C., Williams, H.: Short representation of quadratic integers. In: Computational algebra and number theory, Sydney (1992); Math. Appl., vol. 325, pp. 159–185. Kluwer Acad. Publ., Dordrecht (1995)
Cohn, H.: Advanced number theory. Dover Publications Inc., New York (1980); Reprint of A second course in number theory, Dover Books on Advanced Mathematics (1962)
Damgård, I.B., Frandsen, G.S.: Efficient algorithms for gcd and cubic residuosity in the ring of Eisenstein integers. J. Symb. Comput. 39(6), 643–652 (2005)
Hungerford, T.W.: Algebra. Graduate Texts in Mathematics, vol. 73. Springer, New York (1980); Reprint of the 1974 original
Ireland, K., Rosen, M.: A classical introduction to modern number theory, 2nd edn. Graduate Texts in Mathematics, vol. 84. Springer-Verlag, New York (1990)
Kaltofen, E., Rolletschek, H.: Computing greatest common divisors and factorizations in quadratic number fields. Math. Comp. 53(188), 697–720 (1989)
Knuth, D.E.: The art of computer programming, 2nd edn., vol. 2. Addison-Wesley Publishing Co., Reading (1981)
Lagarias, J.C.: Worst-case complexity bounds for algorithms in the theory of integral quadratic forms. J. Algorithms 1(2), 142–186 (1980)
Lang, S.: Algebra, 3rd edn. Addison-Wesley Publishing Company, Reading (1993)
Lehmer, D.H.: Euclid’s algorithm for large numbers. American Mathematical Monthly 45, 227–233 (1938)
Lemmermeyer, F.: The Euclidean algorithm in algebraic number fields. Exposition. Math. 13(5), 385–416 (1995); Updated version (February 2004), http://www.fen.bilkent.edu.tr/~franz/publ/survey.pdf
Lenstra Jr., H.W.: On the calculation of regulators and class numbers of quadratic fields. In: Number theory days, 1980 (Exeter, 1980). London Math. Soc. Lecture Note Ser., vol. 56, pp. 123–150. Cambridge Univ. Press, Cambridge (1982)
Schonhage, A.: Schnelle berechnung von kettenbruchentwicklungen. Acta Informatica 1, 139–144 (1971)
Schönhage, A., Strassen, V.: Schnelle Multiplikation grosser Zahlen. Computing (Arch. Elektron. Rechnen) 7, 281–292 (1971)
Schönhage, A.: Fast reduction and composition of binary quadratic forms. In: ISSAC 1991: Proceedings of the 1991 international symposium on Symbolic and algebraic computation, pp. 128–133. ACM Press, New York (1991)
Shoup, V.: A Computational Introduction to Number Theory and Algebra. Cambridge University Press, Cambridge (2005)
Stein, J.: Computational problems associated with Racah algebra. J. Comput. Phys. (1), 397–405 (1967)
Weilert, A.: (1+ i)-ary GCD computation in Z[i] as an analogue to the binary GCD algorithm. J. Symbolic Comput. 30(5), 605–617 (2000)
Weilert, A.: Asymptotically fast GCD computation in Z[i]. In: Bosma, W. (ed.) ANTS 2000. LNCS, vol. 1838, pp. 595–613. Springer, Heidelberg (2000)
Wikström, D.: On the l-ary gcd-algorithm in rings of integers. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 1189–1201. Springer, Heidelberg (2005)
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Agarwal, S., Frandsen, G.S. (2006). A New GCD Algorithm for Quadratic Number Rings with Unique Factorization. In: Correa, J.R., Hevia, A., Kiwi, M. (eds) LATIN 2006: Theoretical Informatics. LATIN 2006. Lecture Notes in Computer Science, vol 3887. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11682462_8
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DOI: https://doi.org/10.1007/11682462_8
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