Modular Arithmetic on Elements of Small Norm in Quadratic Fields

  • Authors:

  • M. J. Jacobson, Jr.;H. C. Williams

  • Affiliations:

  • Department of Computer Science, University of Manitoba, Winnipeg, MB, R3T 2N2 jacobs@cs.umanitoba.ca;Dept. of Mathematics and Statistics, University of Calgary, Calgary, AB, T2N 1N4 williams@math.ucalgary.ca

  • Venue:
  • Designs, Codes and Cryptography
  • Year:
  • 2002

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Abstract

We describe an algorithm which rapidly computes the coefficients of elements of small norm in quadraticfields modulo a positive integer. Our method requires that an approximation of the natural logarithm of thatquadratic field element is known to sufficient accuracy. To demonstrate the efficiency and utility of our method,we apply it to eliminate a number of exceptional cases of a theorem of Dujella and Pethő [9]involving Diophantine triples. In particular, we are able to show that Theorem 1.2 of [9] isunconditionally true for all k ≤ 100 with the possible exception of k = 37, for whichthe theorem holds under the assumption of the Extended Riemann Hypothesis.