Maximal quotient rational reconstruction: an almost optimal algorithm for rational reconstruction

  • Authors:

  • Michael Monagan

  • Affiliations:

  • Simon Fraser University, Burnaby, B.C., Canada

  • Venue:
  • ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
  • Year:
  • 2004

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Abstract

Let n/d ∈ Q, m be a positive integer and let u = n/d mod m. Thus $u$ is the image of a rational number modulo m. The rational reconstruction problem is; given u and m find n/d. A solution was first given by Wang in 1981. Wang's algorithm outputs n/d when m 2 M2 where M = max(|n|,d). Because of the wide application of this algorithm in computer algebra, several authors have investigated its practical efficiency and asymptotic time complexity.In this paper we present a new solution which is almost optimal in the following sense; with controllable high probability, our algorithm will output n/d when m is a modest number of bits longer than 2 |n| d. This means that in a modular algorithm where m is a product of primes, the modular algorithm will need one or two primes more than the minimum necessary to reconstruct n/d; thus if |n| ⇐ d or d ⇐ |n| the new algorithm saves up to half the number of primes. Further, our algorithm will fail with high probability when m