Fast rational function reconstruction

  • Authors:

  • Sara Khodadad;Michael Monagan

  • Affiliations:

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

  • Venue:
  • Proceedings of the 2006 international symposium on Symbolic and algebraic computation
  • Year:
  • 2006

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Abstract

Let F be a field and let f and g be polynomials in F[t] satisfying deg f deg g. Recall that on input of f and g the extended Euclidean algorithm computes a sequence of polynomials (si, ti, ri) satisfying sif + tig = ri. Thus for i with gcd(ti, f) = 1, we obtain rational functions ri/ti ∈ F(t) satisfying ri/ti ≡ g (mod f).In this paper we modify the fast extended Euclidean algorithm to compute the smallest ri/ti, that is, an ri/ti minimizing deg ri + deg ti. This means that in an output sensitive modular algorithm when we are recovering rational functions in F(t) from their images modulo f(t) where f(t) is increasing in degree, we can recover them as soon as the degree of f is large enough and we can do this fast.We have implemented our modified fast Euclidean algorithm for F = Zp, p a word sized prime, in Java. Our fast algorithm beats the ordinary Euclidean algorithm around degree 200. This has application to polynomial gcd computation and linear algebra over Zp(t).