, Volume 64, Issue 3, pp 454–480

Computing Sparse Multiples of Polynomials

  • Mark Giesbrecht
  • Daniel S. Roche
  • Hrushikesh Tilak


We consider the problem of finding a sparse multiple of a polynomial. Given fF[x] of degree d over a field F, and a desired sparsity t, our goal is to determine if there exists a multiple hF[x] of f such that h has at most t non-zero terms, and if so, to find such an h. When F=ℚ and t is constant, we give an algorithm which requires polynomial-time in d and the size of coefficients in h. When F is a finite field, we show that the problem is at least as hard as determining the multiplicative order of elements in an extension field of F (a problem thought to have complexity similar to that of factoring integers), and this lower bound is tight when t=2.


Sparse polynomial Sparsest multiple 


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Copyright information

© Springer Science+Business Media, LLC (outside the USA) 2012

Authors and Affiliations

  • Mark Giesbrecht
    • 1
  • Daniel S. Roche
    • 2
  • Hrushikesh Tilak
    • 1
  1. 1.Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.United States Naval AcademyAnnapolisUSA

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