Sub-linear root detection, and new hardness results, for sparse polynomials over finite fields

  • Authors:

  • Jingguo Bi;Qi Cheng;J. Maurice Rojas

  • Affiliations:

  • Tsinghua University, Beijing, China;University of Oklahoma, Norman, OK, USA;Texas A & M University, College Station, TX, USA

  • Venue:
  • Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
  • Year:
  • 2013

Quantified Score

Hi-index 0.00

Visualization

Abstract

We present a deterministic 2O(t)qt-2/t-1 +o(1) algorithm to decide whether a univariate polynomial f, with exactly t monomial terms and degree q, has a root in Fq. Our method is the first with complexity sub-linear in q when tis fixed. We also prove a structural property for the nonzero roots in Fq of any t-nomial: the nonzero roots always admit a partition into no more than 2√t-1(q-1)t-2/t-1 cosets of two subgroups S1 ⊆ S2 of F*q. This can be thought of as a finite field analogue of Descartes' Rule. A corollary of our results is the first deterministic sub-linear algorithm for detecting common degree one factors of k-tuples of t-nomials in Fq[x when k and t are fixed. When t is not fixed we show that, for p prime, detecting roots in Fp for f is NP-hard with respect to $BPP-reductions. Finally, we prove that if the complexity of root detection is sub-linear (in a refined sense), relative to the straight-line program encoding, then NEXP⊆P/poly.