SubLinear Root Detection, and New Hardness Results, for Sparse Polynomials Over Finite Fields
Abstract
We present a deterministic 2^O(t)q^{(t2)(t1)+o(1)} algorithm to decide whether a univariate polynomial f, with exactly t monomial terms and degree <q, has a root in F_q. A corollary of our method  the first with complexity sublinear in q when t is fixed  is that the nonzero roots in F_q can be partitioned into at most 2 \sqrt{t1} (q1)^{(t2)(t1)} cosets of two subgroups S_1,S_2 of F^*_q, with S_1 in S_2. Another corollary is the first deterministic sublinear algorithm for detecting common degree one factors of ktuples of tnomials in F_q[x] when k and t are fixed. When t is not fixed we show that each of the following problems is NPhard with respect to BPPreductions, even when p is prime: (1) detecting roots in F_p for f, (2) deciding whether the square of a degree one polynomial in F_p[x] divides f, (3) deciding whether the discriminant of f vanishes, (4) deciding whether the gcd of two tnomials in F_p[x] has positive degree. Finally, we prove that if the complexity of root detection is sublinear (in a refined sense), relative to the straightline program encoding, then NEXP is not in P/Poly.
 Publication:

arXiv eprints
 Pub Date:
 April 2012
 arXiv:
 arXiv:1204.1113
 Bibcode:
 2012arXiv1204.1113B
 Keywords:

 Mathematics  Number Theory;
 Computer Science  Computational Complexity
 EPrint:
 15 pages total (cover page, 10 pages, references, and 3 short appendices). This version corrects various minor typos, and improves the statement of the first main theorem