Fast Parallel Algorithms for Sparse Multivariate Polynomial Interpolation over Finite Fields

The authors consider the problem of reconstructing (i.e., interpolating) a t-sparse multivariate polynomial given a black box which will produce the value of the polynomial for any value of the arguments. It is shown that, if the polynomial has coefficients in a finite field $GF[q]$ and the black box can evaluate the polynomial in the field $GF[q^{\ulcorner 2\log_{q}(nt)+3 \urcorner}]$, where n is the number of variables, then there is an algorithm to interpolate the polynomial in $O(\log^3 (nt))$ boolean parallel time and $O(n^2 t^6 \log^2 nt)$ processors.This algorithm yields the first efficient deterministic polynomial time algorithm (and moreover boolean $NC$-algorithm) for interpolating t-sparse polynomials over finite fields and should be contrasted with the fact that efficient interpolation using a black box that only evaluates the polynomial at points in $GF[q]$ is not possible (cf. [M. Clausen, A. Dress, J. Grabmeier, and M. Karpinski, Theoret. Comput. Sci., 1990, to appear]). This algorithm, tog...