Fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields
Reads0
Chats0
TLDR
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 polynometric at points in $GF[q]$ is not possible.Abstract:
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...read more
Citations
More filters
Book
Limits to Parallel Computation: P-Completeness Theory
TL;DR: In providing an up-to-date survey of parallel computing research from 1994, Topics in Parallel Computing will prove invaluable to researchers and professionals with an interest in the super computers of the future.
Book
Arithmetic Circuits: A Survey of Recent Results and Open Questions
Amir Shpilka,Amir Yehudayoff +1 more
TL;DR: The goal of this monograph is to survey the field of arithmetic circuit complexity, focusing mainly on what it finds to be the most interesting and accessible research directions, with an emphasis on works from the last two decades.
Journal ArticleDOI
Derandomizing polynomial identity tests means proving circuit lower bounds
TL;DR: If Permanent requires superpolynomial-size arithmetic circuits, then one can test in subexponential time whether a given arithmetic circuit of polynomially bounded degree computes an identically zero polynomial.
Journal Article
Derandomizing Polynomial Identity Tests Means Proving Circuit Lower Bounds
TL;DR: In this paper, it was shown that derandomizing Polynomial Identity Testing is equivalent to proving arithmetic circuit lower bounds for NEXP, and that if one can test in polynomial time (or even non-deterministic subexponential time, infinitely often) whether a given arithmetic circuit over integers computes an identically zero poynomial, then either NEXP ⊄ P/poly or Permanent is not computable by polynomially-size arithmetic circuits.
Journal ArticleDOI
Learning read-once formulas with queries
TL;DR: The main results are a polynomial-time algorithm for exact identification of monotone read-once formulas using only membership queries, and a protocol based on the notion of a minimally adequate teacher using equivalence and membership queries.
References
More filters
Journal ArticleDOI
Fast Probabilistic Algorithms for Verification of Polynomial Identities
TL;DR: Vanous fast probabdlsttc algonthms, with probability of correctness guaranteed a prion, are presented for testing polynomial ldentmes and propemes of systems of polynomials and ancdlary fast algorithms for calculating resultants and Sturm sequences are given.
Book ChapterDOI
Probabilistic algorithms for sparse polynomials
TL;DR: This work has tried to demonstrate how sparse techniques can be used to increase the effectiveness of the modular algorithms of Brown and Collins and believes this work has finally laid to rest the bad zero problem.
Book
The Complexity of Boolean Functions
TL;DR: This chapter discusses Circuits and other Non-Uniform Computation Methods vs. Turing Machines and other Uniform Computation Models, and the Design of Efficient Circuits for Some Fundamental Functions.
Journal ArticleDOI
A taxonomy of problems with fast parallel algorithms
TL;DR: An attempt is made to identify important subclasses of NC and give interesting examples in each subclass, and a new problem complete for deterministic polynomial time is given, namely, finding the lexicographically first maximal clique in a graph.
Journal ArticleDOI
Factoring polynomials over large finite fields
TL;DR: In this paper, the authors present a deterministic procedure for factoring polynomials over finite fields, which reduces the problem of factoring an arbitrary polynomial over the Galois field GF(p m) to finding the roots in GF(m) of certain other polynomorphisms over GF (m).