Sparse interpolation over finite fields via low-order roots of unity

  title={Sparse interpolation over finite fields via low-order roots of unity},
  author={Andrew Arnold and Mark Giesbrecht and Daniel S. Roche},
We present a new Monte Carlo algorithm for the interpolation of a straight-line program as a sparse polynomial f over an arbitrary finite field of size q. We assume a priori bounds D and T are given on the degree and number of terms of f. The approach presented in this paper is a hybrid of the diversified and recursive interpolation algorithms, the two previous fastest known probabilistic methods for this problem. By making effective use of the information contained in the coefficients… 

Tables from this paper

On Exact Division and Divisibility Testing for Sparse Polynomials
This work proposes a new randomized algorithm that computes the quotient of two sparse polynomials when the division is exact and identifies some structure patterns in the divisor G for which it can efficiently compute such a polynomial S.
Asymptotically Optimal Monte Carlo Sparse Multivariate Polynomial Interpolation Algorithms of Straight-Line Program
Based on the Monte Carlo interpolation algorithm, this paper gives an asymptotically optimal algorithm for the multiplication of several multivariate polynomials, whose complexity is softly linear in the input size plus the output size, if the logarithm factors are ignored.
Output-Sensitive Algorithms for Sumset and Sparse Polynomial Multiplication
We present randomized algorithms to compute the sumset (Minkowski sum) of two integer sets, and to multiply two univariate integer polynomials given by sparse representations. Our algorithm for
Sparse Polynomial Interpolation Based on Derivative
Two new interpolation algorithms for sparse multivariate polynomials represented by a straight-line program(SLP) have been proposed and one has better complexity than existing deterministic algorithms over a field with large characteristic.
Sparse Polynomial Interpolation over Fields with Large or Zero Characteristic
A new interpolation algorithm for a sparse multivariate polynomial represented by a straight-line program (SLP) that is a Monte Carlo randomized algorithm and works over fields with large or zero characteristic.
Sparse Polynomial Interpolation and Testing
Two methods for the interpolation of a sparse polynomial modelled by a straight-line program (SLP): a sequence of arithmetic instructions and an alternative method of randomized Kronecker substitutions that can more efficiently reconstruct a sparse interpolant f from multiple univariate images of considerably reduced degree.
Sparse polynomial interpolation and division in soft-linear time
This work presents a new Monte Carlo randomized algorithm to recover a polynomial 𝑓 (𝑥) with integer coefficients given a way to evaluate 𝐓(𝜃) mod 𝓃 for any chosen integers 𝜁 and 𝚚 .
Sparse Interpolation in Terms of Multivariate Chebyshev Polynomials
  • E. Hubert, M. Singer
  • Mathematics, Computer Science
    Foundations of Computational Mathematics
  • 2021
A deterministic algorithm is presented to recover a function that is the linear combination of at most r such polynomials from the knowledge of r and an explicitly bounded number of evaluations of this function.
Probably faster multiplication ofsparse polynomials
In this framework, the unknown polynomial R is given through a blackbox functions that can be evaluated at points in suitable extensions of the coefficient ring and the idea to “exploit colliding terms” in section 6.6 forms the starting point of the work.
Sparse Polynomial Interpolation Based on Diversification
A new Monte Carlo algorithm is developed for interpolating a sparse multivariate polynomial over a finite field by doing additional probes and is the first one to achieve the complexity of fractional power about $D$ while keeping linear in $n,T$.


Approximate formulas for some functions of prime numbers
Faster Sparse Interpolation of Straight-Line Programs
A new probabilistic algorithm for interpolating a "sparse" polynomial f given by a straight-line program is given, which is asymptotically more efficient in terms of the total cost of the probes required than previous methods.
Diversification improves interpolation
This work improves on the best-known algorithm for interpolation over large finite fields by presenting a Las Vegas randomized algorithm that uses fewer black box evaluations and provides the first provably stable algorithm for this problem, at the cost of modestly more evaluations.
Explaining the wheel sieve
A simple mathematical framework is developed, which leads to a smoother and more insightful derivation of the new algorithm, and which may be of independent interest to the number theorist.
Interpolation of polynomials given by straight-line programs
Parallel sparse polynomial interpolation over finite fields
We present a probabilistic algorithm to interpolate a sparse multivariate polynomial over a finite field, represented with a black box. Our algorithm modifies the algorithm of Ben-Or and Tiwari from
Early termination in sparse interpolation algorithms
Multivariate sparse interpolation using randomized Kronecker substitutions
A new algorithm for multivariate interpolation is given which uses these new techniques for reducing a multivariate sparse polynomial to a univariatePolynomial along with any existing univariate interpolations algorithm.
Probability Inequalities for sums of Bounded Random Variables
Abstract Upper bounds are derived for the probability that the sum S of n independent random variables exceeds its mean ES by a positive number nt. It is assumed that the range of each summand of S
Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity
A portable container carrying rack that is adaptable for carrying several various size containers concurrently on any flat bed delivery vehicle and is particularly suitable for transporting floral objects, such as potted plants, cut flowers, and various other floral arrangements that are held in a container.