A new deterministic algorithm for sparse multivariate polynomial interpolation

Abstract

We present a deterministic algorithm to interpolate an msparse n-variate polynomial which uses poly(n, m, log H, log d) bit operations. Our algorithm works over the integers. Here H is a bound on the magnitude of the coefficient values of the given polynomial. The degree of given polynomial is bounded by d and m is upper bound on number of monomials. This running time is polynomial in the output size. Our algorithm only requires modular black box access to the given polynomial, as introduced in [12]. As an easy consequence, we obtain an algorithm to interpolate polynomials represented by arithmetic circuits.

Revisit Sparse Polynomial Interpolation Based on Randomized Kronecker Substitution

“…Assume m = x e 1 1 x e 2 2 • • • x en n . Since cm is not a collision in f mod (p) (x s ), without loss of generality, assume cm(x s ) mod (x p − 1) = a 1 x d 1 , where a 1 x d 1 is defined in (6). It is easy to show that cm is also not a collision in f (x s ) and in f (x s+pI k ).…”

“…In [9], Ben-Or and Tiwari gave a deterministic algorithm over the field of complex numbers, which needs an upper bound of the number of terms in f . After these work, many interesting algorithms were given, such as the computational complexity enhancement [25,34], the interpolation with nonstandard bases [30], the interpolation over finite fields [29,18,16,20,24], the early termination algorithm [27,19], the hybrid interpolation algorithm [17,28,11,17], the interpolation for modular black-box polynomials [10], and the reduction based methods for black-box and SLP polynomials [3,4,7,13,16,22].…”

“…The second model is the modular black-box model [5,6], which works for the polynomials in Q[x 1 , . .…”

Deterministic Interpolation of Sparse Black-box Multivariate Polynomials using Kronecker Type Substitutions

“…Even for polynomials over the finite field F q , there exist no interpolation algorithms whose complexity is polynomial in log D and log q for the standard black-box model. For special models such as the SLP model [3,4,5,9,10,17], the precision accuracy black-box model [1,8,26,13], and the modular black-box model [11,12], there exist interpolation algorithms whose complexity is polynomial in log D.…”

Deterministic sparse interpolation of black-box multivariate polynomials using Kronecker type substitutions

“…从表 2 中可看出, 我们的第一个算法比文献 [13] 中的算法复杂度低. 文献 [18] 可以看出 [7-9, 12, 15-17] 、有限精度黑盒模型 [2,[19][20][21] 和模黑盒模型 [22,23] , 存在复杂度关于 log D 是多项式的 插值算法.…”