arXiv subjects

Qiao-Long Huang

Publications and source records attributed to Qiao-Long Huang.

15 recordsLinked to original sources

arXiv2026

A New Sparse Algorithm for Polynomial GCD over Integers

We describe a new greatest common divisor (GCD) algorithm for polynomials with integer coefficients. The bit complexity of the new algorithm is polynomial in the input and output sizes and the individual degree bounds.Our algorithm follows the standard approach by reducing multivariate polynomial GCD to univariate polynomial GCD. Our algorithm reduces a multivariate polynomial GCD to a single univariate polynomial GCD. The main idea of our algorithm is a new variable substitution which reduces a multivariate polynomial to a separated one, that is, the coefficients in a main variable are all monomials. The explicit bit complexity is analyzed and we have implemented our algorithm in Maple. It is shown that our algorithm is efficient for polynomials with high degree, large number of variables, but small number of terms in GCD.

math.NT↗

arXiv2026

Sparse Polynomial GCD Algorithms Asymptotically Linear in All Fundamental Parameters

Let $A, B \in \mathbb{Z}[x_1, \dots, x_n]$ be multivariate polynomials with integer coefficients and let $G = \gcd(A, B)$. We present an algorithm for computing $G$ whose expected bit complexity is asymptotically linear in all fundamental parameters: the number of variables $n$, the term count $T = \max\{\|A\|_0, \|B\|_0, \|G\|_0\}$, the total degree $D$, and the logarithmic coefficient sizes $\log\Hi$ and $\log\Ho$, where $\Hi$ bounds the coefficients of the inputs and $\Ho$ bounds those of the GCD. The bit complexity is characterized by the clean bound \[ \widetilde{O}\bigl( n \cdot T \cdot D \cdot \log\Hi \cdot \log\Ho \bigr). \] To our knowledge, this is the first sparse GCD algorithm over the integers that achieves linear complexity in all these parameters simultaneously. The integer algorithm is built upon a new field GCD algorithm. For $A, B \in \K[x_1, \dots, x_n]$ over a field $\K$ with $\operatorname{char}(\K) = 0$ or $\operatorname{char}(\K) > °G$, we give the first algorithm that computes $G = \gcd(A,B)$ with expected \[ \widetilde{O}\bigl( n \cdot T \cdot D \bigr) \] field operations, which is both input- and output-sensitive. The key technical contribution behind both algorithms is a derivative-aided separated Hensel lifting technique introduced in this paper. By introducing an auxiliary variable and leveraging derivative information, our scheme extracts all partial exponents via a single $z^2$-lift per variable, achieving constant sequential depth $O(1)$. This stands in sharp contrast to classical Hensel lifting, which requires $O(D)$ sequential lifting steps and suffers from representation densification in the sparse setting. The field algorithm is then extended to the integer case through modular reduction and rational reconstruction.

math.AC↗

arXiv2026

Deterministic Polynomial-time Exact-root Computation for Sparse Polynomials with Bounded Total Degree

We study the problem of deterministically computing the exact root of a sparse polynomial in the multivariate setting. Let $f \in \F[x_1,\ldots,x_n]$ be a nonzero polynomial that is an exact $e$-th power, say $f = g^e$. Suppose $f$ is $s$-sparse, has an individual degree of at most $d$, and a total degree of $D = \tdeg(f)$. We prove a sparsity bound on the base polynomial $g$: \[ \|g\|_0 \le s^{D(2d+2)/e + 1}. \] Based on this bound, we develop a deterministic algorithm that computes the base $g$. % In contrast to the general deterministic factorization algorithm of Bhargava, Saraf, and Volkovich \cite{BhargavaSarafVolkovich2020}, which achieves only a quasi-polynomial dependence on the input parameters, our algorithm is \emph{polynomial-time} in the setting where the total degree $D$ is bounded. Specifically, the overall complexity is \[ \mathrm{poly}\left(s^{O(Dd)}, n, d, D\right) + s\cdot R(e), \] % where $R(e)$ denotes the cost of constructing a single $e$-th root of a scalar in the base field $\F$, and, when $\operatorname{char}(\F)\mid e$, the cost of computing a single Frobenius root of a scalar. % This term is field-dependent, and over finite fields, $\mathbb{Q}$, or number fields with a suitable representation, it is absorbed into the polynomial complexity bound. % Within the bounded total-degree regime, this yields a deterministic polynomial-time algorithm for exact-root computation.

cs.DS↗

arXiv2026

Quasi-linear Time Multiplication of Sparse Polynomials with Integer Coefficients

Sparse polynomial multiplication is a fundamental problem in computer algebra and the theory of computation, and the development of a quasi-linear time output-sensitive multiplication algorithm has been posed as an open challenge. In this paper, a counterexample is provided to a previously claimed solution to this open problem for integer coefficients. By employing the existing quasi-linear modular-black-box interpolation algorithm, we are able to provide an algorithm with quasi-linear bit complexity for the integer coefficients setting. Furthermore, in the case of coefficients over a finite field, we obtain an algorithm whose bit complexity is linear in the number of terms, the logarithm of the degree, and the logarithm of the size of the finite field.

cs.SC↗

arXiv2026

Sparse Polynomial Divisibility Test over Finite Field is CoNP-hard

In this paper, we show that deciding whether a sparse polynomial does not divide another sparse polynomial exactly over finite fields is NP-hard under BPP many-one reductions. Equivalently, the sparse polynomial divisibility test over finite fields is CoNP-hard. This resolves the long-standing open problem concerning the computational complexity of the divisibility test for sparse polynomials in the setting of finite fields.

cs.SC↗

arXiv2026

Output-sensitive Sparse Polynomial GCD over Finite Fields is NP-hard

In this paper, we prove that output-sensitive sparse polynomial GCD computation over finite fields is NP-hard under BPP many-one reduction. More precisely, for two sparse univariate polynomials $f,g$ with finite field coefficients, there exists no randomized algorithm to compute $\mathrm{gcd}(f,g)$, which is polynomial-time in the sizes of $f,g,\gcd(f,g)$ under the standard complexity assumption $\mathrm{NP}\nsubseteq\mathrm{BPP}$. This settles the open problem posed as Challenge 5 in The Sparsity Challenges in the finite field setting. Furthermore, we show that the Roots of Unity Detection problem over finite fields is NP-hard; that is, determining whether the GCD of a sparse univariate polynomial and $x^n - 1$ has nonzero degree is NP-hard.

cs.SC↗

arXiv2022

Bit Complexity of Polynomial GCD on Sparse Representation

An input- and output-sensitive GCD algorithm for multi-variate polynomials over finite fields is proposed by combining the modular method with the Ben-Or/Tiwari sparse interpolation. The bit complexity of the algorithm is given and is sensitive to the sparse representation, while for previous sparse GCD algorithms, the complexities were given only in some special cases. It is shown that the new algorithm is superior both in theory and in practice comparing with existing GCD algorithms: the complexity in the degree is decreased from quadratic to linear and the running times are decreased by 1-3 orders of magnitude in various benchmarks.

cs.SC↗

arXiv2022

Skew-sparse matrix multiplication

Based on the observation that $\mathbb{Q}^{(p-1) \times (p-1)}$ is isomorphic to a quotient skew polynomial ring, we propose a new method for $(p-1)\times (p-1)$ matrix multiplication over $\mathbb{Q}$, where $p$ is a prime number. The main feature of our method is the acceleration for matrix multiplication if the product is skew-sparse. Based on the new method, we design a deterministic algorithm with complexity $O(T^{ω-2} p^2)$, where $T\le p-1$ is a parameter determined by the skew-sparsity of input matrices and $ω$ is the asymptotic exponent of matrix multiplication. Moreover, by introducing randomness, we also propose a probabilistic algorithm with complexity $O^\thicksim(t^{ω-2}p^2+p^2\log\frac{1}ν)$, where $t\le p-1$ is the skew-sparsity of the product and $ν$ is the probability parameter.

cs.CC↗

arXiv2020

Sparse Polynomial Interpolation Based on Diversification

We consider the problem of interpolating a sparse multivariate polynomial over a finite field, represented with a black box. Building on the algorithm of Ben-Or and Tiwari for interpolating polynomials over rings with characteristic zero, we develop a new Monte Carlo algorithm over the finite field by doing additional probes. To interpolate a polynomial $f\in F_q[x_1,\dots,x_n]$ with a partial degree bound $D$ and a term bound $T$, our new algorithm costs $O^\thicksim(nT\log ^2q+nT\sqrt{D}\log q)$ bit operations and uses $2(n+1)T$ probes to the black box. If $q\geq O(nT^2D)$, it has constant success rate to return the correct polynomial. Compared with previous algorithms over general finite field, our algorithm has better complexity in the parameters $n,T,D$ and is the first one to achieve the complexity of fractional power about $D$, while keeping linear in $n,T$. A key technique is a randomization which makes all coefficients of the unknown polynomial distinguishable, producing a diverse polynomial. This approach, called diversification, was proposed by Giesbrecht and Roche in 2011. Our algorithm interpolates each variable independently using $O(T)$ probes, and then uses the diversification to correlate terms in different images. At last, we get the exponents by solving the discrete logarithms and obtain coefficients by solving a linear system. We have implemented our algorithm in Maple. Experimental results shows that our algorithm can applied to sparse polynomials with large degree. We also analyze the success rate of the algorithm.

cs.SC↗

arXiv2020

Sparse Polynomial Interpolation Based on Derivative

In this paper, we propose two new interpolation algorithms for sparse multivariate polynomials represented by a straight-line program(SLP). Both of our algorithms work over any finite fields $F_q$ with large characteristic. The first one is a Monte Carlo randomized algorithm. Its arithmetic complexity is linear in the number $T$ of non-zero terms of $f$, in the number $n$ of variables. If $q$ is $O((nTD)^{(1)})$, where $D$ is the partial degree bound, then our algorithm has better complexity than other existing algorithms. The second one is a deterministic algorithm. It has better complexity than existing deterministic algorithms over a field with large characteristic. Its arithmetic complexity is quadratic in $n,T,\log D$, i.e., quadratic in the size of the sparse representation. And we also show that the complexity of our deterministic algorithm is the same as the one of deterministic zero-testing of Bläser et al. for the polynomial given by an SLP over finite field (for large characteristic).

cs.SC↗

arXiv2018

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

In this paper, we propose two new deterministic interpolation algorithms for a sparse multivariate polynomial given as a standard black-box by introducing new Kronecker type substitutions. Let $f\in \RB[x_1,\dots,x_n]$ be a sparse black-box polynomial with a degree bound $D$. When $\RB=\C$ or a finite field, our algorithms either have better bit complexity or better bit complexity in $D$ than existing deterministic algorithms. In particular, in the case of deterministic algorithms for standard black-box models, our second algorithm has the current best complexity in $D$ which is the dominant factor in the complexity.

cs.SC↗

arXiv2018

Faster Interpolation Algorithms for Sparse Multivariate Polynomials Given by Straight-Line Programs\

In this paper, we propose new deterministic and Monte Carlo interpolation algorithms for sparse multivariate polynomials represented by straight-line programs. Let $f$ be an $n$-variate polynomial given by a straight-line program, which has a degree bound $D$ and a term bound $T$. Our deterministic algorithm is quadratic in $n,T$ and cubic in $\log D$ in the Soft-Oh sense, which has better complexities than existing deterministic interpolation algorithms in most cases. Our Monte Carlo interpolation algorithms have better complexities than existing Monte Carlo interpolation algorithms and are the first algorithms whose complexities are linear in $nT$ in the Soft-Oh sense. Since $nT$ is a factor of the size of $f$, our Monte Carlo algorithms are optimal in $n$ and $T$ in the Soft-Oh sense.

cs.SC↗

arXiv2018

Revisit Sparse Polynomial Interpolation based on Randomized Kronecker Substitution

In this paper, a new reduction based interpolation algorithm for black-box multivariate polynomials over finite fields is given. The method is based on two main ingredients. A new Monte Carlo method is given to reduce black-box multivariate polynomial interpolation to black-box univariate polynomial interpolation over any ring. The reduction algorithm leads to multivariate interpolation algorithms with better or the same complexities most cases when combining with various univariate interpolation algorithms. We also propose a modified univariate Ben-or and Tiwarri algorithm over the finite field, which has better total complexity than the Lagrange interpolation algorithm. Combining our reduction method and the modified univariate Ben-or and Tiwarri algorithm, we give a Monte Carlo multivariate interpolation algorithm, which has better total complexity in most cases for sparse interpolation of black-box polynomial over finite fields.

cs.SC↗

arXiv2017

Sparse Polynomial Interpolation with Finitely Many Values for the Coefficients

In this paper, we give new sparse interpolation algorithms for black box polynomial f whose coefficients are from a finite set. In the univariate case, we recover f from one evaluation of f(a) for a sufficiently large number a. In the multivariate case, we introduce the modified Kronecker substitution to reduce the interpolation of a multivariate polynomial to the univariate case. Both algorithms have polynomial bit-size complexity.

cs.SC↗

arXiv2017

Sparse Rational Function Interpolation with Finitely Many Values for the Coefficients

In this paper, we give new sparse interpolation algorithms for black box univariate and multivariate rational functions h=f/g whose coefficients are integers with an upper bound. The main idea is as follows: choose a proper integer beta and let h(beta) = a/b with gcd(a,b)=1. Then f and g can be computed by solving the polynomial interpolation problems f(beta)=ka and g(beta)=ka for some integer k. It is shown that the univariate interpolation algorithm is almost optimal and multivariate interpolation algorithm has low complexity in T but the data size is exponential in n.

cs.SC↗