Faster interpolation algorithms for sparse multivariate polynomials given by straight-line programs

Huang, Qiao-Long; Gao, Xiao-Shan

doi:10.1016/j.jsc.2019.10.005

Cited by 9 publications

(15 citation statements)

References 17 publications

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“…Ln 2 T 2 log 2 D log q + LnT 2 log 3 D log q Deterministic Huang & Gao [12] LnT log 2 D(log q + log D) Monte Carlo This paper(Th.3.5)(char(Fq) ≥ D)…”

Section: Summary Of Resultsmentioning

confidence: 99%

“…The sparse interpolation for multivariate polynomials has received considerable interest. There are two basic models for this problem: the polynomial is either given as a straight-line program (SLP) [1,2,3,4,8,10,12,14,15] or a more general black-box [6,13,16,17,20]. As pointed out in [8], for black-box polynomials with degree d, there exist no algorithms yet, which works for arbitrary fields and have complexities polynomial in log d, while for SLP polynomials, it is possible to give algorithms with complexities polynomial in log d.…”

Section: Previous Workmentioning

confidence: 99%

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Sparse Polynomial Interpolation Based on Derivative

Huang¹

2020

Preprint

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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((nT D) (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. [7] for the polynomial given by an SLP over finite field (for large characteristic). * More explicitly, char(Fq) > 2npD, where p is the first N -th prime and N = 4 max{1, ⌈n(T − 1) log 2 D⌉}.

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“…Ln 2 T 2 log 2 D log q + LnT 2 log 3 D log q Deterministic Huang & Gao [12] LnT log 2 D(log q + log D) Monte Carlo This paper(Th.3.5)(char(Fq) ≥ D)…”

Section: Summary Of Resultsmentioning

confidence: 99%

Section: Previous Workmentioning

confidence: 99%

Sparse Polynomial Interpolation Based on Derivative

Huang¹

2020

Preprint

Self Cite

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“…Next lemma revamps the core of Huang's result [18]. Similar results are used in several interpolation algorithms [11,1,19]. Lemma 2.5.…”

Section: Case Of Large Characteristicmentioning

confidence: 99%

On exact division and divisibility testing for sparse polynomials

Giorgi,

Grenet,

Cray

2021

Preprint

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Assessing that a sparse polynomial G divides another sparse polynomial F is not yet known to admit a polynomial time algorithm. While computing the quotient Q = F quo G can be done in polynomial time with respect to the sparsities of F, G and Q, it is not yet sufficient to get a polynomial time divisibility test in general. Indeed, the sparsity of the quotient Q can be exponentially larger than the ones of F and G. In the favorable case where the sparsity #Q of the quotient is polynomial, the best known algorithm to compute Q has a non-linear factor #G#Q in the complexity, which is not optimal.In this work, we are interested in the two aspects of this problem. First, we propose a new randomized algorithm that computes the quotient of two sparse polynomials when the division is exact. Its complexity is quasi-linear in the sparsities of F , G and Q. Our approach relies on sparse interpolation and it works over any finite fields or the ring of integers. Then, as a step toward faster divisibility testing, we provide a new polynomial time algorithm when the divisor has some specific shapes. More precisely, we reduce the problem to finding a polynomial S such that QS is sparse and testing divisibility by S can be done in polynomial time. We identify some structure patterns in the divisor G for which we can efficiently compute such a polynomial S.

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“…Next lemma revamps the core of Huang's result [19]. Similar results are used in several interpolation algorithms [1,11,20].…”

Section: Case Of Large Characteristicmentioning

confidence: 99%

On Exact Division and Divisibility Testing for Sparse Polynomials

Giorgi

Grenet

Cray

2021

Proceedings of the 2021 International Symposium on Symbolic and Algebraic Computation

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No polynomial-time algorithm is known to test whether a sparse polynomial divides another sparse polynomial . While computing the quotient = quo can be done in polynomial time with respect to the sparsities of , and , this is not yet sufficient to get a polynomial-time divisibility test in general. Indeed, the sparsity of the quotient can be exponentially larger than the ones of and . In the favorable case where the sparsity # of the quotient is polynomial, the best known algorithm to compute has a non-linear factor # # in the complexity, which is not optimal.In this work, we are interested in the two aspects of this problem. First, we propose a new randomized algorithm that computes the quotient of two sparse polynomials when the division is exact. Its complexity is quasi-linear in the sparsities of , and . Our approach relies on sparse interpolation and it works over any finite field or the ring of integers. Then, as a step toward faster divisibility testing, we provide a new polynomial-time algorithm when the divisor has a specific shape. More precisely, we reduce the problem to finding a polynomial such that is sparse and testing divisibility by can be done in polynomial time. We identify some structure patterns in the divisor for which we can efficiently compute such a polynomial . CCS CONCEPTS• Computing methodologies → Algebraic algorithms; • Theory of computation → Design and analysis of algorithms; • Mathematics of computing → Probabilistic algorithms.

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Faster interpolation algorithms for sparse multivariate polynomials given by straight-line programs

Cited by 9 publications

References 17 publications

Sparse Polynomial Interpolation Based on Derivative

Sparse Polynomial Interpolation Based on Derivative

On exact division and divisibility testing for sparse polynomials

On Exact Division and Divisibility Testing for Sparse Polynomials

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