Essentially optimal sparse polynomial multiplication

Giorgi, Pascal; Grenet, Bruno; Cray, Armelle Perret du

doi:10.1145/3373207.3404026

Cited by 13 publications

(32 citation statements)

References 21 publications

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“…For multiplication, though many algorithms have been proposed [5,8,15,16,21,24,26,29], none of them was quasi-optimal in the general case. Only recently, we proposed a quasi-optimal algorithm for the multiplication of sparse polynomials over finite fields of large characteristic or over the integers [12]. We note that similar results have been given in a more recent preprint, assuming some heuristics [14].…”

Section: Introductionmentioning

confidence: 64%

“…Both interpolation algorithms presented in the previous sections require a bound on the sparsity of the quotient. To overcome this difficulty, we use the same strategy as for sparse polynomial multiplication [12]. The idea is to guess the sparsity bound as we go using a fast verification procedure.…”

Section: Output Sensitive Algorithmmentioning

confidence: 99%

“…The idea is to guess the sparsity bound as we go using a fast verification procedure. For verifying exact quotient, we can directly reuse Algorithm VerifySP from [12] that verifies if = , with an error probability at most if ≠ . It requiresÕ ( (log + log )) bit operations over F and O ( (log + log )) bit operations over Z where is a bound on the heights of , and .…”

Section: Output Sensitive Algorithmmentioning

confidence: 99%

“…Depending on the characteristic, using Theorems 2.7 or 2.10 and the complexity of [12,Algorithm VerifySP], we obtain the claimed complexity with probability at least 1 − 2 . □…”

Section: Algorithm 3 Sparseexactdivisionmentioning

confidence: 99%

“…For instance, the product of two polynomials and has at most # # nonzero coefficients. But it may have as few as 2 nonzero coefficients [12]. The size growth can be even more dramatic for sparse polynomial division.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 64%

Section: Output Sensitive Algorithmmentioning

confidence: 99%

Section: Output Sensitive Algorithmmentioning

confidence: 99%

“…Depending on the characteristic, using Theorems 2.7 or 2.10 and the complexity of [12,Algorithm VerifySP], we obtain the claimed complexity with probability at least 1 − 2 . □…”

Section: Algorithm 3 Sparseexactdivisionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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