Nearly Optimal Sparse Polynomial Multiplication

Nakos, Vasileios

doi:10.1109/tit.2020.2989385

Cited by 21 publications

(13 citation statements)

References 25 publications

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“…Here the task is to compute the convolution of two sparse vectors much faster than performing FFT, ideally in near-linear time in terms of the input plus output size (i.e., the number of non-zero entries of the input and output vectors). Near-linear in the input plus output size running time was achieved for vectors with non-negative entries by Cole and Hariharan [CH02] and for general vectors in [Nak20], see also [GGdC20] for additional log m factors improvements. Very recently, a Monte Carlo O(k log k)-time algorithm has been achieved in [BFN21] for non-negative convolution, where k is the input plus output size.…”

Section: -Fold Casementioning

confidence: 99%

Fast $n$-fold Boolean Convolution via Additive Combinatorics

Bringmann,

Nakos

2021

Preprint

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We consider the problem of computing the Boolean convolution (with wraparound) of n vectors of dimension m, or, equivalently, the problem of computing the sumsetBoolean convolution formalizes the frequent task of combining two subproblems, where the whole problem has a solution of size k if for some i the first subproblem has a solution of size i and the second subproblem has a solution of size k − i. Our problem formalizes a natural generalization, namely combining solutions of n subproblems subject to a modular constraint. This simultaneously generalises Modular Subset Sum and Boolean Convolution (Sumset Computation). Although nearly optimal algorithms are known for special cases of this problem, not even tiny improvements are known for the general case.We almost resolve the computational complexity of this problem, shaving essentially a factor of n from the running time of previous algorithms. Specifically, we present a deterministic algorithm running in almost linear time with respect to the input plus output size k. We also present a Las Vegas algorithm running in nearly linear expected time with respect to the input plus output size k. Previously, no deterministic or randomized o(nk) algorithm was known.At the heart of our approach lies a careful usage of Kneser's theorem from Additive Combinatorics, and a new deterministic almost linear output-sensitive algorithm for non-negative sparse convolution. In total, our work builds a solid toolbox that could be of independent interest.

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Section: -Fold Casementioning

confidence: 99%

Fast $n$-fold Boolean Convolution via Additive Combinatorics

Bringmann,

Nakos

2021

Preprint

Self Cite

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“…Hsu and Shieh proposed a method with less addition and multiplication [19] in 2020. Besides, there are also some research studies on reducing the space complexity of polynomial multiplication [20,21]. However, the complexity of some research studies based on the acceleration of FFT remains at O(nlog(n)).…”

Section: Related Workmentioning

confidence: 99%

Blockchain-Based Secure Outsourcing of Polynomial Multiplication and Its Application in Fully Homomorphic Encryption

Song

Sang

Zeng

et al. 2021

Security and Communication Networks

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The efficiency of fully homomorphic encryption has always affected its practicality. With the dawn of Internet of things, the demand for computation and encryption on resource-constrained devices is increasing. Complex cryptographic computing is a major burden for those devices, while outsourcing can provide great convenience for them. In this paper, we firstly propose a generic blockchain-based framework for secure computation outsourcing and then propose an algorithm for secure outsourcing of polynomial multiplication into the blockchain. Our algorithm for polynomial multiplication can reduce the local computation cost to O n . Previous work based on Fast Fourier Transform can only achieve O n log n for the local cost. Finally, we integrate the two secure outsourcing schemes for polynomial multiplication and modular exponentiation into the fully homomorphic encryption using hidden ideal lattice and get an outsourcing scheme of fully homomorphic encryption. Through security analysis, our schemes achieve the goals of privacy protection against passive attackers and cheating detection against active attackers. Experiments also demonstrate our schemes are more efficient in comparisons with the corresponding nonoutsourcing schemes.

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“…Another difference with the dense case is that studying the complexity in a pure algebraic model remains meaningless, unless you assume a transdichotomous model on the degree, meaning that the integer computation on the exponents is always (1) [3,25]. The classical approach for computing the product of two polynomials of sparsity T is to generate all the T 2 possible monomials, and to sort them and merge those of equal degree to collect the monomials of the result.…”

Section: Dense Polynomial Multiplicationmentioning

confidence: 99%

Polynomial modular product verification and its implications

Giorgi,

Grenet,

Cray

2021

Preprint

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Polynomial multiplication is known to have quasi-linear complexity in both the dense and the sparse cases. Yet no truly linear algorithm has been given in any case for the problem, and it is not clear whether it is even possible. This leaves room for a better algorithm for the simpler problem of verifying a polynomial product. While finding deterministic methods seems out of reach, there exist probabilistic algorithms for the problem that are optimal in number of algebraic operations.We study the generalization of the problem to the verification of a polynomial product modulo a sparse divisor. We investigate its bit complexity for both dense and sparse multiplicands. In particular, we are able to show the primacy of the verification over modular multiplication when the divisor has a constant sparsity and a second highest-degree monomial that is not too large. We use these results to obtain new bounds on the bit complexity of the standard polynomial multiplication verification. In particular, we provide optimal algorithms in the bit complexity model in the dense case by improving a result of Kaminski and develop the first quasi-optimal algorithm for verifying sparse polynomial product.

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Nearly Optimal Sparse Polynomial Multiplication

Cited by 21 publications

References 25 publications

Fast $n$-fold Boolean Convolution via Additive Combinatorics

Fast $n$-fold Boolean Convolution via Additive Combinatorics

Blockchain-Based Secure Outsourcing of Polynomial Multiplication and Its Application in Fully Homomorphic Encryption

Polynomial modular product verification and its implications

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