Essentially Optimal Sparse Polynomial Multiplication

Exact Sparse Matrix-Vector Multiplication on GPU's and Multicore Architectures

We propose different implementations of the sparse matrix--dense vector multiplication (\spmv{}) for finite fields and rings $\Zb/m\Zb$. We take advantage of graphic card processors (GPU) and multi-core architectures. Our aim is to improve the speed of \spmv{} in the \linbox library, and henceforth the speed of its black box algorithms. Besides, we use this and a new parallelization of the sigma-basis algorithm in a parallel block Wiedemann rank implementation over finite fields.

Highly Scalable Multiplication for Distributed Sparse Multivariate Polynomials on Many-core Systems

We present a highly scalable algorithm for multiplying sparse multivariate polynomials represented in a distributed format. This algo- rithm targets not only the shared memory multicore computers, but also computers clusters or specialized hardware attached to a host computer, such as graphics processing units or many-core coprocessors. The scal- ability on the large number of cores is ensured by the lacks of synchro- nizations, locks and false-sharing during the main parallel step.

Error correction in fast matrix multiplication and inverse

We present new algorithms to detect and correct errors in the product of two matrices, or the inverse of a matrix, over an arbitrary field. Our algorithms do not require any additional information or encoding other than the original inputs and the erroneous output. Their running time is softly linear in the number of nonzero entries in these matrices when the number of errors is sufficiently small, and they also incorporate fast matrix multiplication so that the cost scales well when the number of errors is large. These algorithms build on the recent result of Gasieniec et al (2017) on correct…

用于秩23的$3\ imes3$矩阵乘法的51次加法交替基核

A 51-Addition Alternative-Basis Kernel for Rank-23 $3\times3$ Matrix Multiplication

本文提出一种秩23的3×3矩阵乘法算法，在交替基下用51次加法，加上转换共56次，结合线性规划与稀疏基搜索，并给出机器可验证的证书。

Chunky and Equal-Spaced Polynomial Multiplication

Finding the product of two polynomials is an essential and basic problem in computer algebra. While most previous results have focused on the worst-case complexity, we instead employ the technique of adaptive analysis to give an improvement in many "easy" cases. We present two adaptive measures and methods for polynomial multiplication, and also show how to effectively combine them to gain both advantages. One useful feature of these algorithms is that they essentially provide a gradient between existing "sparse" and "dense" methods. We prove that these approaches provide significant improveme…

Prover efficient public verification of dense or sparse/structured matrix-vector multiplication

With the emergence of cloud computing services, computationally weak devices (Clients) can delegate expensive tasks to more powerful entities (Servers). This raises the question of verifying a result at a lower cost than that of recomputing it. This verification can be private, between the Client and the Server, or public, when the result can be verified by any third party. We here present protocols for the verification of matrix-vector multiplications, that are secure against malicious Servers. The obtained algorithms are essentially optimal in the amortized model: the overhead for the Server…

Asymptotically fast polynomial matrix algorithms for multivariable systems

We present the asymptotically fastest known algorithms for some basic problems on univariate polynomial matrices: rank, nullspace, determinant, generic inverse, reduced form. We show that they essentially can be reduced to two computer algebra techniques, minimal basis computations and matrix fraction expansion/reconstruction, and to polynomial matrix multiplication. Such reductions eventually imply that all these problems can be solved in about the same amount of time as polynomial matrix multiplication.

Polynomial modular product verification and its implications

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…

比矩阵乘法更快地计算稀疏矩阵模\(p^2\)的史密斯标准型

Computing Smith Forms Modulo $p^2$ of Sparse Matrices Faster Than Matrix Multiplication

研究在\(R = \mathbb{Z}/p^2\mathbb{Z}\)环中，当矩阵\(A\)稀疏或有结构时，快速确定其史密斯标准型相关参数\(r_0,r_1,r_2\)的问题。提出随机算法，利用迭代块 - 维德曼技术和结构化预条件器，运算次数优于现有稠密算法。