Nearly Optimal Sparse Polynomial Multiplication

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.

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 introduci…

整数系数稀疏多项式的拟线性时间乘法

Quasi-linear Time Multiplication of Sparse Polynomials with Integer Coefficients

针对整数系数稀疏多项式乘法，通过模块化黑盒插值算法实现拟线性位复杂度，并反驳了此前声称的解决方案。

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…

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…

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。

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。

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…