Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation最新文献_第2页

The GKR Protocol Revisited: Nearly Optimal Prover-Complexity for Polynomial-Time Wiring Algorithms and for Primality Testing in n1/2+o(1) Rounds 重新审视GKR协议:多项式时间连线算法和n /2+o(1)轮素数检验的近最优证明复杂度

Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation Pub Date : 2022-07-04 DOI: 10.1145/3476446.3536183

E. Kaltofen

{"title":"The GKR Protocol Revisited: Nearly Optimal Prover-Complexity for Polynomial-Time Wiring Algorithms and for Primality Testing in n1/2+o(1) Rounds","authors":"E. Kaltofen","doi":"10.1145/3476446.3536183","DOIUrl":"https://doi.org/10.1145/3476446.3536183","url":null,"abstract":"The proof-of-work interactive protocol by Shafi Goldwasser, Yael T. Kalai and Guy N. Rothblum (GKR) [STOC 2008, JACM 2015] certifies the execution of an algorithm via the evaluation of a corresponding boolean or arithmetic circuit whose structure is known to the verifier by circuit wiring algorithms that define the uniformity of the circuit. Here we study protocols whose prover time- and space-complexities are within a poly-logarithmic factor of the time- and space-complexity of the algorithm; we call those protocols 'prover-nearly-optimal.' We show that the uniformity assumptions can be relaxed from LOGSPACE to polynomial-time in the bit-lengths of the labels which enumerate the nodes in the circuit. Our protocol applies GKR recursively to the arising sumcheck problems on each level of the circuit whose values are verified, and deploys any of the prover-nearly-optimal versions of GKR on the constructed sorting/prefix circuits with log-depth wiring functions. The verifier time-complexity of GKR grows linearly in the depth of the circuit. For deep circuits such as the Miller-Rabin integer primality test of an n-bit integer, the large number of rounds may interfere with soundness guarantees after the application of the Fiat-Shamir heuristic. We re-arrange the circuit evaluation problem by the baby-steps/giant-steps method to achieve a depth of n1/2+o(1), at prover cost n2+o(1) bit complexity and communication and verifier cost n3/2+o(1).","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130822686","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Simple C2-finite Sequences: a Computable Generalization of C-finite Sequences 简单c2 -有限序列:c -有限序列的可计算推广

Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation Pub Date : 2022-07-04 DOI: 10.1145/3476446.3536174

P. Nuspl, V. Pillwein

引用次数: 2

Bounding the Number of Roots of Multi-Homogeneous Systems 多齐次系统的根数边界

Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation Pub Date : 2022-07-04 DOI: 10.1145/3476446.3536189

Evangelos Bartzos, I. Emiris, I. Kotsireas, C. Tzamos

{"title":"Bounding the Number of Roots of Multi-Homogeneous Systems","authors":"Evangelos Bartzos, I. Emiris, I. Kotsireas, C. Tzamos","doi":"10.1145/3476446.3536189","DOIUrl":"https://doi.org/10.1145/3476446.3536189","url":null,"abstract":"Determining the number of solutions of a multi-homogeneous polynomial system is a fundamental problem in algebraic geometry. The multi-homogeneous Bézout (m-Bézout) number bounds from above the number of non-singular solutions of a multi-homogeneous system, but its computation is a #P>-hard problem. Recent work related the m-Bézout number of certain multi-homogeneous systems derived from rigidity theory with graph orientations, cf Bartzos et al. (2020). A first generalization applied graph orientations for bounding the root count of a multi-homogeneous system that can be modeled by simple undirected graphs, as shown by three of the authors (Bartzos et al., 2021). Here, we prove that every multi-homogeneous system can be modeled by hypergraphs and the computation of its m-Bézout bound is related to constrained hypergraph orientations. Thus, we convert the algebraic problem of bounding the number of roots of a polynomial system to a purely combinatorial problem of analyzing the structure of a hypergraph. We also provide a formulation of the orientation problem as a constraint satisfaction problem (CSP), hence leading to an algorithm that computes the multi-homogeneous bound by finding constrained hypergraph orientations.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125508951","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Computing a Basis for an Integer Lattice: A Special Case 计算整数格的基:一种特殊情况

Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation Pub Date : 2022-07-04 DOI: 10.1145/3476446.3536184

Haomin Li, A. Storjohann

引用次数: 0

Modular Techniques for Intermediate Primary Decomposition 中间初等分解的模块化技术

Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation Pub Date : 2022-07-04 DOI: 10.1145/3476446.3535488

Yuki Ishihara

引用次数: 1

Non-commutative Optimization - Where Algebra, Analysis and Computational Complexity Meet 非交换优化-代数，分析和计算复杂性相遇

Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation Pub Date : 2022-07-04 DOI: 10.1145/3476446.3535489

A. Wigderson

引用次数: 0

A Property of Modules Over a Polynomial Ring With an Application in Multivariate Polynomial Matrix Factorizations 多项式环上模的一个性质及其在多元多项式矩阵分解中的应用

Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation Pub Date : 2022-07-04 DOI: 10.1145/3476446.3535470

Dong Lu, Dingkang Wang, Fanghui Xiao, Xiaopeng Zheng

{"title":"A Property of Modules Over a Polynomial Ring With an Application in Multivariate Polynomial Matrix Factorizations","authors":"Dong Lu, Dingkang Wang, Fanghui Xiao, Xiaopeng Zheng","doi":"10.1145/3476446.3535470","DOIUrl":"https://doi.org/10.1145/3476446.3535470","url":null,"abstract":"This paper is concerned with a property of modules over a polynomial ring and its application in multivariate polynomial matrix factorizations. We construct a specific polynomial such that the product of the polynomial and a nonzero vector in a module over a polynomial ring can be represented by the elements in a maximum linearly independent vector set of the module over the polynomial ring. Based on this property, a relationship between a rank-deficient matrix and any of its full row rank submatrices is presented. By this result, we show that the problem for general factorizations of rank-deficient matrices can be translated into that of any of their full row rank submatrices in the regular case. Then many results on factorizations of full row rank matrices, such as zero prime factorizations, minor prime factorizations and factor prime factorizations, can be extended to the rank-deficient case. We implement the algorithm of general factorizations for rank-deficient matrices on the computer algebra system Maple, and two examples are given to illustrate the algorithm.","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133462605","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Finer Complexity Estimates for the Change of Ordering of Gröbner Bases for Generic Symmetric Determinantal Ideals 广义对称行列式理想Gröbner基序变化的精细复杂性估计

Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation Pub Date : 2022-07-04 DOI: 10.1145/3476446.3536182

A. Ferguson, H. P. Le

{"title":"Finer Complexity Estimates for the Change of Ordering of Gröbner Bases for Generic Symmetric Determinantal Ideals","authors":"A. Ferguson, H. P. Le","doi":"10.1145/3476446.3536182","DOIUrl":"https://doi.org/10.1145/3476446.3536182","url":null,"abstract":"Polynomial matrices and ideals generated by their minors appear in various domains such as cryptography, polynomial optimization and effective algebraic geometry. When the given matrix is symmetric, this additional structure on top of the determinantal structure, affects computations on the derived ideals. Thus, understanding the complexity of these computations is important. Moreover, this study serves as a stepping stone towards further understanding the effects of structure in determinantal systems, such as those coming from moment matrices. In this paper, we focus on the Sparse-FGLM algorithm, the state-of-the-art for changing ordering of Gröbner bases of zero-dimensional ideals. Under a variant of Fröberg's conjecture, we study its complexity for symmetric determinantal ideals and identify the gain of exploiting sparsity in the Sparse-FGLM algorithm compared with the classical FGLM algorithm. For an n×n symmetric matrix with polynomial entries of degree d, we show that the complexity of Sparse-FGLM for zero-dimensional determinantal ideals obtained from this matrix over that of the FGLM algorithm is at least O(1/d). Moreover, for some specific sizes of minors, we prove finer results of at least O(1/nd) and O(1/m3d).","PeriodicalId":130499,"journal":{"name":"Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation","volume":" 36","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114060703","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 1

Rado Numbers and SAT Computations 无线电数字和SAT计算

Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation Pub Date : 2022-07-04 DOI: 10.1145/3476446.3535494

Yuan Chang, J. D. Loera, W. J. Wesley

引用次数: 4

Rational Univariate Representation of Zero-Dimensional Ideals with Parameters 带参数的零维理想的有理单变量表示

Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation Pub Date : 2022-07-04 DOI: 10.1145/3476446.3535496

Dingkang Wang, Jingjing Wei, Fanghui Xiao, Xiaopeng Zheng

引用次数: 0