A univariate polynomial f is decomposable if it is the composition f = g(h) of polynomials g and h whose degrees are at least two. We consider the nearest decomposable polynomial to a given polynomial f in the Hamming distance. We propose a polynomial-time approximation algorithm for the nearest decomposable polynomial and analyze the quality of the output.
如果单变量多项式 f 是阶数至少为 2 的多项式 g 和 h 的组合 f = g(h)，则该多项式 f 是可分解的。我们考虑与给定多项式 f 在汉明距离上最近的可分解多项式。我们为最近可分解多项式提出了一种多项式时间近似算法，并分析了输出结果的质量。
{"title":"An Approximation Algorithm for the Nearest Decomposable Polynomial in the Hamming Distance","authors":"Hiroshi Sekigawa","doi":"10.1145/3637529.3637532","DOIUrl":"https://doi.org/10.1145/3637529.3637532","url":null,"abstract":"A univariate polynomial f is decomposable if it is the composition f = g(h) of polynomials g and h whose degrees are at least two. We consider the nearest decomposable polynomial to a given polynomial f in the Hamming distance. We propose a polynomial-time approximation algorithm for the nearest decomposable polynomial and analyze the quality of the output.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"1 1","pages":"119 - 125"},"PeriodicalIF":0.1,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139346476","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}
Snehal Bhayani, Praneeth Susarla, S. S. Krishna, Chaitanya Bulusu, Olli Silvén, Markku J. Juntti, Janne Heikkila
Beamforming is a signal processing technique where an array of antenna elements can be steered to transmit and receive radio signals in a specific direction. The usage of millimeter wave (mmWave) frequencies and multiple input multiple output (MIMO) beamforming are considered as the key innovations of 5th Generation (5G) and beyond communication systems. The mmWave radio waves enable high capacity and directive communication, but suffer from many challenges such as rapid channel variation, blockage effects, atmospheric attenuations, etc. The technique initially performs beam alignment procedure, followed by data transfer in the aligned directions between the transmitter and the receiver [1]. Traditionally, beam alignment involves periodical and exhaustive beam sweeping at both transmitter and the receiver, which is a slow process causing extra communication overhead with MIMO and massive MIMO radio units. In applications such as beam tracking, angular velocity, beam steering etc. [2], beam alignment procedure is optimized by estimating the beam directions using first order polynomial approximations. Recent learning-based SOTA strategies [3] for fast mmWave beam alignment also require exploration over exhaustive beam pairs during the training procedure, causing overhead to learning strategies for higher antenna configurations. Therefore, our goal is to optimize the beam alignment cost functions e.g., data rate, to reduce the beam sweeping overhead by applying polynomial approximations of its partial derivatives which can then be solved as a system of polynomial equations. Specifically, we aim to reduce the beam search space by estimating approximate beam directions using the polynomial solvers. Here, we assume both transmitter (TX) and receiver (RX) to be equipped with uniform linear array (ULA) configuration, each having only one degree of freedom (d.o.f.) with Nt and Nr antennas, respectively.
波束成形是一种信号处理技术，可将天线元件阵列转向特定方向发射和接收无线电信号。毫米波（mmWave）频率的使用和多输入多输出（MIMO）波束成形被认为是第五代（5G）及以后通信系统的关键创新。毫米波无线电波可实现高容量和直接通信，但也面临许多挑战，如信道快速变化、阻塞效应、大气衰减等。该技术首先执行波束对准程序，然后在发射器和接收器之间按对准的方向传输数据[1]。传统上，波束对准需要在发射机和接收机上进行周期性和详尽的波束扫描，这是一个缓慢的过程，会给多输入多输出和大规模多输入多输出无线电设备带来额外的通信开销。在波束跟踪、角速度、波束转向等应用中[2]，波束对准过程是非常重要的。[2]，波束对准过程是通过使用一阶多项式近似估计波束方向来优化的。最近用于毫米波波束快速对准的基于学习的 SOTA 策略[3]也需要在训练过程中对所有波束对进行探索，这给更高天线配置的学习策略带来了开销。因此，我们的目标是优化波束对准成本函数（如数据率），通过对其偏导数进行多项式近似来减少波束扫描开销，然后将其作为多项式方程组来求解。具体来说，我们的目标是利用多项式求解器估计近似波束方向，从而减少波束搜索空间。在此，我们假设发射器（TX）和接收器（RX）都采用均匀线性阵列（ULA）配置，每个阵列只有一个自由度（d.o.f.），分别有 Nt 和 Nr 个天线。
{"title":"A Novel Application of Polynomial Solvers in mmWave Analog Radio Beamforming","authors":"Snehal Bhayani, Praneeth Susarla, S. S. Krishna, Chaitanya Bulusu, Olli Silvén, Markku J. Juntti, Janne Heikkila","doi":"10.1145/3637529.3637537","DOIUrl":"https://doi.org/10.1145/3637529.3637537","url":null,"abstract":"Beamforming is a signal processing technique where an array of antenna elements can be steered to transmit and receive radio signals in a specific direction. The usage of millimeter wave (mmWave) frequencies and multiple input multiple output (MIMO) beamforming are considered as the key innovations of 5th Generation (5G) and beyond communication systems. The mmWave radio waves enable high capacity and directive communication, but suffer from many challenges such as rapid channel variation, blockage effects, atmospheric attenuations, etc. The technique initially performs beam alignment procedure, followed by data transfer in the aligned directions between the transmitter and the receiver [1]. Traditionally, beam alignment involves periodical and exhaustive beam sweeping at both transmitter and the receiver, which is a slow process causing extra communication overhead with MIMO and massive MIMO radio units. In applications such as beam tracking, angular velocity, beam steering etc. [2], beam alignment procedure is optimized by estimating the beam directions using first order polynomial approximations. Recent learning-based SOTA strategies [3] for fast mmWave beam alignment also require exploration over exhaustive beam pairs during the training procedure, causing overhead to learning strategies for higher antenna configurations. Therefore, our goal is to optimize the beam alignment cost functions e.g., data rate, to reduce the beam sweeping overhead by applying polynomial approximations of its partial derivatives which can then be solved as a system of polynomial equations. Specifically, we aim to reduce the beam search space by estimating approximate beam directions using the polynomial solvers. Here, we assume both transmitter (TX) and receiver (RX) to be equipped with uniform linear array (ULA) configuration, each having only one degree of freedom (d.o.f.) with Nt and Nr antennas, respectively.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"7 1","pages":"148 - 151"},"PeriodicalIF":0.1,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139343636","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}
We present an algorithm to compute the global minimum of a trigonometric polynomial, when it is invariant under the exponential action of a Weyl group. This is based on a common relaxation technique that leads to a semi-definite program (SDP). It is then shown how to exploit the invariance in order to reduce the number of variables of the SDP and to simplify its structure significantly. This approach complements the one that was proposed as a poster at the recent ISSAC 2022 conference [HMMR22] and later extended to [HMMR23]. In the previous work, we first used the invariance of the objective function to obtain a classical polynomial optimization problem on the orbit space and subsequently relaxed the problem to an SDP. In the present work, we first apply the relaxation and then exploit symmetry. We show that the Weyl group action is induced by an orthogonal representation and describe its isotypic decomposition.
{"title":"Symmetry Adapted Bases for Trigonometric Optimization","authors":"Tobias Metzlaff","doi":"10.1145/3637529.3637535","DOIUrl":"https://doi.org/10.1145/3637529.3637535","url":null,"abstract":"We present an algorithm to compute the global minimum of a trigonometric polynomial, when it is invariant under the exponential action of a Weyl group. This is based on a common relaxation technique that leads to a semi-definite program (SDP). It is then shown how to exploit the invariance in order to reduce the number of variables of the SDP and to simplify its structure significantly. This approach complements the one that was proposed as a poster at the recent ISSAC 2022 conference [HMMR22] and later extended to [HMMR23]. In the previous work, we first used the invariance of the objective function to obtain a classical polynomial optimization problem on the orbit space and subsequently relaxed the problem to an SDP. In the present work, we first apply the relaxation and then exploit symmetry. We show that the Weyl group action is induced by an orthogonal representation and describe its isotypic decomposition.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"63 1","pages":"137 - 140"},"PeriodicalIF":0.1,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139345339","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}
Two-layer organization of modular arithmetic is considered. Lower layer uses many moduli at hardware precision and simultaneous conversion to/from RNS as described in [2]. Upper layer uses specially selected large moduli allowing for fast reduction and/or reconstruction. Implementation of two different strategies for selecting moduli on the upper layer confirms practicality of proposed approach.
{"title":"On a Two-Layer Modular Arithmetic","authors":"Benjamin Chen, Yu Li, Eugene Zima","doi":"10.1145/3637529.3637534","DOIUrl":"https://doi.org/10.1145/3637529.3637534","url":null,"abstract":"Two-layer organization of modular arithmetic is considered. Lower layer uses many moduli at hardware precision and simultaneous conversion to/from RNS as described in [2]. Upper layer uses specially selected large moduli allowing for fast reduction and/or reconstruction. Implementation of two different strategies for selecting moduli on the upper layer confirms practicality of proposed approach.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"1 1","pages":"133 - 136"},"PeriodicalIF":0.1,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139345506","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}
Tereso del Río, AmirHosein Sadeghimanesh, Matthew England
Cylindrical Algebraic Decomposition (CAD) is an algorithm originally proposed by Collins in [4]. Given a set of multivariate polynomials, CAD decomposes the multidimensional real space into connected subsets called cells, within which those polynomials are sign-invariant.
{"title":"Clustering in the Lazard method for Cylindrical Algebraic Decomposition","authors":"Tereso del Río, AmirHosein Sadeghimanesh, Matthew England","doi":"10.1145/3637529.3637533","DOIUrl":"https://doi.org/10.1145/3637529.3637533","url":null,"abstract":"Cylindrical Algebraic Decomposition (CAD) is an algorithm originally proposed by Collins in [4]. Given a set of multivariate polynomials, CAD decomposes the multidimensional real space into connected subsets called cells, within which those polynomials are sign-invariant.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"24 1","pages":"126 - 132"},"PeriodicalIF":0.1,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139343982","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}
An effective computation of a basis of a nontrivial centralizer of a differential operator is the first step towards a Picard-Vessiot theory for spectral problems of ordinary differential operators. The set of almost-commuting operators enjoys a richer structure that allows the computation of these centralizers. We present a method to calculate a basis of almost-commuting operators. Its application to the computation of nontrivial centralizers is illustrated by examples.
{"title":"Computing Almost-Commuting Basis of Ordinary Differential Operators","authors":"Antonio Jiménez-Pastor, Sonia L. Rueda, M. Zurro","doi":"10.1145/3637529.3637531","DOIUrl":"https://doi.org/10.1145/3637529.3637531","url":null,"abstract":"An effective computation of a basis of a nontrivial centralizer of a differential operator is the first step towards a Picard-Vessiot theory for spectral problems of ordinary differential operators. The set of almost-commuting operators enjoys a richer structure that allows the computation of these centralizers. We present a method to calculate a basis of almost-commuting operators. Its application to the computation of nontrivial centralizers is illustrated by examples.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"67 1","pages":"111 - 118"},"PeriodicalIF":0.1,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139344871","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}
Keiper [1] and Li [2] published independent investigations of the connection between the Riemann hypothesis and the properties of sums over powers of zeros of the Riemann zeta function. Here we consider a reframing of the criterion, to work with higher-order derivatives ξr of the symmetrized function ξ(s) at s = 1/2, with all ξr known to be positive. The reframed criterion requires knowledge of the asymptotic properties of two terms, one being an infinite sum over the ξr. This is studied using known asymptotic expansions for the ξr, which give the location of the summand as a relationship between two parameters. This relationship needs to be inverted, which we show can be done exactly using a generalized Lambert function. The result enables an accurate asymptotic expression for the value of the infinite sum.
Keiper [1] 和 Li [2] 分别就黎曼假设与黎曼zeta 函数零点幂上和的性质之间的联系进行了研究。在此，我们考虑对这一准则进行重构，以处理对称函数 ξ(s) 在 s = 1/2 处的高阶导数ξr，已知所有ξr 均为正数。重构准则要求了解两个项的渐近特性，其中一个是ξr 的无限和。这需要利用已知的 ξr 的渐近展开来研究，渐近展开给出了和的位置，即两个参数之间的关系。这种关系需要反演，我们用广义朗伯函数证明了这一点。因此，我们可以精确地得到无限和值的渐近表达式。
{"title":"The Keiper-Li Criterion for the Riemann Hypothesis and Generalized Lambert Functions","authors":"Ross McPhedran, Tony C. Scott, A. Maignan","doi":"10.1145/3637529.3637530","DOIUrl":"https://doi.org/10.1145/3637529.3637530","url":null,"abstract":"Keiper [1] and Li [2] published independent investigations of the connection between the Riemann hypothesis and the properties of sums over powers of zeros of the Riemann zeta function. Here we consider a reframing of the criterion, to work with higher-order derivatives ξr of the symmetrized function ξ(s) at s = 1/2, with all ξr known to be positive. The reframed criterion requires knowledge of the asymptotic properties of two terms, one being an infinite sum over the ξr. This is studied using known asymptotic expansions for the ξr, which give the location of the summand as a relationship between two parameters. This relationship needs to be inverted, which we show can be done exactly using a generalized Lambert function. The result enables an accurate asymptotic expression for the value of the infinite sum.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"19 1","pages":"85 - 110"},"PeriodicalIF":0.1,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139345782","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}
Frédéric Bihan, Erika Croy, Weixun Deng, Kaitlyn Phillipson, Robert J. Rennie, J. M. Rojas
Fewnomial Theory [Kho91] has established bounds on the number of connected components (a.k.a. pieces) of a broad class of real analytic sets as a function of a particular kind of input complexity, e.g., the number of distinct exponent vectors among a generating set for the underlying ideal. Here, we pursue the algorithmic side: We show how to efficiently compute the exact isotopy type of certain (possibly singular) real zero sets, instead of just estimating their number of pieces. While we focus on the circuit case, our results form the foundation for an approach to the general case that we will pursue later.
Fewnomial Theory [Kho91]已经建立了一大类实数解析集合的连通成分（又称片段）的数量边界，作为一种特定输入复杂度的函数，例如，底层理想的生成集合中不同指数向量的数量。在这里，我们追求的是算法方面：我们展示了如何高效计算某些（可能是奇异的）实零集的精确等式类型，而不仅仅是估算它们的个数。虽然我们关注的是电路情况，但我们的结果为我们以后研究一般情况的方法奠定了基础。
{"title":"Quickly Computing Isotopy Type for Exponential Sums over Circuits (Extended Abstract)","authors":"Frédéric Bihan, Erika Croy, Weixun Deng, Kaitlyn Phillipson, Robert J. Rennie, J. M. Rojas","doi":"10.1145/3637529.3637538","DOIUrl":"https://doi.org/10.1145/3637529.3637538","url":null,"abstract":"Fewnomial Theory [Kho91] has established bounds on the number of connected components (a.k.a. pieces) of a broad class of real analytic sets as a function of a particular kind of input complexity, e.g., the number of distinct exponent vectors among a generating set for the underlying ideal. Here, we pursue the algorithmic side: We show how to efficiently compute the exact isotopy type of certain (possibly singular) real zero sets, instead of just estimating their number of pieces. While we focus on the circuit case, our results form the foundation for an approach to the general case that we will pursue later.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"15 1","pages":"152 - 155"},"PeriodicalIF":0.1,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139346032","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}
For alternate Cantor real base numeration systems we generalise the result of Frougny and Solomyak on arithmetics on the set of numbers with finite expansions. We generalise the notion of finiteness property, denoted (F). We also provide several necessary conditions, and comment on a sufficient condition of this property. The sufficient condition allows us to find a set of generalised Cantor bases with Property (F). The used construction also provides a method for performing addition of finite expansions in Cantor real bases.
{"title":"Arithmetics in Generalised Cantor Base Systems","authors":"Katarína Studenicová","doi":"10.1145/3637529.3637539","DOIUrl":"https://doi.org/10.1145/3637529.3637539","url":null,"abstract":"For alternate Cantor real base numeration systems we generalise the result of Frougny and Solomyak on arithmetics on the set of numbers with finite expansions. We generalise the notion of finiteness property, denoted (F). We also provide several necessary conditions, and comment on a sufficient condition of this property. The sufficient condition allows us to find a set of generalised Cantor bases with Property (F). The used construction also provides a method for performing addition of finite expansions in Cantor real bases.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"31 1","pages":"156 - 159"},"PeriodicalIF":0.1,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139346531","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}
With the rise of new technologies, the demand for efficient cryptographic hardware is rising. This work is focusing on finite field based cryptography. To explore the vast design space obtained from possible finite field parameters, automated generation of hardware submodules using designer-specified algorithms for finite field arithmetic is needed. This work presents a hardware design automation framework exploiting symbolic computation capabilities of GAP to generate the expressions, needed for hardware implementations, on-the-fly, and compiles them into synthesizable datapaths, test-vectors, and testbences.
随着新技术的兴起，对高效加密硬件的需求也在不断增加。这项工作的重点是基于有限域的密码学。为了探索可能的有限域参数所带来的广阔设计空间，需要使用设计者指定的有限域运算算法自动生成硬件子模块。这项工作提出了一个硬件设计自动化框架，利用 GAP 的符号计算功能即时生成硬件实现所需的表达式，并将其编译成可综合的数据路径、测试向量和测试平台。
{"title":"How to use a CAS for Hardware Design Automation","authors":"N. Zidarič","doi":"10.1145/3637529.3637536","DOIUrl":"https://doi.org/10.1145/3637529.3637536","url":null,"abstract":"With the rise of new technologies, the demand for efficient cryptographic hardware is rising. This work is focusing on finite field based cryptography. To explore the vast design space obtained from possible finite field parameters, automated generation of hardware submodules using designer-specified algorithms for finite field arithmetic is needed. This work presents a hardware design automation framework exploiting symbolic computation capabilities of GAP to generate the expressions, needed for hardware implementations, on-the-fly, and compiles them into synthesizable datapaths, test-vectors, and testbences.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"46 1","pages":"141 - 147"},"PeriodicalIF":0.1,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139343765","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}
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