The classical fast Fourier transform (FFT) allows to compute in quasi-linear time the product of two polynomials, in the circular convolution ring R[x]/(xd -1) --- a task that naively requires quadratic time. Equivalently, it allows to accelerate matrix-vector products when the matrix is circulant. In this work, we discover that the ideas of the FFT can be applied to speed up the orthogonalization process of matrices with circulant blocks of size d x d. We show that, when d is composite, it is possible to proceed to the orthogonalization in an inductive way ---up to an appropriate re-indexation of rows and columns. This leads to a structured Gram-Schmidt decomposition. In turn, this structured Gram-Schmidt decomposition accelerates a cornerstone lattice algorithm: the nearest plane algorithm. The complexity of both algorithms may be brought down to Θ(d log d). Our results easily extend to cyclotomic rings, and can be adapted to Gaussian samplers. This finds applications in lattice-based cryptography, improving the performances of trapdoor functions.
{"title":"Fast Fourier Orthogonalization","authors":"L. Ducas, Thomas Prest","doi":"10.1145/2930889.2930923","DOIUrl":"https://doi.org/10.1145/2930889.2930923","url":null,"abstract":"The classical fast Fourier transform (FFT) allows to compute in quasi-linear time the product of two polynomials, in the circular convolution ring R[x]/(xd -1) --- a task that naively requires quadratic time. Equivalently, it allows to accelerate matrix-vector products when the matrix is circulant. In this work, we discover that the ideas of the FFT can be applied to speed up the orthogonalization process of matrices with circulant blocks of size d x d. We show that, when d is composite, it is possible to proceed to the orthogonalization in an inductive way ---up to an appropriate re-indexation of rows and columns. This leads to a structured Gram-Schmidt decomposition. In turn, this structured Gram-Schmidt decomposition accelerates a cornerstone lattice algorithm: the nearest plane algorithm. The complexity of both algorithms may be brought down to Θ(d log d). Our results easily extend to cyclotomic rings, and can be adapted to Gaussian samplers. This finds applications in lattice-based cryptography, improving the performances of trapdoor functions.","PeriodicalId":169557,"journal":{"name":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129050029","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}
The algebraic and algorithmic study of integro-differential algebras and operators has only started in the past decade. Integro-differential operators allow us in particular to study initial value and boundary problems for linear ODEs from an algebraic point of view. Differential operators already provide a rich algebraic structure with a wealth of results and algorithmic methods. Adding integral operators and evaluations, many new phenomena appear, including zero devisors and non-finitely generated ideals. In this tutorial, we give an introduction to symbolic methods for integro-differential operators and boundary problems developed over the last years. In particular, we discuss normal forms, basic algebraic properties, and the computation of polynomial solutions for ordinary integro-differential equations with polynomial coefficients. We will also outline methods for manipulating and solving linear boundary problems and illustrate them with an implementation.
{"title":"Symbolic Computation with Integro-Differential Operators","authors":"G. Regensburger","doi":"10.1145/2930889.2930942","DOIUrl":"https://doi.org/10.1145/2930889.2930942","url":null,"abstract":"The algebraic and algorithmic study of integro-differential algebras and operators has only started in the past decade. Integro-differential operators allow us in particular to study initial value and boundary problems for linear ODEs from an algebraic point of view. Differential operators already provide a rich algebraic structure with a wealth of results and algorithmic methods. Adding integral operators and evaluations, many new phenomena appear, including zero devisors and non-finitely generated ideals. In this tutorial, we give an introduction to symbolic methods for integro-differential operators and boundary problems developed over the last years. In particular, we discuss normal forms, basic algebraic properties, and the computation of polynomial solutions for ordinary integro-differential equations with polynomial coefficients. We will also outline methods for manipulating and solving linear boundary problems and illustrate them with an implementation.","PeriodicalId":169557,"journal":{"name":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131163403","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}
It has been recently discovered by Bell, Heinle and Levandovskyy that a large class of algebras, including the ubiquitous G-algebras, are finite factorization domains (FFD for short). Utilizing this result, we contribute an algorithm to find all distinct factorizations of a given element f ∈ G, where G is any G-algebra, with minor assumptions on the underlying field. Moreover, the property of being an FFD, in combination with the factorization algorithm, enables us to propose an analogous description of the factorized Gröbner basis algorithm for G-algebras. This algorithm is useful for various applications, e.g. in analysis of solution spaces of systems of linear partial functional equations with polynomial coefficients, coming from G. Additionally, it is possible to include inequality constraints for ideals in the input.
{"title":"A Factorization Algorithm for G-Algebras and Applications","authors":"A. Heinle, V. Levandovskyy","doi":"10.1145/2930889.2930906","DOIUrl":"https://doi.org/10.1145/2930889.2930906","url":null,"abstract":"It has been recently discovered by Bell, Heinle and Levandovskyy that a large class of algebras, including the ubiquitous G-algebras, are finite factorization domains (FFD for short). Utilizing this result, we contribute an algorithm to find all distinct factorizations of a given element f ∈ G, where G is any G-algebra, with minor assumptions on the underlying field. Moreover, the property of being an FFD, in combination with the factorization algorithm, enables us to propose an analogous description of the factorized Gröbner basis algorithm for G-algebras. This algorithm is useful for various applications, e.g. in analysis of solution spaces of systems of linear partial functional equations with polynomial coefficients, coming from G. Additionally, it is possible to include inequality constraints for ideals in the input.","PeriodicalId":169557,"journal":{"name":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","volume":"495 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129889798","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 propose a general algorithmic approach to noncommutative operator algebras generated by linear operators. Ore algebras are a well-established tool covering many cases arising in applications. However, integro-differential operators, for example, do not fit this structure. Instead of using (parametrized) Gröbner bases in noncommutative polynomial algebras as has been used so far in the literature, we use Bergman's basis-free analog in tensor algebras. This allows for a finite reduction system with unique normal forms. To have a smaller reduction system, we develop a generalization of Bergman's setting, which also makes the algorithmic verification of the confluence criterion more efficient. We provide an implementation in Mathematica and we illustrate both versions of the tensor setting using integro-differential operators as an example.
{"title":"Algorithmic Operator Algebras via Normal Forms for Tensors","authors":"Jamal Hossein Poor, C. Raab, G. Regensburger","doi":"10.1145/2930889.2930900","DOIUrl":"https://doi.org/10.1145/2930889.2930900","url":null,"abstract":"We propose a general algorithmic approach to noncommutative operator algebras generated by linear operators. Ore algebras are a well-established tool covering many cases arising in applications. However, integro-differential operators, for example, do not fit this structure. Instead of using (parametrized) Gröbner bases in noncommutative polynomial algebras as has been used so far in the literature, we use Bergman's basis-free analog in tensor algebras. This allows for a finite reduction system with unique normal forms. To have a smaller reduction system, we develop a generalization of Bergman's setting, which also makes the algorithmic verification of the confluence criterion more efficient. We provide an implementation in Mathematica and we illustrate both versions of the tensor setting using integro-differential operators as an example.","PeriodicalId":169557,"journal":{"name":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","volume":"102 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130857207","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 describe an asymptotically fast variant of the LLL lattice reduction algorithm. It takes as input a basis B ∈ Zn x n and returns a (reduced) basis C of the Euclidean lattice L spanned by B, whose first vector satisfies |c1| ≤ (1+c) (4/3)(n-1)/4 (det L)1/n for any fixed c>0. It terminates within O(n4+ε β1+ε) bit operations for any ε >0, with β = log maxi |bi|. It does rely on fast integer arithmetic but does not make use of fast matrix multiplication.
{"title":"Faster LLL-type Reduction of Lattice Bases","authors":"A. Neumaier, D. Stehlé","doi":"10.1145/2930889.2930917","DOIUrl":"https://doi.org/10.1145/2930889.2930917","url":null,"abstract":"We describe an asymptotically fast variant of the LLL lattice reduction algorithm. It takes as input a basis B ∈ Zn x n and returns a (reduced) basis C of the Euclidean lattice L spanned by B, whose first vector satisfies |c1| ≤ (1+c) (4/3)(n-1)/4 (det L)1/n for any fixed c>0. It terminates within O(n4+ε β1+ε) bit operations for any ε >0, with β = log maxi |bi|. It does rely on fast integer arithmetic but does not make use of fast matrix multiplication.","PeriodicalId":169557,"journal":{"name":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","volume":"103 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115368576","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}
M. Barkatou, T. Cluzeau, Jacques-Arthur Weil, L. D. Vizio
We consider a linear differential system [A] : y'=A, y}, where A has with coefficients in C(x). The differential Galois group G of [A] is a linear algebraic group which measures the algebraic relations among solutions. Although there exist general algorithms to compute $G$, none of them is either practical or implemented. This paper proposes an algorithm to compute the Lie algebra g of G when [A] is absolutely irreducible. The algorithm is implemented in Maple.
我们考虑一个线性微分系统[a]: y'= a, y}，其中a在C(x)中有系数。[A]的微分伽罗瓦群G是测量解间代数关系的线性代数群。虽然存在计算$G$的通用算法，但没有一个是实用的或可实现的。本文提出了在[A]绝对不可约时计算g的李代数g的一种算法。该算法在Maple中实现。
{"title":"Computing the Lie Algebra of the Differential Galois Group of a Linear Differential System","authors":"M. Barkatou, T. Cluzeau, Jacques-Arthur Weil, L. D. Vizio","doi":"10.1145/2930889.2930932","DOIUrl":"https://doi.org/10.1145/2930889.2930932","url":null,"abstract":"We consider a linear differential system [A] : y'=A, y}, where A has with coefficients in C(x). The differential Galois group G of [A] is a linear algebraic group which measures the algebraic relations among solutions. Although there exist general algorithms to compute $G$, none of them is either practical or implemented. This paper proposes an algorithm to compute the Lie algebra g of G when [A] is absolutely irreducible. The algorithm is implemented in Maple.","PeriodicalId":169557,"journal":{"name":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115569353","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}
Let P and Q be two polynomials in K[x,y] with degree at most d, where K is a field. Denoting by R ∈ K[x] the resultant of P and Q with respect to y, we present an algorithm to compute R mod xk in O~(kd) arithmetic operations in K, where the ~O notation indicates that we omit polylogarithmic factors. This is an improvement over state-of-the-art algorithms that require to compute R in O~(d3) operations before computing its first k coefficients.
设P和Q是K[x,y]中阶不超过d的两个多项式，其中K是一个域。用R∈K[x]表示P和Q关于y的结果，我们给出了在K的O~(kd)算术运算中计算R mod xk的算法，其中的~O符号表示我们省略了多对数因子。这是对最先进的算法的改进，这些算法需要在计算前k个系数之前在O~(d3)次操作中计算R。
{"title":"A Fast Algorithm for Computing the Truncated Resultant","authors":"G. Moroz, É. Schost","doi":"10.1145/2930889.2930931","DOIUrl":"https://doi.org/10.1145/2930889.2930931","url":null,"abstract":"Let P and Q be two polynomials in K[x,y] with degree at most d, where K is a field. Denoting by R ∈ K[x] the resultant of P and Q with respect to y, we present an algorithm to compute R mod xk in O~(kd) arithmetic operations in K, where the ~O notation indicates that we omit polylogarithmic factors. This is an improvement over state-of-the-art algorithms that require to compute R in O~(d3) operations before computing its first k coefficients.","PeriodicalId":169557,"journal":{"name":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","volume":"108 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121232488","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 focus on succinct data structures, that is on time and space efficient representations of trees and other combinatorial objects that dominate the memory requirements of most sophisticated programs and systems.
{"title":"Succinct Data Structures ... Potential for Symbolic Computation?","authors":"J. Munro","doi":"10.1145/2930889.2930944","DOIUrl":"https://doi.org/10.1145/2930889.2930944","url":null,"abstract":"We focus on succinct data structures, that is on time and space efficient representations of trees and other combinatorial objects that dominate the memory requirements of most sophisticated programs and systems.","PeriodicalId":169557,"journal":{"name":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126160213","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}
Processes to automate the selection of appropriate algorithms for various matrix computations are described. In particular, processes to check for, and certify, various matrix properties of black-box matrices are presented. These include sparsity patterns and structural properties that allow "superfast" algorithms to be used in place of black-box algorithms. Matrix properties that hold generically, and allow the use of matrix preconditioning to be reduced or eliminated, can also be checked for and certified --- notably including in the small-field case, where this presently has the greatest impact on the efficiency of the computation.
{"title":"Selecting Algorithms for Black Box Matrices: Checking For Matrix Properties That Can Simplify Computations","authors":"W. Eberly","doi":"10.1145/2930889.2930894","DOIUrl":"https://doi.org/10.1145/2930889.2930894","url":null,"abstract":"Processes to automate the selection of appropriate algorithms for various matrix computations are described. In particular, processes to check for, and certify, various matrix properties of black-box matrices are presented. These include sparsity patterns and structural properties that allow \"superfast\" algorithms to be used in place of black-box algorithms. Matrix properties that hold generically, and allow the use of matrix preconditioning to be reduced or eliminated, can also be checked for and certified --- notably including in the small-field case, where this presently has the greatest impact on the efficiency of the computation.","PeriodicalId":169557,"journal":{"name":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116600475","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}
The p-curvature of a system of linear differential equations in positive characteristic p is a matrix that measures how far the system is from having a basis of polynomial solutions. We show that the similarity class of the p-curvature can be determined without computing the p-curvature itself. More precisely, we design an algorithm that computes the invariant factors of the p-curvature in time quasi-linear in √ p. This is much less than the size of the p-curvature, which is generally linear in p. The new algorithm allows to answer a question originating from the study of the Ising model in statistical physics.
{"title":"Computation of the Similarity Class of the p-Curvature","authors":"A. Bostan, X. Caruso, É. Schost","doi":"10.1145/2930889.2930897","DOIUrl":"https://doi.org/10.1145/2930889.2930897","url":null,"abstract":"The p-curvature of a system of linear differential equations in positive characteristic p is a matrix that measures how far the system is from having a basis of polynomial solutions. We show that the similarity class of the p-curvature can be determined without computing the p-curvature itself. More precisely, we design an algorithm that computes the invariant factors of the p-curvature in time quasi-linear in √ p. This is much less than the size of the p-curvature, which is generally linear in p. The new algorithm allows to answer a question originating from the study of the Ising model in statistical physics.","PeriodicalId":169557,"journal":{"name":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","volume":"322 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122166527","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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