Showing papers in "ACM Communications in Computer Algebra in 2015"
TL;DR: This work presents a few techniques which allow to make better use of hardware integer arithmetic when implementing algorithms for sparse polynomial interpolation.
Abstract: We present a few techniques which allow to make better use of hardware integer arithmetic when implementing algorithms for sparse polynomial interpolation.
26 citations
TL;DR: It is proved that dim VLM = dim VL+dim VM for arbitrary operators L and M of the considered form, and some algorithms based on this property of operators are proposed.
Abstract: We consider linear ordinary differential or difference systems of the form L(y) = 0 where L is an operator with matrix coefficients, the unknown vector y has m components y1, . . . , ym, m > 1. The matrix coefficients are of size m x m, their entries belong to a differential or difference field K of characteristic 0. For any such a system the solution space VL is considered, and the components of each solution are in a fixed appropriate differential or difference extension of K (e.g., in the universal Picard-Vessiot extension). We prove that dim VLM = dim VL+dim VM for arbitrary operators L and M of the considered form, and discuss some algorithms based on this property of operators. In particular, we propose an algorithm to compute dim VL, as well as a new algorithm having a low complexity for recognizing unimodular operators and constructing the inverse of a unimodular operator.
18 citations
TL;DR: Signature-based Grobner basis algorithms have been studied extensively in the literature as mentioned in this paper, with a general description of how the idea of signatures is quite natural when performing the reduction process using linear algebra.
Abstract: This paper is a survey on the area of signature-based Grobner basis algorithms that was initiated by Faugere's F5 algorithm in 2002. We explain the general ideas behind the usage of signatures. We show how to classify the various known variants by 3 different orderings. For this we give translations between different notations and show that besides notations many approaches are just the same. Moreover, we give a general description of how the idea of signatures is quite natural when performing the reduction process using linear algebra. This survey shall help to outline this field of active research.
16 citations
TL;DR: A particular method for solving linear systems with noise, due to Arora-Ge, which reduces the problem to non-linear but noise-free system solving, and shows under a natural algebraic assumption that BinaryError-LWE can be solved in subexponential time as soon as the number of samples is quasi-linear.
Abstract: We analyse the complexity of algebraic algorithms for solving systems of linear equations with \emph{noise}. Such systems arise naturally in the theory of error-correcting codes as well as in computational learning theory. More recently, linear systems with noise have found application in cryptography. The \emph{Learning with Errors} (LWE) problem has proven to be a rich and versatile source of innovative cryptosystems, such as fully homomorphic encryption schemes. Despite the popularity of the LWE problem, the complexity of algorithms for solving it is not very well understood, particularly when variants of the original problem are considered. Here, we focus on and generalise a particular method for solving these systems, due to Arora \& Ge, which reduces the problem to non-linear but noise-free system solving. Firstly, we provide a refined complexity analysis for the original Arora-Ge algorithm for LWE. Secondly, we study the complexity of applying algorithms for computing Grobner basis, a fundamental tool in computational commutative algebra, to solving Arora-Ge-style systems of non-linear equations. We show positive and negative results. On the one hand, we show that the use of Grobner bases yields an exponential speed-up over the basic Arora-Ge approach. On the other hand, we give a negative answer to the natural question whether the use of such techniques can yield a subexponential algorithm for the LWE problem. Under a mild algebraic assumption, we show that it is highly unlikely that such an improvement exists. We also consider a variant of LWE known as BinaryError-LWE introduced by Micciancio and Peikert recently. By combining Grobner basis algorithms with the Arora-Ge modelling, we show under a natural algebraic assumption that BinaryError-LWE can be solved in subexponential time as soon as the number of samples is quasi-linear, e.g.\ m=O(nloglogn)m=O(n \log \log n). We also derive precise complexity bounds for BinaryError-\LWE with m=O(n)m=O(n), showing that this new approach yields better results than best currently-known generic (exact) CVP solver as soon as m/n≥6.6m/n \geq 6.6. More generally, our results provide a good picture of the hardness degradation of BinaryError-LWE for a number of samples ranging from m=n(1+Ω(1/log(n))m=n\left(1+\Omega\big(1/{\rm log}(n)\right) (a case for which BinaryError-\LWE{} is as hard as solving some lattice problem in the worst case) to m=O(n2)m=O(n^2) (a case for which it can be solved in polynomial-time). This addresses an open question from Micciancio and Peikert. Whilst our results do not contradict the hardness results obtained by Micciancio and Peikert, they should rule out BinaryError-\LWE for many cryptographic applications. The results in this work depend crucially on the assumption the algebraic systems considered systems are not easier and not harder to solve than a random system of equations. We have verified experimentally such hypothesis. We also have been able to prove formally the assumptions is several restricted situations. We emphasize that these issues are highly non-trivial since proving our assumptions in full generality would allow to prove a famous conjecture in commutative algebra known as Froberg's Conjecture.
16 citations
TL;DR: The techniques are based on the combinatorics of the noncommutative generating series in the quasi-shu e Hopf algebra, which study the multi-indexed harmonic sums and polylogarithms.
Abstract: Extending the Faulhaber's formula, the Bernoulli polynomials and the Eulerian polynomials, we study the multi-indexed harmonic sums and polylogarithms. Our techniques are based on the combinatorics of the noncommutative generating series in the quasi-shu e Hopf algebra.
14 citations
TL;DR: SDeval is a software project that contains tools for creating and running benchmarks with a focus on problems in computer algebra, built on top of the Symbolic Data project, able to translate problems in the database into executable code for various computer algebra systems.
Abstract: In this paper we will present SDeval, a software project that contains tools for creating and running benchmarks with a focus on problems in computer algebra. It is built on top of the Symbolic Data project, able to translate problems in the database into executable code for various computer algebra systems. The included tools are designed to be very flexible to use and to extend, such that they can be easily deployed even in contexts of other communities. We also address particularities of benchmarking in the field of computer algebra.Furthermore, with SDEval, we provide a feasible and automatable way of reproducing benchmarks published in current research works, which appears to be a difficult task in general due to the customizability of the available programs.
14 citations
TL;DR: A new data structure called POLY is described that was added to Maple 17 to improve performance for polynomials in Maple, and recent work done for Maple 18 (released in 2014) is described.
Abstract: The principal data structure Maple uses to represent polynomials and general mathematical expressions involving functions like sin x, e2x, y'(x),(n/k) etc., is known to the Maple developers as the sum-of-products data structure. Gaston Gonnet, as the primary author of the Maple kernel, designed and implemented this data structure in the early 1980s. As part of the process of simplifying a mathematical formula, he represented every Maple object and every sub-object uniquely in memory. This makes testing for equality of expressions very fast. In this article, on occasion of Gonnet's retirement, we present details of his design, its pros and cons, and changes we and others have made to it over the years. One of the cons of the sum-of-products data structure is it is not as efficient at multiplying multivariate polynomials as other special purpose computer algebra systems. We describe a new data structure called POLY that we added to Maple 17 (released in 2013) to improve performance for polynomials in Maple, and recent work done for Maple 18 (released in 2014).
13 citations
TL;DR: In this paper, the authors investigated the decomposition of Jacobians of superelliptic curves based on their automorphisms and provided necessary and sufficient conditions in terms of $m$ and n$ for decomposition.
Abstract: We investigate the decomposition of Jacobians of superelliptic curves based on their automorphisms. For curve with equation $y^n=f(x^m)$ we provide an necessary and sufficient condition in terms of $m$ and $n$ for the decomposition of the Jacobian induced by the automorphisms of the curve. Moreover, we generalize a construction in \cite{Ya} of a family of non-hyperelliptic curves $\mathcal X_{r,s} $ and determine arithmetic conditions on $r$ and $s$ that the Jacobians $\mbox{Jac} (\mathcal X_{r, s})$ decomposes.
12 citations
12 citations
TL;DR: A relaxed representation for the inverse is introduced and how to obtain an algorithm for OnlineSystem that allows for fast matrix multiplication into the inherently iterative algorithm for row rank prole computation is shown.
Abstract: We consider a variation of the well known problem of computing the unique solution to a nonsingular systemAx = b of n linear equations over a eld K. The variation assumes that A has generic rank prole and requires as output not only the single solution vector A 1 b2 K n 1 , but rather the solution to all leading principle subsystems. Most importantly, the rows of the augmented system A b are given one at a time from rst to last, and as soon as the next row is given the solution to the next leading principal subsystem should be produced. We call this problem OnlineSystem. The obvious iterative algorithm for OnlineSystem has a cost in terms of eld operations that is cubic in the dimension of A. In this paper we introduce a relaxed representation for the inverse and show how to obtain an algorithm for OnlineSystem that allows us to incorporate matrix multiplication. As an application we show how to introduce fast matrix multiplication into the inherently iterative algorithm for row rank prole computation presented previously by the authors.
12 citations
TL;DR: In this article, the authors apply the methods of rewriting systems and Grobner-Shirshov bases to give a unified approach to a class of linear operators on associative algebras, called Rota-Baxter type operators.
Abstract: In this paper we apply the methods of rewriting systems and Grobner-Shirshov bases to give a unified approach to a class of linear operators on associative algebras. These operators resemble the classic Rota-Baxter operator, and they are called Rota-Baxter type operators. We characterize a Rota-Baxter type operator by the convergency of a rewriting system associated to the operator. By associating such an operator to a Grobner-Shirshov basis, we obtain a canonical basis for the free algebras in the category of associative algebras with that operator. This construction include as special cases several previous ones for free objects in similar categories, such as those of Rota-Baxter algebras and Nijenhuis algebras.
TL;DR: In this article, the authors extended the work of Lisonek and Singh on construction X for quantum error-correcting codes to finite fields of order p2 where p is prime and gave some new results on the Hermitian dual of repeated root cyclic codes.
Abstract: In this paper we extend the work of Lisonek and Singh on construction X for quantum error-correcting codes to finite fields of order p2 where p is prime. Further, we give some new results on the Hermitian dual of repeated root cyclic codes. These results are used to construct new quantum error-correcting codes.
TL;DR: An algorithm is derived which, under genericity assumptions, computes the intersection multiplicity of a zero-dimensional variety at any of its points, thus providing an alternative to the standard approaches based on Groebner bases and standard bases.
Abstract: Traditionally, Groebner bases and cylindrical algebraic decomposition are the fundamental tools of computational algebraic geometry. Recent progress in the theory of regular chains has exhibited efficient algorithms for doing local analysis on algebraic varieties. In this note, we present the implementation of these new ideas within the module AlgebraicGeometryTools of the RegularChains library. The functionalities of this new module include the computation of the (non-trivial) limit points of the quasi-component of a regular chain. This type of calculation has several applications like computing the Zarisky closure of a constructible set as well as computing tangent cones of space curves, thus providing an alternative to the standard approaches based on Groebner bases and standard bases, respectively. From there, we have derived an algorithm which, under genericity assumptions, computes the intersection multiplicity of a zero-dimensional variety at any of its points. This algorithm relies only on the manipulations of regular chains.
TL;DR: The algorithm generalizes Welch/Berlekamp decoding of Reed/Solomon error correcting codes and their numeric floating point counterparts and gives an algorithm that computes the unique solution x = f(u)/g(u), which is a vector of rational functions, by evaluating the parameter u at distinct points.
Abstract: We consider the problem of solving a full rank consistent linear system A(u)x = b(u) where the m x n matrix A and the m-dimensional vector b has entries that are polynomials in u over a field. We give an algorithm that computes the unique solution x = f(u)/g(u), which is a vector of rational functions, by evaluating the parameter u at distinct points. Those points ξλ where the matrix A evaluates to a matrix A(ξλ), with entries over the scalar field, of lower rank, or in the numeric setting to an ill-conditioned matrix, are not identified but accounted for by error-correcting code techniques. We also correct true errors where the evaluation at some u = ξλ results in an erroneous, possibly full rank consistent and well-conditioned scalar linear system. Our algorithm generalizes Welch/Berlekamp decoding of Reed/Solomon error correcting codes and their numeric floating point counterparts.We have implemented our algorithms with floating point arithmetic. For the determination of the exact numerator and denominator degrees and number of errors we use singular values based numeric rank computations. The arising linear systems for the error-corrected parametric solution are demonstrated to be well-conditioned even when the input scalars have noise. In several initial experiments we have shown that our approach is numerically stable even for larger systems m = n = 100, provided the degrees in the solution are small (≤ 2). For smaller systems m = n = 10 with higher degrees (≤ 20) the algorithm works similarly to rational function recovery. Our implementation can correct 13 true errors in both settings.
TL;DR: The notion of counting polynomials is used, to dene the (extended) rank weight enumerator, since in this generality the set of codewords of a given rank weight is no longer nite.
Abstract: This paper investigates the rank weight enumerator of a code over L, where L is a nite extension of a eld K. This is a generalization of the case where K = Fq and L = Fqm of Gabidulin codes to arbitrary characteristic. We use the notion of counting polynomials, to dene the (extended) rank weight enumerator, since in this generality the set of codewords of a given rank weight is no longer nite. Also the extended and generalized rank weight enumerator are studied in analogy with previous work on codes with respect to the Hamming metric.
TL;DR: An efficient decision procedure for positivity on a class of holonomic sequences satisfying recurrences of arbitrary order is presented.
Abstract: We present an efficient decision procedure for positivity on a class of holonomic sequences satisfying recurrences of arbitrary order.
TL;DR: Joux’s algorithm is used to compute discrete logarithms in the 1303-bit finite field F36·137 and illustrated for the first time its effectiveness in ‘general’ small-characteristic finite fields with very modest computational resources.
Abstract: In 2013, Joux presented a new algorithm for solving the discrete logarithm problem in finite fields of small characteristic with a main novelty involving the resolution of bilinear equation systems. The algorithm improved significantly all previous methods for this purpose. We used Joux’s algorithm to compute discrete logarithms in the 1303-bit finite field F36·137 and illustrated for the first time its effectiveness in ‘general’ small-characteristic finite fields with very modest computational resources.
TL;DR: In this paper, the authors considered the structure of multivariable modular codes in an ambient space which is a principal ideal ring, and showed that these codes admit a relatively simple description, quite close to that of semisimple.
Abstract: Multivariable codes over a finite field are a natural generalization of several classes of codes, including cyclic, negacyclic, constancyclic, polycyclic and abelian codes. Since these particular families have been also considered in the context of codes over a finite chain ring, we proposed constructions of multivariable codes over such a class of finite rings. As in the case of traditional cyclic codes over finite fields the modular case (i.e., codes with repeated roots) is much more difficult to handle than the semisimple case (i.e., codes with non-repeated roots). In this sense, different authors have dedicated their efforts to provide a better understanding of the properties of cyclic, negacyclic, constancylic and polycyclic modular codes over a finite chain ring. Among these codes, those contained in an ambient space which is a principal ideal ring admit a relatively simple description, quite close to that of semisimple. This feature has been recently used in the description of abelian codes over a finite field, and in the description of modular additive cyclic codes over F4. As a natural continuation of these works, in this paper we consider the structure of multivariable modular codes in an ambient space which is a principal ideal ring.
TL;DR: This talk will see that the Wronskian bracket gives rise to the so-called non-commutative version of a Lie algebra, namely a Leibniz algebra, and presents the construction of the free differential algebra over a (differential) LeibNiz algebra.
Abstract: It is well-known that any associative algebra becomes a Lie algebra under the commutator bracket. This relation is actually functorial, and this functor, as any algebraic functor, is known to admit a left adjoint, namely the universal enveloping algebra of a Lie algebra. This correspondence may be lifted to the setting of differential (Lie) algebras. In this contribution it is shown that, also in the differential context, there is another, similar, but somewhat different, correspondence. Indeed any commutative differential algebra becomes a Lie algebra under the Wronskian bracket W ( a , b ) = a b ′ − a ′ b . It is proved that this correspondence again is functorial, and that it admits a left adjoint, namely the differential enveloping (commutative) algebra of a Lie algebra. Other standard functorial constructions, such as the tensor and symmetric algebras, are studied for algebras with a given derivation.
TL;DR: This paper proposes an O(n3 log n) QFT algorithm over Sn by relieving bottlenecks of the previous algorithm, which was faster than the existing algorithm.
Abstract: This paper improves the main result in our paper "Quantum Fourier Transform over Symmetric Groups, Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC), pp. 227--234, Northeastern University, Boston, USA (June 26-29, 2013)." In that paper, we presented an O(n4) quantum Fourier transform (QFT) algorithm over symmetric group Sn, which was faster than the existing algorithm. This paper proposes an O(n3 log n) QFT algorithm over Sn by relieving bottlenecks of our previous algorithm.
TL;DR: This paper obtains linear codes over Z4 with bounded GC content and gives upper bounds for the edit distance and construct codes with bounded edit distance.
Abstract: In this paper we construct codes for DNA computing using the greedy algorithm over Z4. We obtain linear codes over Z4 with bounded GC content. We also consider the edit distance, we gave upper bounds for the edit distance and construct codes with bounded edit distance.
TL;DR: Some changes of the traditional scheme for finding rational solutions of linear difference equations (homogeneous or inhomogeneous) with rational coefficients allow one to detect the absence of rational solutions in an early stage of computation and to stop the work.
Abstract: We discuss some changes of the traditional scheme for finding rational solutions of linear difference equations (homogeneous or inhomogeneous) with rational coefficients. These changes in some cases allow one to detect the absence of rational solutions in an early stage of computation and to stop the work. The corresponding strategy may be useful for the algorithms based on finding rational solutions of equations of considered form, e.g., for some symbolic summation algorithms.
TL;DR: An open public "MQ Challenge" is established to stimulate progress in the design of efficient algorithms to solve MQ problem, and thus test limit parameters guaranteeing security of MPKC, a candidate for Post-Quantum Cryptography.
Abstract: Multivariate polynomials over finite fields have found applications in Public Key Cryptography (PKC) where the hardness to find solutions provides the "one-way function" indispensable to such cryptosystems. Several schemes for both encryption and signature have been proposed, many of which are using quadratic (degree 2) polynomials. Finding a solution to such systems in general is called MQ problem, which easiest "generic" instances are NP-hard. An important feature of this Multivariate Pubic Key Cryptography (MPKC) is the resistance to quantum computers: no faster quantum algorithm than classical ones to solve MQ problem is known. Besides being thereby a candidate for Post-Quantum Cryptography, signatures are much shorter than to other candidates. We have established an open public "MQ Challenge" (https://www.mqchallenge.org) to stimulate progress in the design of efficient algorithms to solve MQ problem, and thus test limit parameters guaranteeing security of MPKC.
TL;DR: This paper introduces a software product which generates a model for computing the time response of LTI circuits.
Abstract: When users analyze symbolically physical phenomena, software products which partially help them write a symbolic equation to the phenomena are available. In electric engineering, there are a few software products which compute symbolically voltages and currents of electric circuits. They cover the frequency response analysis on behavior of linear time-invariant (LTI) circuits. However, they do not cover symbolic computation on the time response of circuits. This paper introduces a software product which generates a model for computing the time response of LTI circuits.
TL;DR: The Basic Polynomial Algebra Subprograms provides arithmetic operations for univariate and multivariate polynomials over prime fields or with integer coefficients to take advantage of hardware accelerators in the development of polynomial systems solvers.
Abstract: The Basic Polynomial Algebra Subprograms (BPAS) provides arithmetic operations (multiplication, division, root isolation, etc.) for univariate and multivariate polynomials over prime fields or with integer coefficients. The code is mainly written in CilkPlus [11] targeting multicore processors. The current distribution focuses on dense polynomials and the sparse case is work in progress. A strong emphasis is to put on adaptive algorithms as the library aims at supporting a wide variety of situations in terms of problem sizes and available computing resources. One of the purposes of the BPAS project is to take advantage of hardware accelerators in the development of polynomial systems solvers. The BPAS library is publicly available in source at www.bpaslib.org.
TL;DR: This work proves that ideal codes defined over Ore polynomial rings A[z;σ, δ], where A is a finite (possibly non-commutative) semisimple algebra, are generated by an idempotent element.
Abstract: Most of the codes used in engineering support a vector space structure (linear block codes) or becomes a submodule of a polynomial ring (convolutional codes). In the linear case, the benefits are amplified if we also consider the notion of cyclicity, since the ambient space is also endowed with an algebra structure and cyclic codes come to be ideals. Over convolutional codes, this notion requires something more sophisticated than a simple extension of the block one [1], and the underlying working algebra is no longer a polynomial ring but an skew polynomial ring A[z;σ] over a finite commutative semisimple algebra A. Very recently, in [3], these codes are called ideal codes and they are defined over Ore polynomial rings A[z;σ, δ], where A is a finite (possibly non-commutative) semisimple algebra. Nevertheless, effective algorithms are provided whenever A is a separable group algebra of a finite group over a finite field. In this work we aim to cover more examples, see Example 3, than the former papers by assuming certain mild conditions of separability in A, see Corollary 2. In particular, we prove that they are generated by an idempotent element. We also provide an algorithm for computing a generating idempotent, which in particular is applicable the ideal codes from [1]. For brevity, we only shall consider Ore polynomial rings A[z;σ, δ] where the σ-derivation δ = 0, albeit, under suitable conditions, our results remain true for an arbitrary δ. In our examples, except for 0 and 1, we shall write the elements of a finite field F as powers of a primitive element.
TL;DR: A new version of the McEliece cryptosystem based on punctured convolutional codes and the pseudo-random generators and the modified self-shrinking generator to fill the punctured pattern.
Abstract: The purpose of this paper is to present a new version of the McEliece cryptosystem based on punctured convolutional codes and the pseudo-random generators. We use the modified self-shrinking generator to fill the punctured pattern. More precisely we propose to fill out the pattern punctured by the bits generated using a pseudo random generator LFSR.
TL;DR: The problem of indefinite summation (anti-differencing) in general is: given a closed form expression F (x) to find aclosed form expression G(x), which satisfies the first order linear difference equation.
Abstract: Let K be a field of characteristic zero, x – an independent variable, E – the shift operator with respect to x, i.e., Ef(x) = f(x+1) for an arbitrary function f(x), ∆ = E−1 – the difference operator with respect to x. The problem of indefinite summation (anti-differencing) in general is: given a closed form expression F (x) to find a closed form expression G(x), which satisfies the first order linear difference equation
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TL;DR: Celestials in 3-space that are obtained from translating a circle along a circle, in either Euclidean or elliptic space are considered, and a classically avored theorem in elliptic geometry is obtained: if the authors translate a line along acircle but not along a line then exactly 2 translated lines will coincide.
Abstract: The sphere in 3-space has an innite number of circles through any closed point. The torus has 4 circles through any closed point. Two of these circles are known as Villarceau circles ([0]). We dene a \celestial" to be a real surface with at least 2 real circles through a generic closed point. Equivalently, a celestial is a surface with at least 2 families of real circles. In 1980 Blum [1] conjectured that a real surface has either at most 6 families of circles or an innite number. For compact surfaces this conjecture has been proven by Takeuchi [2] in 1987 using topological methods. In 2001 Schicho [3] classied complex surfaces with at least 2 families of conics. This result together with Moebius geometry led to a classication of celestials in 3-space [4]. In 2012 Pottmann et al. [5] conjectured that a surface in 3-space with exactly 3 circles through a closed point is a Darboux Cyclide. We conrm this conjecture as a corollary from our classication in [4]. We recall that a translation is an isometry where every point moves with the same distance. In this talk we consider celestials in 3-space that are obtained from translating a circle along a circle, in either Euclidean or elliptic space. This is a natural extension of classical work by William Kingdon Cliord and Felix Klein on the Cliord torus. Krasauskas, Pottmann and Skopenkov conjectured, that celestials in 3-space of Moebius degree 8 are Moebius equivalent to an Euclidean or Elliptic translational celestial. This conjecture is true if its Moebius model has a family of great circles ([6]). Moreover, its real singular locus consist of a great circle. As a corollary we obtain a classically avored theorem in elliptic geometry: if we translate a line along a circle but not along a line then exactly 2 translated lines will coincide ([6]).
TL;DR: This talk shows how to compute a minimal generating set of invariants along with the substitution to rewrite any invariant in terms of this generating set, which can be chosen to consist of polynomial invariants.
Abstract: In this talk we study the field of rational invariants of the linear action of a finite abelian group in the non modular case. By diagonalization, the group is accurately described by an integer matrix of exponents. Making use of integer linear algebra we show how to compute a minimal generating set of invariants along with the substitution to rewrite any invariant in terms of this generating set. This generating set can be chosen to consist of polynomial invariants. As an application, we provide a symmetry reduction scheme for dynamical and polynomial systems whose solution set is invariant by the group action. In addition we provide an algorithm to find such symmetries given a dynamical or polynomial system. This is joint work with Evelyne Hubert ( INRIA Méditerranée, France)