Computation Of Greatest Common Divisor Research Articles

Approximate GCD of several multivariate sparse polynomials based on SLRA interpolation

To compute the greatest common divisor (GCD) of a set of multivariate polynomials, modular algorithms are typically employed to prevent any growth in the coefficient polynomials in the intermediate expressions. However, when dealing with multivariate polynomials with a priori errors on their coefficients, using such modular algorithms becomes challenging. This is because any resulting approximate GCD computed in one variable may have perturbations depending on the evaluation point and may not be an image of the same desired multivariate approximate GCD. This necessitates computing it as given multivariate polynomials, and operating with large matrices whose size is exponential in the number of variables. In this paper, we present a new modular algorithm, suitable for dense cases and effective for sparse ones, called “SLRA interpolation”. This algorithm uses the multidimensional fast Fourier transform (FFT) and the structured low-rank approximation (SLRA) of non-square block diagonal matrices. The SLRA interpolation technique may reduce the time-complexity for one iteration in the computation of approximate GCD of several multivariate polynomials, especially for the sparse case.

Fast Detection of Degenerate Predicates in Free Space Construction

An implementation of a computational geometry algorithm is robust if the combinatorial output is correct for every input. Robustness is achieved by ensuring that the predicates in the algorithm are evaluated correctly. A predicate is the sign of an algebraic expression whose variables are input parameters. The hardest case is detecting degenerate predicates where the value of the expression equals zero. We encounter this case in constructing the free space of a polyhedron that rotates around a fixed axis and translates freely relative to a stationary polyhedron. Each predicate involved in the construction is expressible as the sign of a univariate polynomial [Formula: see text] evaluated at a zero [Formula: see text] of a univariate polynomial [Formula: see text], where the coefficients of [Formula: see text] and [Formula: see text] are polynomials in the coordinates of the polyhedron vertices. A predicate is degenerate when [Formula: see text] is a zero of a common factor of [Formula: see text] and [Formula: see text]. We present an efficient degeneracy detection algorithm based on a one-time factoring of all the univariate polynomials over the ring of multivariate polynomials in the vertex coordinates. Our algorithm is 3500 times faster than the standard algorithm based on greatest common divisor computation. It reduces the share of degeneracy detection in our free space computations from 90% to 0.5% of the running time.

The computation of the degree of the greatest common divisor of three Bernstein basis polynomials

This paper considers the application of the Sylvester resultant matrix to the computation of the degree of the greatest common divisor (GCD) of three Bernstein basis polynomials f(y),g(y) and h(y). It is shown that the governing equations can be written in two forms, which lead to different Sylvester matrices. The first form requires that the polynomials be considered in pairs, but different pairs of polynomials may yield different computational answers, for example, the solution of the computations GCD(f,g) and GCD(g,h) may differ from the solution of the computations GCD(f,g) and GCD(f,h), depending on f(y),g(y) and h(y). This problem does not arise when the second form is considered, which requires that the three polynomials be considered simultaneously. Complications arise in both forms because of the combinatorial terms in the Bernstein basis functions, which cause the entries of the matrices to span several orders of magnitude, even if the coefficients of the polynomials are of the same order of magnitude. It is shown that the adverse effects of this wide range of magnitudes can be mitigated by postmultiplying both forms of the Sylvester matrix by a diagonal matrix of combinatorial terms and preprocessing f(y),g(y) and h(y) by three operations. Results of GCD computations from the two forms of the Sylvester matrix when f(y),g(y) and h(y) are perturbed by noise, and with the omission and inclusion of the preprocessing operations, are shown.

Computing Sparse GCD of Multivariate Polynomials via Polynomial Interpolation

The problem of computing the greatest common divisor (GCD) of multivariate polynomials, as one of the most important tasks of computer algebra and symbolic computation in more general scope, has been studied extensively since the beginning of the interdisciplinary of mathematics with computer science. For many real applications such as digital image restoration and enhancement, robust control theory of nonlinear systems, L1-norm convex optimization in compressed sensing techniques, as well as algebraic decoding of Reed-Solomon and BCH codes, the concept of sparse GCD plays a core role where only the greatest common divisors with much fewer terms than the original polynomials are of interest due to the nature of problems or data structures. This paper presents two methods via multivariate polynomial interpolation which are based on the variation of Zippel’s method and Ben-Or/Tiwari algorithm, respectively. To reduce computational complexity, probabilistic techniques and randomization are employed to deal with univariate GCD computation and univariate polynomial interpolation. The authors demonstrate the practical performance of our algorithms on a significant body of examples. The implemented experiment illustrates that our algorithms are efficient for a quite wide range of input.

Reversible circuit design for GCD computation in cryptography algorithms

SummaryThe implementation of cryptography algorithms is constantly observing threat from multiple techniques in breaking secure system. The current trend in breaking the secure system of cryptography is by power analysis technique. In order to break the secure key, the energy consumed in the digital circuit during computation is measured. This technique is commonly known as differential power analysis, explored by the hackers to break the secure systems. To circumvent these type of attacks, it is necessary to explore different designs, which dissipate less energy. Ideally, reversible circuits dissipate zero energy. We present a new reversible architecture for greatest common divisor (GCD) computation using modified Binary GCD algorithm. We present the generalized design methodology of reversible GCD computation unit. We compare the proposed GCD computation design with the existing design. The proposed reversible GCD architecture takes less number of iterations compared with the existing GCD architecture in the literature. The proposed design outperforms the existing GCD design in terms of Quantum Cost, Gate Count, and Ancilla Inputs. Copyright © 2016 John Wiley & Sons, Ltd.

Efficient Implementation of FDFM Approach for Euclidean Algorithms on the FPGA

The FDFM (Few DSP slices and Few block Memories) approach is an efficient approach which implements a processor core executing a particular algorithm using few DSP slices and few block RAMs in a single FPGA. Since a processor core based on the FDFM approach uses few hardware resources, hundreds of processor cores working in parallel can be implemented in an FPGA. The main contribution of this paper is to develop a processor core that executes Euclidean algorithm computing the GCD (Greatest Common Divisor) of two large numbers in an FPGA. This processor core that we call GCD processor core uses only one DSP slice and one block RAM, and 1280 GCD processors can be implemented in a Xilinx Virtex-7 family FPGA XC7VX485T-2. The experimental results show that the performance of this FPGA implementation using 1280 GCD processor cores is 0.0904μs per one GCD computation for two 1024-bit integers. Quite surprisingly, it is 3.8 times faster than the best GPU implementation and 316 times faster than a sequential implementation on the Intel Xeon CPU.

A rigorous version of R.P. Brent's model for the binary Euclidean algorithm

The binary Euclidean algorithm is a modification of the classical Euclidean algorithm for computation of greatest common divisors which avoids ordinary integer division in favour of division by powers of two only. The expectation of the number of steps taken by the binary Euclidean algorithm when applied to pairs of integers of bounded size was first investigated by R.P. Brent in 1976 via a heuristic model of the algorithm as a random dynamical system. Based on numerical investigations of the expectation of the associated Ruelle transfer operator, Brent obtained a conjectural asymptotic expression for the mean number of steps performed by the algorithm when processing pairs of odd integers whose size is bounded by a large integer. In 1998 B. Vallée modified Brent's model via an induction scheme to rigorously prove an asymptotic formula for the average number of steps performed by the algorithm; however, the relationship of this result with Brent's heuristics remains conjectural. In this article we establish previously conjectural properties of Brent's transfer operator, showing directly that it possesses a spectral gap and preserves a unique continuous density. This density is shown to extend holomorphically to the complex right half-plane and to have a logarithmic singularity at zero. By combining these results with methods from classical analytic number theory we prove the correctness of three conjectured formulae for the expected number of steps, resolving several open questions promoted by D.E. Knuth in The Art of Computer Programming.

A novel fast hybrid GCD computation algorithm

We propose a novel algorithm for integer’s greatest common divisor (GCD) computation that hybridises both Euclidian and binary algorithms according to a new schema, in order to accelerate the GCD computation especially in the case of large bit difference between the two inputs. The proposed algorithm run slightly faster than existing algorithms and is very much easier to implement. We provide a simple proof of correctness for the algorithm, and we show that best performances can be achieved for large integers when subtraction operator complexity is relatively low with respect to the division one.

Structured matrix methods for the computation of multiple roots of a polynomial

This paper considers the application of structured matrix methods for the computation of multiple roots of a polynomial. In particular, the given polynomial f(y) is formed by the addition of noise to the coefficients of its exact form fˆ(y), and the noise causes multiple roots of fˆ(y) to break up into simple roots. It is shown that structured matrix methods enable the simple roots of f(y) that originate from the same multiple root of fˆ(y) to be ‘sewn’ together, which therefore allows the multiple roots of fˆ(y) to be computed. The algorithm that achieves these results involves several greatest common divisor computations and polynomial deconvolutions, and special care is required for the implementation of these operations because they are ill-posed. Computational examples that demonstrate the theory are included, and the results are compared with the results from MultRoot, which is a suite of Matlab programs for the computation of multiple roots of a polynomial.

A New Systolic Architecture for Modular Division

A new systolic architecture for modular division is proposed in this paper. The architecture is based on a new hardware algorithm for modular division that is extended from the plus-minus algorithm for the greatest common divisor computation. Both the area complexity and the time complexity of the new architecture are linear with respect to the operand bit length. Compared to the architecture using a redundant number representation proposed by Kaihara and Takagi, the new architecture allows a doubled throughput due primarily to the greatly reduced critical path delay, while the area is only about 20 percent larger. Moreover, it is shown that, with a small addition to the control logic, the architecture can also be used to perform Montgomery modular multiplication, thereby efficiently realizing a unified modular multiplier/divider

Matrix pencil methodologies for computing the greatest common divisor of polynomials: hybrid algorithms and their performance

The computation of the greatest common divisor (GCD) of several polynomials is a problem that emerges in many fields of applications. The GCD computation has a non-generic nature and thus its numerical computation is a hard problem. In this paper we examine the family of matrix pencil methods for GCD computation and investigate their performance as far as their complexity, error analysis and their effectiveness for evaluating approximate solutions. The relative merits of the various variants of such methods are examined for the different cases of sets of polynomials with varying number of elements and degree. The developed algorithms combine symbolical and numerical programming and this is what we define here as hybrid computations. The combination of numerical operations with symbolical programming can improve the nature of the methods and guarantees the stability of the algorithm. Furthermore, it emphasizes the significance of hybrid computations in complex problems such as the computation of GCD. All methods are tested thoroughly for several sets of polynomials and the results are presented in tables.

Computation of Approximate Polynomial GCDs and an Extension

Computation of approximate polynomial greatest common divisors (GCDs) is important both theoretically and due to its applications to control linear systems, network theory, and computer-aided design. We study two approaches to the solution so far omitted by the researchers, despite intensive recent work in this area. Correlation to numerical Padé approximation enabled us to improve computations for both problems (GCDs and Padé). Reduction to the approximation of polynomial zeros enabled us to obtain a new insight into the GCD problem and to devise effective solution algorithms. In particular, unlike the known algorithms, we estimate the degree of approximate GCDs at a low computational cost, and this enables us to obtain certified correct solution for a large class of input polynomials. We also restate the problem in terms of the norm of the perturbation of the zeros (rather than the coefficients) of the input polynomials, which leads us to the fast certified solution for any pair of input polynomials via the computation of their roots and the maximum matchings or connected components in the associated bipartite graph.

Numerical computation of the least common multiple of a set of polynomials

The paper presents a number of properties of the least common multiple (LCM) m(s) of a given set of polynomials P. These results lead to the formulation of a new procedure for computing the LCM that avoids the computation of roots. This procedure involves the computation of the greatest common divisor (GCD) z(s) of a set of polynomials T derived from P, and the factorisation of the product of the original set P, p(s) as p(s) = m(s)·z(s). The symbolic procedure leads to a numerical one, where robust methods for the computation of GCD are first used. In this numerical method the approximate factorisation of polynomials is an important part of the overall algorithm. The latter problem is handled by studying two associated problems: evaluation of order of approximation and the optimal completion problem. The new method provides a robust procedure for the computation of LCM and enables the computation of approximate values, when the original data are inaccurate.

On locally invertible rate-1/n convolutional encoders

A locally invertible convolutional encoder has a local inverse defined as a full rank w/spl times/w matrix that specifies a one-to-one mapping between equal-length blocks of information and encoded bits. In this correspondence, it is shown that a rate-1/n convolutional encoder is nondegenerate and noncatastrophic if and only if it is locally invertible. Local invertibility is used to obtain upper and lower bounds on the number of consecutive zero-weight branches in a convolutional codeword. Further, existence of a local inverse can be used as an alternate test for noncatastrophicity instead of the usual approach involving computation of the greatest common divisor of n polynomials.

A generalized Euclidean algorithm for geometry theorem proving

In the first part of this paper, we present an algorithm that computes an unmixed-dimensional decomposition of a varietyV. EachVi in the decompositionV=V1U...UVm is given by a finite set of polynomials which represents the generic points of the irreducible components ofVi. The basic operation in our algorithm is the computation of greatest common divisors of univariate polynomials over extension fields given by regular chains. No factorization is needed. In the second part, this algorithm is applied to geometry theorem proving. We show that it can be used for deciding whether geometry statements are generically true or whether they are true under given nondegeneracy conditions. If a geometry statement is generically true, the simplest nondegeneracy condition with respect to a lexicographical degree ordering can be constructed by means of our algorithm.

A Generalized Euclidean Algorithm for Computing Triangular Representations of Algebraic Varieties

We present an algorithm that computes an unmixed-dimensional decomposition of an arbitrary algebraic variety V. Each V i in the decomposition V = V 1 ∩ … ∩ V m is given by a finite set of polynomials which represents the generic points of the irreducible components of V i . The basic operation in our algorithm is the computation of greatest common divisors of univariate polynomials over extension fields. No factorization is needed. Some of the main problems in polynomial ideal theory can be solved by means of our algorithm: we show how the dimension of an ideal can be computed, systems of algebraic equations can be solved, and radical membership can be decided. Our algorithm has been implemented in the computer algebra system MAPLE. Timings on well-known examples from computer algebra literature are given.

Practical cellular dividers

A discussion is presented of parallel division algorithms that can be classified among modified higher radix nonrestoring online division methods, where redundant representations are extensively utilized to speed up the operation. The network realizations of these algorithms are cellular, or even systolic with exclusively local control; they have both size (area) and time of O(n), where n is the length of the dividend representation. The same structures can also be used as a signed, digit-serial multiplier. When suitably equipped with some control and a few registers, the divider/multiplier brings remarkable performance to large modular arithmetic, RSA cryptography, and greatest common divisor computations. They are also of interest for the design of floating-point units and signal processing applications.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Systolic VLSI Arrays for Polynomial GCD Computation

The problem of finding a greatest common divisor (GCD) of any two nonzero polynomials is fundamental to algebraic and symbolic computations, as well as to the decoder implementation for a variety of error-correcting codes. This paper describes new systolic arrays that can lead to efricient VLSI solutions to both the GCD problem and the extended GCD problem.

P-adic reconstruction of rational numbers

In a recent paper, Wang [1981] introduces a p-adic algorithm for the construction of partial fraction decompositions. This differs from the usual p-adic algorithms for factorisation or the computation of greatest common divisors ([Wang, 1978], [Wang, 1980], [Moore &amp; Norman, 1981]) in that the p-adic image is used to reconstruct rational numbers, rather than integers.

Computation of the GCD of polynomials of several variables on a MPC

The article is devoted to computer calculations with polynomials of several variables, in particular, to the construction of sets of large-block polynomial operations and to the concretization of the generalized algorithm of Euclid for computing the greatest common divisor (GCD) of two polynomials of several variables.