An asymptotically fast probabilistic algorithm for computing polynomial GCD's over an algebraic number field

In this correspondence, a new viewpoint is proposed for estimating an image from its distorted versions in presence of noise without the a priori knowledge of the distortion functions. In z-domain, the desired image can be regarded as the greatest common polynomial divisor among the distorted versions. With the assumption that the distortion filters are finite impulse response (FIR) and relatively coprime, in the absence of noise, this becomes a problem of taking the greatest common divisor (GCD) of two or more two-dimensional (2-D) polynomials. Exact GCD is not desirable because even extremely small variations due to quantization error or additive noise can destroy the integrity of the polynomial system and lead to a trivial solution. Our approach to this blind deconvolution approximation problem introduces a new robust interpolative 2-D GCD method based on a one-dimensional (1-D) Sylvester-type GCD algorithm. Experimental results with both synthetically blurred images and real motion-blurred pictures show that it is computationally efficient and moderately noise robust.

We present new methods for computing the greatest common right divisor of polynomial matrices. These methods involve the recently studied generalized Sylvester and generalized Bezoutian resultant matrices, which require no polynomial operations. They can provide a row proper greatest common right divisor, test for coprimeness and calculate dual dynamical indices. The generalized resultant matrices are developments of the scalar Sylvester and Bezoutian resultants and many of the familiar properties of these latter matrices are demonstrated to have analogs with the properties of the generalized resultant matrices for matrix polynomials.

A matrix Euclidean algorithm induced by state space realization

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.

We propose an algorithm to compute a greatest common divisor (GCD) of univariate polynomials with large integer coefficients on Graphics Processing Units (GPUs). At the highest level, our algorithm relies on modular techniques to decompose the problem into subproblems that can be solved separately. Next, we employ resultant-based or matrix algebra methods to compute a GCD of each modular image in parallel. Our approach exhibits block structure to distribute the computation of a single modular GCD over several thread blocks, and thus to remove any hardware limitations on the maximal size of polynomials that can be handled. To "combine" all modular results, we have adopted Mixed-Radix Conversion (MRC) algorithm running on the GPU. Our approach shows a significant speed-up over host-based GCD algorithm from Maple 13.

The study of the Least Common Multiple (LCM) and the Greatest Common Divisor (GCD) explores the fundamental field of number theory. In addition to being essential to theoretical number theory, these mathematical ideas have extensive applications in computer science, algebra, and real-world settings like network routing and scheduling. Using techniques like the Euclidean algorithm for GCD calculation and prime factorization for LCM, the study starts by examining the fundamental concepts and notations of GCD and LCM. Then the author goes further into the detail about how these ideas are used to polynomial equations, highlighting how they may be used to solve challenging issues and simplify algebraic expressions. The studys practical applications, which demonstrate how GCD and LCM optimize scheduling tasks and boost network routing efficiency, are presented in the conclusion. The results of this study highlight the broad impact and adaptability of LCM and GCD in theoretical and applied mathematics, providing valuable insights for further research in these domains.

The objective of this is to show the application of Euclid's Lemma to calculate the Greatest Common Divisor in Elementary Education and evaluate the receptivity that the students had the method. We studied and wrote about the division in integers, divisibility and properties involved, the theorem euclidian division, the greatest common divisor, Euclid's Lemma, the euclidean algorithm, the Fundamental Theorem of Arithmetic and also on the Diophantine Equations they are an interesting application of the Greatest Common Divisor. We use a short course on Greatest Common Divisor in two elementary school classes and present some results obtained.

This paper presents a preliminary report on a new algorithm for computing the Greatest Common Divisor (GCD) of two multivariate polynomials over the integers. The algorithm is strongly influenced by the method used for factoring multivariate polynomials over the integers. It uses an extension of the Hensel lemma approach originally suggested by Zassenhaus for factoring univariate polynomials over the integers. We point out that the cost of the Modular GCD algorithm applied to sparse multivariate polynomials grows at least exponentially in the number of variables appearing in the GCD. This growth is largely independent of the number of terms in the GCD. The new algorithm, called the EZ (Extended Zassenhaus) GCD Algorithm, appears to have a computing bound which in most cases is a polynomial function of the number of terms in the original polynomials and the sum of the degrees of the variables in them. Especially difficult cases for the EZ GCD Algorithm are described. Applications of the algorithm to the computation of contents and square-free decompositions of polynomials are indicated.

Modular inversion, the multiplicative inverse of an integer in the ring of integers modulo a prime number, is widely used in public-key cryptography. However, it is one of the most computationally intensive operations, thus, it remains the main performance bottleneck for many cryptographic algorithms.This paper presents to the best of the author’s knowledge, the first FPGA-based hardware design for computing the multiplicative inverse using the recently proposed fast constant-time Greatest Common Divisor (GCD) algorithm. This paper introduces two distinct design architectures targeting different applications: (a) a full-width design and (b) a sequential design. The presented designs are compact, parameterizable, and scalable in terms of area and speed. The evaluation shows the proposed designs, which are constant-time and protect against timing-based attacks, outperform existing software and hardware implementations that use other modular inversion techniques. As a specific example, this work presents an evaluation focusing on the use of the multiplicative inverse hardware module to accelerate the ElGamal cryptosystem. The proposed design achieves a speed-up of 90% in the modular inverse calculation and a speed-up of 45% in the overall ElGamal decryption algorithm using our sequential hardware design of fast constant-time GCD algorithm.In addition to developing the fast hardware implementation, this work potentially opens up a new direction for designing cryptosystems: the inverse operation is often avoided when designing algorithms, due to its complexity. With the new hardware module, using the inverse becomes more tractable, making it more appealing to use in the design of new cryptosystems.

A new numerical method for the computation of the greatest common divisor (GCD) of an m-set of polynomials of R [s], Pm,d of maximal degree d, is presented. This method is based on a recently developed theoretical algorithm (Karcanias 1987) that uses elementary transformations and shifting operations; the present algorithm takes into account the non-generic nature of GCD and thus uses steps, which minimize the introduction of additional errors and defines the GCD in an approximate sense. For a given set Pm,d with a basis matrix Pm, the method defines first, the most orthogonal uncorrupted base Pr from the rows of Pm, where r = rank(Pm) ≤ m. By applying successively gaussian transformations and shifting, on the basis matrix Pr ε Rr×(d+1) we produce each time a new basis matrix Pz with z = rank(Pz) < r. The method terminates when the rank of Pz is approximately equal to 1; the coefficient vector of the GCD is then defined as a row of the unit rank matrix Pz. The method defines the exact degree of the GCD, successfully evaluates an approximate solution and works satisfactorily with large numbers of polynomials of any fixed degree.

Finding the greatest common divisor (GCD) of two given polynomials is a basic problem in algebraic computing. The problem is usually stated as follows: given the (real or complex) coefficients of two polynomials, compute the coefficients of their greatest common divisor.

Greatest common divisor matrices

We establish consequences of the moving form of Schmidt’s Subspace Theorem. Indeed, we obtain inequalities that bound the logarithmic greatest common divisor of moving multivariable polynomials evaluated at moving S S -unit arguments. In doing so, we complement recent work of Levin. As an additional application, we obtain results that pertain to the greatest common divisor problem for algebraic linear recurrence sequences. These observations are motivated by previous related works of Corvaja-Zannier, Levin, and others.

On greatest common divisor matrices and their applications

This research investigates on the numerical methods for computing the greatest common divisors (GCD) of two polynomials in the orthogonal basis without having to convert to the power series form. Previous implementations were conducted using the Gauss Elimination with partial pivoting (GEPP) and QR Householder algorithms, respectively. This work proceeds to seek for a better approximate solution by comparing the results of the implementations with the QR with column pivoting (QRCP) algorithm. The results reveal that QRCP is as competent as the GEPP algorithm, up to a certain degree, giving a reasonably good approximate solution. It is also found that normalizing the columns of the associated coefficient matrix slightly reduces the condition number of the matrix but has no significant effect on the GCD solutions when applying the GEPP and QR Householder algorithms. However equilibration of the columns by computing its ∞-norm is capable to improve the solution when QRCP is applied. Comparing the three algorithms on some test problems, QR Householder outperforms the rest and is able to give a good approximate solution in the worst case condition when the smallest element of the matrix is 1, the entries ranging up to 15 digits integers.