Analysis of the subtractive algorithm for greatest common divisors

On the residue classes of integer sequences satisfying a linear three term recurrence formula

: The relations between approximants of the regular continued fraction expansion for square root of C + L and the approximations given by an extension of Newton's formula are given. The general relationship is given between the approximants of a regular periodic continued fraction expansion for square root of C + L and the approximations given by the extended Newton's formula. These formulas are modified to the case of symmetric periodic continued fraction expansions. Rules are given that relate the approximants of mixed periodic continued fraction expansions for square root of C + L to approximations given by the extended Newton's formula. (Author)

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.

Motivated by problems in random continued fraction expansions, this research describes the design and implementation on θ-expansions of a real number, x in (0, θ) where 0 < θ < 1. Initially, for such a number θ, the convergent of θ-expansions is computed with comparison to the regular continued fraction (RCF)-convergent based on the samples of the approximated value, x. θ-expansions algorithm and continued fraction algorithm are applied in a Maple software to compute the θ-convergent and RCF-convergent. In this respect, the purpose of this research is to examine the growth rate of convergent for both expansions and provide the theoretical behaviours of θ-convergent in θ-expansions. Numerical results based on these samples show 3 conditions of convergence growth rate, which are C2 and C4 give the best decimal approximations in θ-expansions, whereas, Cn be the best approximations in RCF expansions. Therefore, this research reveals that various value of θ affect the performance on their growth rate of convergent for these two expansions. As θ approaching to 1, it reduces their convergence errors. The analysis of the experimental results reveals the RCF-convergent give a better performance compared to θ-convergent as its yield to less convergence errors. In addition, the value of convergent do converge to the approximated value, x and that is how the name of convergent comes.

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.

This paper describes the implementation of recursive algorithms in unique factorization domains, namely multivariate polynomial greatest common divisors (gcd) and factorization into irreducible parts in the Java computer algebra library (JAS). The implementation of gcds, resultants and factorization is part of the essential building blocks for any computation in algebraic geometry, in particular in automated deduction in geometry. There are various implementations of these algorithms in procedural programming languages. Our aim is an implementation in a modern object oriented programming language with generic data types, as it is provided by Java programming language. We exemplify that the type design and implementation of JAS is suitable for the implementation of several greatest common divisor algorithms and factorization of multivariate polynomials. Due to the design we can employ this package in very general settings not commonly seen in other computer algebra systems. As for example, in the coefficient arithmetic for advanced Grobner basis computations like in polynomial rings over rational function fields or (finite, commutative) regular rings. The new package provides factory methods for the selection of one of the several implementations for non experts. Further we introduce a parallel proxy for gcd implementations which runs different implementations concurrently.

In this work we deal with extreme value theory in the context of continued fractions using techniques from probability theory, ergodic theory and real analysis. We give an upper bound for the rate of convergence in the Doeblin–Iosifescu asymptotics for the exceedances of digits obtained from the regular continued fraction expansion of a number chosen randomly from$(0,1)$according to the Gauss measure. As a consequence, we significantly improve the best known upper bound on the rate of convergence of the maxima in this case. We observe that the asymptotics of order statistics and the extremal point process can also be investigated using our methods.

It is well known that the regular continued fraction expansion of a quadratic irrational is symmetric about its centre; we refer to this symmetry as horizontal. However, an additional vertical symmetry is exhibited by the continued fraction expansions arising from a family of quadratics known as Schinzel sleepers. This paper provides a method for generating every Schinzel sleeper and investigates their period lengths as well as both their horizontal and vertical symmetries.

On the metric theory of [formula omitted]-adic continued fractions

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.

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 &amp; 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 obvious way to compute the continued fraction of a real number α > 1 is to compute a very accurate numerical approximation of α, and then to iterate the well-known truncate-and-invert step which computes the next partial quotient a = bαc and the next complete quotient α′ = 1/(α − a). This method is called the basic method. In the course of this process precision is lost, and one has to take precautions to stop before the partial quotients become incorrect. Lehmer [6] gives a safe stopping-criterion, and a trick to reduce the amount of multi-length arithmetic. Schonhage [12] describes an algorithm for computing the greatest common divisor of u and v and the related continued fraction expansion of u/v in O(n log n log log n) steps if both u and v do not exceed 2. A disadvantage of this approach is that if we wish to extend the list of partial quotients computed from an initial approximation of α, we have to compute a more accurate initial approximation of α, compute the new complete quotient using this new approximation and the partial quotients already computed from the old approximation, and then extend the list of partial quotients using that new complete quotient (we notice that Shiu in [13, p.1312] incorrectly states that all the previous calculations have to be repeated, but the partial quotients computed so far don’t have to be recomputed). Bombieri and Van der Poorten [1], and Shiu [13] have recently proposed a remedy for this problem. They give a formula for computing a rational approximation of the next complete convergent from the first n partial quotients. From that complete convergent about n new partial quotients can be computed. So each step gives an approximate doubling of the number of partial quotients. To start the method, a few partial quotients have to be computed with the basic or indirect method. In [1] this method is proposed for algebraic numbers (which are zeros of polynomials) of degree ≥ 3, whereas Shiu also applies it to more general numbers, namely to transcendental numbers that can be defined as the zero of a function for which the logarithmic derivative at a rational point can be computed with arbitrary precision. This includes numbers like π, log π, and log 2. For each of thirteen different numbers, Shiu computes 10000 partial quotients. Their frequency distributions are compared with the one which almost all numbers should obey, according to the Khintchine– Levy theory [3, 7]. No significant deviations from this theory are reported. Shiu calls his method the direct method. Curiously, Shiu does not refer to what we would call the polynomial method for algebraic numbers [2, 5, 11] of degree ≥ 3, which computes the partial quotients of α using only the coefficients of its defining polynomial. Moreover, Shiu gives neither implementational

The paper is related to the classical problem of the rational approximation of analytic functions of one or several variables, particulary the issues that arise in the construction and studying of continued fraction expansions and their multidimensional generalizations—branched continued fraction expansions. We used combinations of three- and four-term recurrence relations of the generalized hypergeometric function 3F2 to construct the branched continued fraction expansions of the ratios of this function. We also used the concept of correspondence and the research method to extend convergence, already known for a small region, to a larger region. As a result, we have established some convergence criteria for the expansions mentioned above. It is proved that the branched continued fraction expansions converges to the functions that are an analytic continuation of the ratios mentioned above in some region. The constructed expansions can approximate the solutions of certain differential equations and analytic functions, which are represented by generalized hypergeometric function 3F2. To illustrate this, we have given a few numerical experiments at the end.

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.