Approximate Greatest Common Divisor Research Articles

Generalized NTRU Algorithms on Algebraic Rings

The NTRU (Number Theory Research Unit) is a prominent post-quantum public key cryptography algorithm and a current focus of research. Although many NTRU variants have been proposed, a comprehensive generalization of these variants is still lacking. To address this issue, we propose the G-NTRU (Generalized NTRU), an algorithmic framework for NTRU variants on algebraic rings. This framework generalizes specific ring extensions to more general algebraic structures, allowing the unification of various NTRU variants. We then analyze the key properties of the G-NTRU, including correctness, lattice-based characteristics, and security. The semantic security of the G-NTRU is demonstrated through the introduction of the G-AGCD (Generalized Approximate Greatest Common Divisor). To validate the generality of the G-NTRU framework, we introduce the CNTRU (Complex Number NTRU), a variant of the NTRU over the ring of complex numbers. The CNTRU shares the same properties as the G-NTRU, further confirming the versatility of the G-NTRU in studying NTRU variants over algebraic rings.

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.

Enhancing Global Blockchain Privacy via a Digital Mutual Trust Mechanism

Blockchain technology, initially developed as a decentralized and transparent mechanism for recording transactions, faces significant privacy challenges due to its inherent transparency, exposing sensitive transaction data to all network participants. This study proposes a blockchain privacy protection algorithm that employs a digital mutual trust mechanism integrated with advanced cryptographic techniques to enhance privacy and security in blockchain transactions. The contribution includes the development of a new dynamic Byzantine consensus algorithm within the Practical Byzantine Fault Tolerance framework, incorporating an authorization mechanism from the reputation model and a proof consensus algorithm for robust digital mutual trust. Additionally, the refinement of homomorphic cryptography using the approximate greatest common divisor technique optimizes the encryption process to support complex operations securely. The integration of a smart contract system facilitates automatic and private transaction execution across the blockchain network. Experimental evidence demonstrates the superior performance of the algorithm in handling privacy requests and transaction receipts with reduced delays and increased accuracy, marking a significant improvement over existing methods.

A novel position estimation method for wayside pass-by noise sources based on Doppler effect correction

The localization of moving noise sources has garnered substantial attention in recent years, prompting researchers to develop a range of methodologies for positioning sound sources across diverse scenarios. Despite this progress, the intricate interplay of complex spatial sound fields and the Doppler effect during motion presents a formidable challenge in achieving accurate sound source localization. To address these challenges, this paper proposes a novel method for estimating positions of wayside pass-by noise sources based on Doppler Effect Correction (Do-bPE). The proposed method capitalizes on a spatial microphone array to capture signals emanating from moving sound sources. Each channel’s sound signal is transformed into the time–frequency domain using Short Time Fourier Transform (STFT). The Approximate Greatest Common Divisor (AGCD) algorithm is then applied to extract the instantaneous frequency of the moving source from the time–frequency spectrogram. By leveraging nonlinear least squares fitting and the acoustic Doppler model, this instantaneous frequency facilitates the kinematic parameters estimation of Doppler effect, which provides the distances between each microphone within the array and the trajectory of moving sound source. Then, by employing the weighted least squares method to solve the relationship between the sound source and the array microphones, an accurate spatial position for the sound source can be obtained. The method successfully applies the Doppler effect to estimate the source distances, subsequently refining position estimation accuracy through microphone array processing. Simulation and experimental results show that the method’s positioning accuracy and robustness are significantly improved compared to existing approaches.

Privacy preserving via multi-key homomorphic encryption in cloud computing

As the world grapples with the COVID-19 and its variants, multi-user collaboration by means of cloud computing is ubiquitous. How to make better use of cloud resources while preventing user privacy leakage has become particularly important. Multi-key homomorphic encryption(MKHE) can effectively deal with the privacy disclosure issue during the multi-user collaboration in the cloud computing setting. Firstly, we improve the DGHV homomorphic scheme by modifying the selection of key and the coefficients in encryption, so as to eliminate the restriction on the parity of the ciphertext modulus in the public key. On this basis, we further propose a DGHV-type MKHE scheme based on the number theory. In our scheme, an extended key is introduced for ciphertext extension, and we prove that it is efficient in performance analysis. The semantic security of our schemes is proved under the assumption of error-free approximate greatest common divisor and the difficulty of large integer factorization. Furthermore, the simulation experiments show the availability and computational efficiency of our MKHE scheme. Therefore, our scheme is suitable for the multi-user scenario in cloud environment.

Approximate greatest common divisor of several polynomials from Hankel matrices

This paper is devoted to present a new method for computing the approximate Greatest Common Divisor (GCD) of several polynomials (not pairwise) from the generalized Hankel matrix. Our approach based on the calculation of cofactors is tested for several sets of polynomials.

Approximate GCD by relaxed NewtonSLRA algorithm

We propose a better algorithm for approximate greatest common divisor (approximate GCD) of univariate polynomials in terms of robustness and distance, based on the NewtonSLRA algorithm that is a solver for the structured low rank approximation (SLRA) problem. Our algorithm mainly enlarges the tangent space in the NewtonSLRA algorithm and adapts it to a certain weighted Frobenius norm. Moreover, we propose some improvement in computing time.

Cryptanalysis of Modular Exponentiation Outsourcing Protocols

Abstract Public-key cryptographic primitives are time consuming for resource-constrained devices. A classical problem is to securely offload group exponentiations from a (comparatively) weak device—the client—to an untrusted more powerful device—the server. A delegation protocol must usually meet two security objectives: privacy—the exponent or the base should not be revealed to a passive adversary—and verifiability—a malicious server should not be able to make the client accept an invalid value as the result of the delegated computation. Most proposed protocols relies on a secret splitting of the exponent and the base, and a considerable amount of literature has been devoted to their analysis. Recently, Su et al. (Su, Q., Zhang, R. and Xue, R. (2020) Secure outsourcing algorithms for composite modular exponentiation based on single untrusted cloud. Comput. J., 63, 1271.) and Rangasamy and Kuppusamy (Rangasamy, J. and Kuppusamy, L. (2018) Revisiting Single-Server Algorithms for Outsourcing Modular Exponentiation. In Chakraborty, D. and Iwata, T. (eds), Progress in Cryptology - INDOCRYPT 2018: 19th International Conference in Cryptology in India, New Delhi, India, December 912, Vol. 11356, Lecture Notes in Computer Science. Springer, Heidelberg, Germany, pp. 320. proposed outsourcing protocols for modular exponentiations. They claim that their protocols achieve security (privacy and verifiability). We show that these claims are flawed and that their schemes are broken beyond repair. They remain insecure even if one increases significantly the proposed parameters (and consequently the protocols computational and communication complexities). Our attacks rely on standard lattice-based cryptanalytic techniques, namely the Coppersmith methods to find small integer zeroes of modular multivariate polynomials and simultaneous Diophantine approximation methods for the so-called approximate greatest common divisor problem.

An approximate factorisation of three bivariate Bernstein basis polynomials defined in a triangular domain

This paper considers an approximate factorisation of three bivariate Bernstein basis polynomials that are defined in a triangular domain. This problem is important for the computation of the intersection points of curves in computer-aided design systems, and it reduces to the determination of an approximate greatest common divisor (AGCD) d(y) of the polynomials. The Sylvester matrix and its subresultant matrices of these three polynomials are formed and it is shown that there are four forms of these matrices. The most difficult part of the computation is the determination of the degree of d(y) because it reduces to the determination of the rank loss of these matrices. This computation is made harder by the presence of trinomial terms in the Bernstein basis functions because they cause the entries of the matrices to span many orders of magnitude. The adverse numerical effects of this wide range of magnitudes of the entries of the four forms of the Sylvester matrix and its subresultant matrices are mitigated by processing the polynomials before these matrices are formed. It is shown that significantly improved results are obtained if the polynomials are processed before computations are performed on their Sylvester matrices and subresultant matrices.

Algorithms for CRT-variant of Approximate Greatest Common Divisor Problem

AbstractThe approximate greatest common divisor problem (ACD) and its variants have been used to construct many cryptographic primitives. In particular, the variants of the ACD problem based on Chinese remainder theorem (CRT) are being used in the constructions of a batch fully homomorphic encryption to encrypt multiple messages in one ciphertext. Despite the utility of the CRT-variant scheme, the algorithms that secures its security foundation have not been probed well enough.In this paper, we propose two algorithms and the results of experiments in which the proposed algorithms were used to solve the variant problem. Both algorithms take the same time complexity $\begin{array}{} \displaystyle 2^{\tilde{O}(\frac{\gamma}{(\eta-\rho)^2})} \end{array}$ up to a polynomial factor to solve the variant problem for the bit size of samples γ, secret primes η, and error bound ρ. Our algorithm gives the first parameter condition related to η and γ size. From the results of the experiments, it has been proved that the proposed algorithms work well both in theoretical and experimental terms.

Approximate square-free part and decomposition

Square-free decomposition is one of fundamental computations for polynomials. However, any conventional algorithm may not work for polynomials with a priori errors on their coefficients. There are mainly two approaches to overcome this empirical situation: approximate polynomial GCD (greatest common divisor) and the nearest singular polynomial. In this paper, we show that these known approaches are not enough for detecting the nearest square-free part (which has no multiple roots) within the given upper bound of perturbations (a priori errors), and we propose a new definition and a new method to detect a square-free part and its decomposition numerically by following a recent framework of approximate polynomial GCD.

Toward the best algorithm for approximate GCD of univariate polynomials

Approximate polynomial GCD (greatest common divisor) of polynomials with a priori errors on their coefficients, is one of interesting problems in Symbolic-Numeric Computations. In fact, there are many known algorithms: QRGCD, UVGCD, STLN based methods, Fastgcd and so on. The fundamental question of this paper is “which is the best?” from the practical point of view, and subsequently “is there any better way?” by any small extension, any effect by pivoting, and any combination of sub-routines along the algorithms. In this paper, we consider a framework that covers those algorithms and their sub-routines, and makes their sub-routines being interchangeable between the algorithms (i.e. disassembling the algorithms and reassembling their parts). By this framework along with/without small new extensions and a newly adapted refinement sub-routine, we have done many performance tests and found the current answer. In summary, 1) UVGCD is the best way to get smaller tolerance, 2) modified Fastgcd is better for GCD that has one or more clusters of zeros with large multiplicity, and 3) modified ExQRGCD is better for GCD that has no cluster of zeros.

Approximate GCD in Bernstein basis

In general, finding the Greatest Common Divisor (GCD) of two exactly-known univariate polynomials is a well understood problem. However, it is also known that the GCD problem for noisy polynomials (polynomials with errors in their coefficients) is ill-posed. More precisely, a small error in coefficients of polynomials P and Q with a non-trivial GCD generically leads to a constant GCD. We note that the choice of basis makes no difference to this difficulty.

The GPGCD algorithm with the Bézout matrix

With the progress of algebraic computations on polynomials and matrices, we are paying more attention to approximate algebraic algorithms. Among approximate algebraic algorithms, those for calculating approximate greatest common divisor (GCD) consider a pair of given polynomials f and g that are relatively prime in general, and find f and g which are close to f and g , respectively, in the sense of polynomial norm, and have the GCD of certain degree. The algorithms can be classified into two categories: 1) for a given tolerance (magnitude) of || f - f|| and || g - g||, make the degree of approximate GCD as large as possible, and 2) for a given degree d , minimize the magnitude of || f - f|| and || g - g||.

Attack and Improvement on a Symmetric Fully Homomorphic Encryption Scheme

Fully homomorphic encryption (FHE) has been regarded as the holy grail of cryptography and supports meaningful computations on encrypted data. Recently, Kumer et al. proposed a symmetric FHE scheme based on the Euler theorem. However, we illustrate in the paper that the secret key can be recovered from the homomorphic computation key and only a pair of known plaintext/ciphertext by computing the greatest common divisor with the Euclidean algorithm. Our observations are well supported by the numerical experiments on the suggested parameters. Furthermore, an improvement in the scheme is presented. We mimic Dijk et al.'s idea to illustrate that the improvement can be reduced to the approximate greatest common divisor (AGCD) problem. Several known attacks, including the proposal in this paper, are also examined.

An ODE-based method for computing the approximate greatest common divisor of polynomials

Computing the greatest common divisor of a set of polynomials is a problem which plays an important role in different fields, such as linear system, control, and network theory. In practice, the polynomials are obtained through measurements and computations, so that their coefficients are inexact. This poses the problem of computing an approximate common factor. We propose an improvement and a generalization of the method recently proposed in Guglielmi et al. (SIAM J. Numer. Anal. 55, 1456–1482, 2017), which restates the problem as a (structured) distance to singularity of the Sylvester matrix. We generalize the algorithm in order to work with more than 2 polynomials and to compute an Approximate GCD (Greatest Common Divisor) of degree k ≥ 1; moreover, we show that the algorithm becomes faster by replacing the eigenvalues by the singular values.

Joint Detection and Localization of an Unknown Number of Sources Using the Algebraic Structure of the Noise Subspace

Source localization and spectral estimation are among the most fundamental problems in statistical and array signal processing. Methods which rely on the orthogonality of the signal and noise subspaces, such as Pisarenko's method, MUSIC, and root-MUSIC are some of the most widely used algorithms to solve these problems. As a common feature, these methods require both apriori knowledge of the number of sources, and an estimate of the noise subspace. Both requirements are complicating factors to the practical implementation of the algorithms, and when not satisfied exactly, can potentially lead to severe errors. In this paper, we propose a new localization criterion based on the algebraic structure of the noise subspace that is described for the first time to the best of our knowledge. Using this criterion and the relationship between the source localization problem and the problem of computing the greatest common divisor (GCD), or more practically approximate GCD, for polynomials, we propose two algorithms which adaptively learn the number of sources and estimate their locations. Simulation results show a significant improvement over root-MUSIC in challenging scenarios such as closely located sources, both in terms of detection of the number of sources and their localization over a broad and practical range of SNRs. Further, no performance sacrifice in simple scenarios is observed.

Blind image deconvolution via Hankel based method for computing the GCD of polynomials

In this paper we present an algorithm, that is based on computing approximate greatest common divisors (GCD) of polynomials, for solving the problem of blind image deconvolution. Specifically, we design a specialized algorithm for computing the GCD of bivariate polynomials corresponding to z-transforms of blurred images to recover the original image. The new algorithm is based on the fast GCD algorithm for univariate polynomials in which the successive transformation matrices are upper triangular Toeplitz matrices. The complexity of our algorithm is O(n2log(n)) where the size of blurred images is n×n. All algorithms have been implemented in Matlab and experimental results with synthetically blurred images are included to illustrate the effectiveness of our approach.

Identification of Forward and Feedback Transfer Functions in Closed-Loop Systems with Feedback Delay

The subject of this paper is the identification of closed-loop continuous-time systems, with delayed feedback action, from sampled input-output measurements. In particular, a method for the identification of both forward and feedback subsystems is presented that requires only the knowledge of their orders and of the time-delay introduced in the feedback loop. The identification procedure is divided in two parts. The first step captures the behavior of the whole closed-loop system, estimating its transfer function. In the second step two different approaches are presented to separate the contributions of the forward and feedback subsystems in the loop. One of these approaches exploits a system theoretical method to compute the approximate greatest common divisor between polynomials. Numerical results validate the effectiveness of the proposed technique.

A non-linear structure-preserving matrix method for the computation of the coefficients of an approximate greatest common divisor of two Bernstein polynomials

This paper describes a non-linear structure-preserving matrix method for the computation of the coefficients of an approximate greatest common divisor (AGCD) of degree t of two Bernstein polynomials f(y) and g(y). This method is applied to a modified form St(f,g)Qt of the tth subresultant matrix St(f,g) of the Sylvester resultant matrix S(f,g) of f(y) and g(y), where Qt is a diagonal matrix of combinatorial terms. This modified subresultant matrix has significant computational advantages with respect to the standard subresultant matrix St(f,g), and it yields better results for AGCD computations. It is shown that f(y) and g(y) must be processed by three operations before St(f,g)Qt is formed, and the consequence of these operations is the introduction of two parameters, α and θ, such that the entries of St(f,g)Qt are non-linear functions of α,θ and the coefficients of f(y) and g(y). The values of α and θ are optimised, and it is shown that these optimal values allow an AGCD that has a small error, and a structured low rank approximation of S(f,g), to be computed.