Approximate Greatest Common Divisor Research Articles

An Efficient Quantum Somewhat Homomorphic Symmetric Searchable Encryption

In 2009, Gentry first introduced an ideal lattices fully homomorphic encryption (FHE) scheme. Later, based on the approximate greatest common divisor problem, learning with errors problem or learning with errors over rings problem, FHE has developed rapidly, along with the low efficiency and computational security. Combined with quantum mechanics, Liang proposed a symmetric quantum somewhat homomorphic encryption (QSHE) scheme based on quantum one-time pad, which is unconditional security. And it was converted to a quantum fully homomorphic encryption scheme, whose evaluation algorithm is based on the secret key. Compared with Liang’s QSHE scheme, we propose a more efficient QSHE scheme for classical input states with perfect security, which is used to encrypt the classical message, and the secret key is not required in the evaluation algorithm. Furthermore, an efficient symmetric searchable encryption (SSE) scheme is constructed based on our QSHE scheme. SSE is important in the cloud storage, which allows users to offload search queries to the untrusted cloud. Then the cloud is responsible for returning encrypted files that match search queries (also encrypted), which protects users’ privacy.

A Public Key Compression Scheme for Fully Homomorphic Encryption Based on Quadratic Parameters With Correction

For efficiency improvement and public key size reduction, a new public key compression scheme is proposed for fully homomorphic encryption based on quadratic parameters with correction (QPC-PKC scheme). Compared with existing public key compression schemes, the size of the public key in the proposed scheme is reduced from $\tilde {\mathrm {O}}({\lambda }^{5})$ to $\tilde {\mathrm {O}}({\lambda }^{3.5})$ by reducing the number of subgroup public key elements and the element bit-lengths. Based on the construction mechanisms of the somewhat fully homomorphic encryption (SWHE), a QPC-PKC SWHE scheme is constructed and the parameter constraints are presented. The correctness and semantical security of the proposed QPC-PKC SWHE scheme are then proved based on the error-free approximate greatest common divisor assumption. Finally, the public key size performance of the QPC-PKC scheme is theoretically analyzed, while the public key sizes and running times of the QPC-PKC SWHE scheme are experimentally evaluated. The results show that the public key size of the proposed scheme is significantly reduced compared with the existing schemes, and the encryption efficiency of the QPC-PKC SWHE scheme is also improved as expected.

The computation of the degree of an approximate greatest common divisor of two Bernstein polynomials

This paper considers the computation of the degree t of an approximate greatest common divisor d(y) of two Bernstein polynomials f(y) and g(y), which are of degrees m and n respectively. The value of t is computed from the QR decomposition of the Sylvester resultant matrix S(f,g) and its subresultant matrices Sk(f,g), k=2,…,min⁡(m,n), where S1(f,g)=S(f,g). It is shown that the computation of t is significantly more complicated than its equivalent for two power basis polynomials because (a) Sk(f,g) can be written in several forms that differ in the complexity of the computation of their entries, (b) different forms of Sk(f,g) may yield different values of t, and (c) the binomial terms in the entries of Sk(f,g) may cause the ratio of its entry of maximum magnitude to its entry of minimum magnitude to be large, which may lead to numerical problems. It is shown that the QR decomposition and singular value decomposition (SVD) of the Sylvester matrix and its subresultant matrices yield better results than the SVD of the Bézout matrix, and that f(y) and g(y) must be processed before computations are performed on these resultant and subresultant matrices in order to obtain good results.

Learning traffic signal phase and timing information from low-sampling rate taxi GPS trajectories

Traffic signal phase and timing (TSPaT) information is valuable for various applications, such as velocity advisory systems, navigation systems, collision warning systems, and so forth. However, the acquisition of the TSPaT information in the city-scale is very challenging. In this paper, we propose a framework to learn the TSPaT information from low-sampling rate taxi GPS trajectories. Specifically, our framework could learn: the phasing scheme, i.e., the number of phases and the assignment of traffic movements to phases; timing plans, including the cycle length and green lengths of phases within a cycle, for each given fixed-time signalized intersection. In our framework, the cycle length is the first important parameters to be learned. We formalize the cycle length estimation problem as a general approximate greatest common divisor (AGCD) problem, and propose the most frequent AGCD (MFAGCD) algorithm to solve the problem. The MFAGCD algorithm is robust to noises and outliers, and could estimate the cycle length with a high accuracy using a small number of green-start times extracted from taxi GPS trajectories. Based the correlation between phases, we propose an all-direction joint determination method to jointly estimate green lengths using green-start times and cross-over times from all phases. The effectiveness of our framework is experimentally evaluated on three selected fixed-time signalized intersections in Shanghai, China.

A new fully homomorphic encryption over the integers using smaller public key

Fully homomorphic encryption scheme with practical time complexity is a widely acknowledged research problem in cryptography. In this work, a new somewhat homomorphic encryption with practical time complexities is proposed, from which fully homomorphic encryption is obtained using the optimisations suggested in the contemporary works. The central idea behind the proposition in achieving such reasonable time complexities lies in employing a small public key containing only two big integers with consequent reduction in the message expansion. The scheme may be considered as a variant of the DGHV's integers-based scheme, which was one of the earlier attempts in devising a conceptually simpler fully homomorphic encryption. The semantic security of the proposition is proved in the standard model by reducing the same to the hard problem of solving partial approximate greatest common divisor of two integers.

AGCD: a robust periodicity analysis method based on approximate greatest common divisor

Periodicity is one of the most common phenomena in the physical world. The problem of periodicity analysis (or period detection) is a research topic in several areas, such as signal processing and data mining. However, period detection is a very challenging problem, due to the sparsity and noisiness of observational datasets of periodic events. This paper focuses on the problem of period detection from sparse and noisy observational datasets. To solve the problem, a novel method based on the approximate greatest common divisor (AGCD) is proposed. The proposed method is robust to sparseness and noise, and is efficient. Moreover, unlike most existing methods, it does not need prior knowledge of the rough range of the period. To evaluate the accuracy and efficiency of the proposed method, comprehensive experiments on synthetic data are conducted. Experimental results show that our method can yield highly accurate results with small datasets, is more robust to sparseness and noise, and is less sensitive to the magnitude of period than compared methods.

Improved Fully Homomorphic Encryption over the Integers with Shorter Public Keys

Fully homomorphic encryption (FHE) is a “holy grail” of cryptography. However, it is not yet adopted in practice because no known scheme is efficient. In this paper, we mainly focus on how to reduce the public key sizes in FHE. Based on Dijk et al.’s FHE scheme (DGHV) and Gentry’s fully homomorhpic technology, we propose two schemes with shorter public keys by encrypting with a combination of the public key elements to replace a linear form in DGHV scheme and a quadratic form in Coron et al.’s scheme. Compared with DGHV scheme and Coron et al.’s scheme, our schemes can greatly reduce the public key sizes, which make our schemes more efficient. Furthermore, we prove the security of our schemes based on difficulties of the approximate greatest common divisor (AGCD) problem and the sparse subset sum problem.

A Practical Fundamental Frequency Extraction Algorithm for Motion Parameters Estimation of Moving Targets

In this paper, a practical method is proposed for a moving target's fundamental frequency (MTFF) extraction from its acoustic signal. This method is developed for the application of motion parameters estimation. Starting from the analysis of the target frequency model and the acoustic Doppler model, the characteristics of moving target's signal are discussed. Based on the signatures of target's acoustic signal, a new approximate greatest common divisor (AGCD) method is developed to obtain an initial fundamental frequency (IFF). Then, the corresponding harmonic number associated with the IFF is determined by maximizing an objective function formulated as an impulse-train-weighted symmetric average magnitude sum function (SAMSF) of the observed signal. The frequency of the SAMSF is determined by target's acoustic signal, the period of the impulse train is controlled by the estimated IFF harmonic, and the maximization of the objective function is carried out through a time-domain matching of periodicity of the impulse train with that of the SAMSF. Finally, a precise fundamental frequency is achieved based on the obtained IFF and its harmonic number. In order to demonstrate the effectiveness of the proposed method, experiments are conducted on wheeled vehicles, tracked vehicles, and propeller-driven aircrafts. Evaluation of the algorithm performance in comparison with other traditional methods indicates that the proposed MTFF is practical for the fundamental frequency extraction of moving targets.

Factorization Approach to Structured Low-Rank Approximation with Applications

We consider the problem of approximating an affinely structured matrix, for example, a Hankel matrix, by a low-rank matrix with the same structure. This problem occurs in system identification, signal processing, and computer algebra, among others. We impose the low rank by modeling the approximation as a product of two factors with reduced dimension. The structure of the low-rank model is enforced by introducing a penalty term in the objective function. The proposed local optimization algorithm is able to solve the weighted structured low-rank approximation problem, as well as to deal with the cases of missing or fixed elements. In contrast to approaches based on kernel representations (in the linear algebraic sense), the proposed algorithm is designed to address the case of small targeted rank. We compare it to existing approaches on numerical examples of system identification, approximate greatest common divisor problem, and symmetric tensor decomposition and demonstrate its consistently good performance.

Fully Homomorphic Encryption Using Hidden Ideal Lattice

All the existing fully homomorphic encryption schemes are based on three different problems, namely the bounded distance decoding problem over ideal lattice, the approximate greatest common divisor problem over integers, and the learning with error problem. In this paper, we unify the first two families of problems by introducing a new class of problems, which can be reduced from both problems. Based on this new problem, namely the bounded distance decoding over hidden ideal lattice, we present a new fully homomorphic encryption scheme. Since it is a combination of the two problems to some extent, the performance of our scheme lies between the ideal lattice based schemes and the integer based schemes. Furthermore, we also show a lower bound and upper bound of the problem that our scheme is based on. Assuming this security conjecture holds, we can incorporate smaller parameters, which will result in a scheme that is more efficient than both lattice based and integer based schemes. Hence, our scheme makes a perfect alternative to the state-of-art ring learning with error based schemes.

A heuristic verification of the degree of the approximate GCD of two univariate polynomials

We consider the problem of computing verified real interval perturbations of the coefficients of two univariate polynomials such that there exist corresponding perturbed polynomials which have an exact greatest common divisor (GCD) of a given degree k. Based on the certification of the rank deficiency of a submatrix of the Bezout matrix of two univariate polynomials, we propose an algorithm to compute verified real perturbations. Numerical experiments show the performance of our algorithm.

An Efficient Somewhat HE scheme over Integers and Its Variation

In 2010, Dijk et al. demonstrated a simple somewhat homomorphic encryption (HE) scheme over the integers of which this simplicity came at the cost of a public key size in O(λ 10 ). Although in 2011 Coron et al. reduced the public key size to O(λ 7 ), it is still too large for practical applications, especially for the cloud computing. In this paper, we propose a new form of somewhat HE scheme to reduce further the public key size and a variation of the scheme to optimize the ciphertext size. First of all, we propose a new somewhat HE scheme which is built on the hardness of the approximate greatest common divisor (GCD) problem of two integers, where the public key size in the scheme is reduced to O(λ 3 ). Furthermore, we can reduce the length of the ciphertext of the new somewhat HE scheme by applying the modular reduction technique. Additionally, we give simulation results for evaluating ability of the proposed scheme.

A geometrical approach to finding multivariate approximate LCMs and GCDs

In this article we present a new approach to compute an approximate least common multiple (LCM) and an approximate greatest common divisor (GCD) of two multivariate polynomials. This approach uses the geometrical notion of principal angles whereas the main computational tools are the Implicitly Restarted Arnoldi method and sparse QR decomposition. Upper and lower bounds are derived for the largest and smallest singular values of the highly structured Macaulay matrix. This leads to an upper bound on its condition number and an upper bound on the 2-norm of the product of two multivariate polynomials. Numerical examples are provided.

Resultant matrices and the computation of the degree of an approximate greatest common divisor of two inexact Bernstein basis polynomials

The computation of the degree d of an approximate greatest common divisor of two Bernstein basis polynomials f(y) and g(y) that are noisy forms of, respectively, the exact polynomials fˆ(y) and gˆ(y) that have a non-constant common divisor is considered using the singular value decomposition of their Sylvester S(f,g) and Bézout B(f,g) resultant matrices. It is shown that the best estimate of d is obtained when S(f,g) is postmultiplied by a diagonal matrix Q that is derived from the vectors that lie in the null space of S(f,g), where the correct value of d is defined as the degree of the greatest common divisor of the exact polynomials fˆ(y) and gˆ(y). The computed value of d is improved further by preprocessing f(y) and g(y), and examples of the computation of d using S(f,g), S(f,g)Q and B(f,g) are presented.

Approximate Greatest Common Divisor of Many Polynomials and Pseudo-Spectrum

The paper is concerned with establishing the links between the approximate GCD of a set of polynomials and the notion of the pseudo-spectrum defined on a set of polynomials. By examining the pseudo-spectrum of the structured matrix we will derive estimates of the area of the approximate roots of the initial polynomial set. We will relate the strength of the GCD to the weighted strength of the pseudo-spectra and we investigate under which conditions the roots of the approximate GCDs are a subset of the pseudo-spectra.

Computing Approximation GCD of Several Polynomials by Structured Total Least Norm

The task of determining the greatest common divisors (GCD) for several polynomials which arises in image compression, computer algebra and speech encoding can be formulated as a low rank approximation problem with Sylvester matrix. This paper demonstrates a method based on structured total least norm (STLN) algorithm for matrices with Sylvester structure. We demonstrate the algorithm to compute an approximate GCD. Both the theoretical analysis and the computational results show that the method is feasible.

Nearest common root of polynomials, approximate greatest common divisor and the structured singular value

In this paper the following problem is considered: given two coprime polynomials, find the smallest perturbation in the magnitude of their coefficients such that the perturbed polynomials have a common root. It is shown that the problem is equivalent to the calculation of the structured singular value of a matrix arising in robust control and a numerical solution to the problem is developed. A simple numerical example illustrates the effectiveness of the method for two polynomials of low degree. Finally, problems involving the calculation of the approximate greatest common divisor of univariate polynomials are considered, by proposing a generalization of the definition of the structured singular value involving additional rank constraints.

More Practical Fully Homomorphic Encryption

In this paper, we first modify the Smart-Vercauteren’s fully homomorphic encryption scheme [SV10] by applying self-loop bootstrappable technique. The security of the modified scheme only depends on the hardness of the polynomial coset problem, removing the assumption of the sparse subset sum problem in the original paper in [SV10]. Moreover, we construct a non-self-loop in FHE by using cycle keys. Then, we further improve our scheme to make it be practical. The securities of our improving FHE’s are respectively based on the hardness of factoring integer problem, solving Diophantine equation problem, and finding approximate greatest common divisor problem.

Blind image deconvolution using a banded matrix method

In this paper we study the blind image deconvolution problem in the presence of noise and measurement errors. We use a stable banded matrix based approach in order to robustly compute the greatest common divisor of two univariate polynomials and we introduce the notion of approximate greatest common divisor to encapsulate the above approach, for blind image restoration. Our method is analyzed concerning its stability and complexity resulting to useful conclusions. It is proved that our approach has better complexity than the other known greatest common divisor based blind image deconvolution techniques. Examples illustrating our procedures are given.

GPGCD: An iterative method for calculating approximate GCD of univariate polynomials

We present an iterative algorithm for calculating approximate greatest common divisor (GCD) of univariate polynomials with the real or the complex coefficients. For a given pair of polynomials and a degree, our algorithm finds a pair of polynomials which has a GCD of the given degree and whose coefficients are perturbed from those in the original inputs, making the perturbations as small as possible, along with the GCD. The problem of approximate GCD is transferred to a constrained minimization problem, then solved with the so-called modified Newton method, which is a generalization of the gradient-projection method, by searching the solution iteratively. We demonstrate that, in some test cases, our algorithm calculates approximate GCD with perturbations as small as those calculated by a method based on the structured total least norm (STLN) method and the UVGCD method, while our method runs significantly faster than theirs by approximately up to 30 or 10 times, respectively, compared with their implementation. We also show that our algorithm properly handles some ill-conditioned polynomials which have a GCD with small or large leading coefficient.