Algorithm For Factoring Polynomials Research Articles

Implementation of a noise-robust quantum algorithm for multivariate polynomial factorization

We implement in an IBM quantum computer a quantum algorithm for multivariate polynomial factoring and propose a noise-avoiding protocol to distill the experimental results in the presence of noise. This algorithm uses single-qubit quantum state tomography (QST) processes to factor a specific type of multivariate polynomials. In one-to-one correspondence, it encodes each multivariate polynomial to one quantum state. While the validity of the algorithm is experimentally verified, the quality of the final results is subjected to the decoherence levels of the preparation of the quantum states. In this paper we propose a protocol to ensure the validity of factors found by our algorithm in the presence of such decoherence and noise. This method might be, in fact, part of a larger class of methods based on that same premise and useful outside the implementation of this specific algorithm. In combination with the noise robustness of the single-qubit QST, our factorization algorithm performs perfectly, even reversing the effects of weak noise, for the second- to fifth-order polynomial cases for which it has been implemented.

An explicit Wiener-Hopf factorization algorithm for matrix polynomials and its exact realizations within ExactMPF package.

We discuss an explicit algorithm for solving the Wiener–Hopf factorization problem for matrix polynomials. By an exact solution of the problem, we understand the one constructed by a symbolic computation. Since the problem is, generally speaking, unstable, this requirement is crucial to guarantee that the result following from the explicit algorithm is indeed a solution of the original factorization problem. We prove that a matrix polynomial over the field of Gaussian rational numbers admits the exact Wiener–Hopf factorization if and only if its determinant is exactly factorable. Under such a condition, we adapt the explicit algorithm to the exact calculations and develop the ExactMPF package realized within the Maple Software. The package has been extensively tested. Some examples are presented in the paper, while the listing is provided in the electronic supplementary material. If, however, a matrix polynomial does not admit the exact factorization, we clarify a notion of the numerical (or approximate) factorization that can be constructed by following the explicit factorization algorithm. We highlight possible obstacles on the way and discuss a level of confidence in the final result in the case of an unstable set of partial indices. The full listing of the package ExactMPF is given in the electronic supplementary material.

Achieving reliable and anti-collusive outsourcing computation and verification based on blockchain in 5G-enabled IoT

Widespread applications of 5G technology have prompted the outsourcing of computation dominated by the Internet of Things (IoT) cloud to improve transmission efficiency, which has created a novel paradigm for improving the speed of common connected objects in IoT. However, although it makes it easier for ubiquitous resource-constrained equipment that outsources computing tasks to achieve high-speed transmission services, security concerns, such as a lack of reliability and collusion attacks, still exist in the outsourcing computation. In this paper, we propose a reliable, anti-collusion outsourcing computation and verification protocol, which uses distributed storage solutions in response to the issue of centralized storage, leverages homomorphic encryption to deal with outsourcing computation and ensures data privacy. Moreover, we embed outsourcing computation results and a novel polynomial factorization algorithm into the smart contract of Ethereum, which not only enables the verification of the outsourcing result without a trusted third party but also resists collusion attacks. The results of the theoretical analysis and experimental performance evaluation demonstrate that the proposed protocol is secure, reliable, and more effective compared with state-of-the-art approaches.

Factoring polynomials over finite fields

In this paper, we describe a new polynomial factorization algorithm over finite fields with odd characteristics. The main ingredient of the algorithm is special singular curves. The algorithm relies on the extension of the Mumford representation and Cantor’s algorithm to these special singular curves.

Formalizing the LLL Basis Reduction Algorithm and the LLL Factorization Algorithm in Isabelle/HOL

The LLL basis reduction algorithm was the first polynomial-time algorithm to compute a reduced basis of a given lattice, and hence also a short vector in the lattice. It approximates an NP-hard problem where the approximation quality solely depends on the dimension of the lattice, but not the lattice itself. The algorithm has applications in number theory, computer algebra and cryptography. In this paper, we provide an implementation of the LLL algorithm. Both its soundness and its polynomial running-time have been verified using Isabelle/HOL. Our implementation is nearly as fast as an implementation in a commercial computer algebra system, and its efficiency can be further increased by connecting it with fast untrusted lattice reduction algorithms and certifying their output. We additionally integrate one application of LLL, namely a verified factorization algorithm for univariate integer polynomials which runs in polynomial time.

Directed evaluation

Let K be a fixed effective field. The most straightforward approach to compute with an element in the algebraic closure of K is to compute modulo its minimal polynomial. The determination of a minimal polynomial from an arbitrary annihilator requires an algorithm for polynomial factorization over K. Unfortunately, such algorithms do not exist over generic effective fields. They do exist over fields that are explicitly generated over their prime sub-field, but they are often expensive. The dynamic evaluation paradigm, introduced by Duval and collaborators in the eighties, offers an alternative algorithmic solution for computations in the algebraic closure of K. This approach does not require an algorithm for polynomial factorization, but it still suffers from a non-trivial overhead due to suboptimal recomputations. For the first time, we design another paradigm, called directed evaluation, which combines the conceptual advantages of dynamic evaluation with a good worst case complexity bound.

Multi-instance iris remote authentication using private multi-class perceptron on malicious cloud server

In recent years, biometric authentication system (BAS) has become the most promising and popular authentication system in identity management. Due to its capability to solve the limitations of unimodal systems, multi-biometric systems (MBS) have been extensively accepted in various fields. The main step in MBS is information fusion. On the other hand, directly storing the fused templates into a centralized server leads to privacy concerns. Recently, many BAS based on homomorphic encryption has been introduced to provide confidentiality for the fused templates. However, most of the existing solutions rely on an implication of the assumption that the server is “Honest-but-Curious”. As a result, the compromise of such server results into entire system vulnerability. To address this, we propose a novel P rivacy P reserving (PP) multi-instance iris remote authentication system to accord with attacks at the malicious server and over the transmission channel. Our scheme uses F ully H omomorphic E ncryption (FHE) to achieve the confidentiality of the fused iris templates and polynomial factorization algorithm to achieve the integrity of the matching result. We propose a PP iris authentication system using P rivate M ulti-C lass P erceptron (PMCP) by using the properties of FHE. Moreover, we propose C ontradistinguish S imilarity A nalysis (CSA), a feature level fusion technique that minimizes the between-class correlations and maximizes the pair-wise correlations. Our method has experimented on IITD and CASIA-V3-Interval iris databases to check the effectiveness and robustness. Experimental results show that our method provides improved accuracy, and eliminates the need to trust the cloud server when compared to the state-of-the-art approaches.

Intrinsic factorization of ideals in Dedekind domains

We present a generalization of a polynomial factorization algorithm that works with ideals in maximal orders of global function fields. The method presented in this paper is intrinsic in the sense that it does not depend on the embedding of the ring of polynomials into the Dedekind domain in question.

Intrinsic Factorization of Ideals in Dedekind Domains

We present a generalization of a polynomial factorization algorithm that works with ideals in maximal orders of global function fields. The method presented in this paper is intrinsic in the sense that it does not depend on the embedding of the ring of polynomials into the Dedekind domain in questi on.

Direct sum decomposability of polynomials and factorization of associated forms

We prove two criteria for direct sum decomposability of homogeneous polynomials. For a homogeneous polynomial with a non-zero discriminant, we interpret direct sum decomposability of the polynomial in terms of factorization properties of the Macaulay inverse system of its Milnor algebra. This leads to an if-and-only-if criterion for direct sum decomposability of such a polynomial, and to an algorithm for computing direct sum decompositions over any field, either of characteristic $0$ or of sufficiently large positive characteristic, for which polynomial factorization algorithms exist. For homogeneous forms over algebraically closed fields, we interpret direct sums and their limits as forms that cannot be reconstructed from their Jacobian ideal. We also give simple necessary criteria for direct sum decomposability of arbitrary homogeneous polynomials over arbitrary fields and apply them to prove that many interesting classes of homogeneous polynomials are not direct sums.

A Verified Implementation of the Berlekamp\u2013Zassenhaus Factorization Algorithm

We formally verify the Berlekamp–Zassenhaus algorithm for factoring square-free integer polynomials in Isabelle/HOL. We further adapt an existing formalization of Yun’s square-free factorization algorithm to integer polynomials, and thus provide an efficient and certified factorization algorithm for arbitrary univariate polynomials. The algorithm first performs factorization in the prime field mathrm {GF}(p){} and then performs computations in the ring of integers modulo p^k, where both p and k are determined at runtime. Since a natural modeling of these structures via dependent types is not possible in Isabelle/HOL, we formalize the whole algorithm using locales and local type definitions. Through experiments we verify that our algorithm factors polynomials of degree up to 500 within seconds.

Toward an optimal quantum algorithm for polynomial factorization over finite fields

We present a randomized quantum algorithm for polynomial factorization over finite fields. For polynomials of degree n over a finite field F_q, the average-case complexity of our algorithm is an expected O(n^{1 + o(1)} \log^{2 + o(1)}q) bit operations. Only for a negligible subset of polynomials of degree $n$ our algorithm has a higher complexity of O(n^{4/3 + o(1)} \log^{2 + o(1)}q) bit operations. This breaks the classical 3/2-exponent barrier for polynomial factorization over finite fields \cite{guo2016alg}.

Toward an optimal quantum algorithm for polynomial factorization over finite fields

We present a randomized quantum algorithm for polynomial factorization over finite fields. For polynomials of degree n over a finite field Fq, the average-case complexity of our algorithm is an exp...

Verifiable outsourced computation over encrypted data

In recent years, cloud computing has become the most popular and promising service platform. A cloud user can outsource its heavy computation overhead to a cloud service provider (CSP) and let the CSP make the computation instead. In order to guarantee the correctness of the outsourced processing (e.g., machine learning and data mining), a proof should be provided by the CSP in order to make sure that the processing is carried out properly. On the other hand, from the security and privacy points of view, users will always encrypt their sensitive data first before they are outsourced to the CSP rather than sending the raw data directly. However, processing and verifying of encrypted data computation has always been a challenging problem. Homomorphic Encryption (HE) has been proposed to tackle this task on computations over encrypted data and ensure the confidentiality of the data. However, original HE cannot provide an efficient approach to verify the correctness of computation over encrypted data that is processed by CSP. In this paper, we propose a verifiable outsourced computation scheme over encrypted data with the help of fully homomorphic encryption and polynomial factorization algorithm. Our scheme protects user data security in outsourced processing and allows public verification on the computation result processed by CSP with zero knowledge. We then prove the security of our scheme and analyze its performance by comparing it with some latest related works. Performances analysis shows that our scheme reduces the overload of both the cloud users and the verifier.

Polynomials in Base $x$ and the Prime-Irreducible Affinty

Arthur Cohn's irreducibility criterion for polynomials with integer coefficients and its generalization connect primes to irreducibles, and integral bases to the variable $x$. As we follow this link, we find that these polynomials are ready to spill two of their secrets: (i) There exists a unique “base-$x$” representation of such polynomials that makes the ring $\mathbb{Z}[x]$ into an ordered domain; and (ii) There is a 1-1 correspondence between positive rational primes $p$ and certain infinite sets of irreducible polynomials $f(x)$ that attain the value $p$ at sufficiently large $x$, each generated in finitely many steps from the $p$th cyclotomic polynomial. The base-$x$ representation provides practical conversion methods among numeric bases (not to mention a polynomial factorization algorithm), while the prime-irreducible correspondence puts a new angle on the Bouniakowsky Conjecture, a generalization of Dirichlet's Theorem on Primes in Arithmetic Progressions.

Antenna Array Design in MIMO Radar Using NSK Polynomial Factorization Algorithm

The work presented here is concerned with the antenna array design in collocated multiple-input multiple-output (MIMO) radars. After knowing the system requirements, the antenna array design problem is formulated as a standard polynomial factorization. In addition, an algorithm based on Newton-Schubert-Kronecker (NSK) polynomial factorization is proposed. The algorithm contains three steps. First, linear factors are extracted by extended Vieta theorem. Then, undermined high-order factors are confirmed with Newton interpolation and certain high-order factors should be searched for within the undermined ones. Finally, the antenna array configurations are determined according to the result of polynomial factorization. Simulations confirm the wide use of the proposed algorithm in MIMO radar antenna array design.

Improvements on the Cantor-Zassenhaus factorization algorithm

The paper presents a careful analysis of the Cantor-Zassenhaus polynomial factorization algorithm, thus obtaining tight bounds on the performances, and proposing useful improvements. In particular, a new simplified version of this algorithm is described, which entails a lower computational cost. The key point is to use linear test polynomials, which not only reduce the computational burden, but can also provide good estimates and deterministic bounds of the number of operations needed for factoring. Specifically, the number of attempts needed to factor a given polynomial, and the least degree of a polynomial such that a factor is found with at most a fixed number of attempts, are computed. Interestingly, the results obtained demonstrate the existence of some sort of duality relationship between these two problems.

On the acceleration of an algorithm for polynomial factorization

On the acceleration of an algorithm for polynomial factorization

Single-factor lifting and factorization of polynomials over local fields

Let f(x) be a separable polynomial over a local field. The Montes algorithm computes certain approximations to the different irreducible factors of f(x), with strong arithmetic properties. In this paper, we develop an algorithm to improve any one of these approximations, till a prescribed precision is attained. The most natural application of this “single-factor lifting” routine is to combine it with the Montes algorithm to provide a fast polynomial factorization algorithm. Moreover, the single-factor lifting algorithm may be applied as well to accelerate the computational resolution of several global arithmetic problems in which the improvement of an approximation to a single local irreducible factor of a polynomial is required.

A note on Gao’s algorithm for polynomial factorization

Shuhong Gao (2003) [6] has proposed an efficient algorithm to factor a bivariate polynomial f over a field F. This algorithm is based on a simple partial differential equation and depends on a crucial fact: the dimension of the polynomial solution space G associated with this differential equation is equal to the number r of absolutely irreducible factors of f. However, this holds only when the characteristic of F is either zero or sufficiently large in terms of the degree of f. In this paper we characterize a vector subspace of G for which the dimension is r, regardless of the characteristic of F, and the properties of Gao’s construction hold. Moreover, we identify a second vector subspace of G that leads to an analogous theory for the rational factorization of f.