Equivalence of Polynomial Identity Testing and Polynomial Factorization

In this paper, we show that the problem of deterministically factoring multivariate polynomials reduces to the problem of deterministic polynomial identity testing. Specifically, we show that given an arithmetic circuit (either explicitly or via black-box access) that computes a multivariate polynomial f, the task of computing arithmetic circuits for the factors of f can be solved deterministically, given a deterministic algorithm for the polynomial identity testing problem (we require either a white-box or a black-box algorithm, depending on the representation of f).Together with the easy observation that deterministic factoring implies a deterministic algorithm for polynomial identity testing, this establishes an equivalence between these two central derandomization problems of arithmetic complexity.Previously, such an equivalence was known only for multilinear circuits (Shpilka & Volkovich, 2010).     

An Efficient Digital Confidentiality Scheme Based on Commutative Chaotic Polynomial

Multimedia Tools and Applications - One of the most critical aspects of this technologically progressive era is the propagation of information through an unsecured communication channel. The...     

Unconditionally Secure Oblivious Polynomial Evaluation: A Survey and New Results

Oblivious polynomial evaluation (OPE) is a two-party protocol that allows a receiver, R to learn an evaluation f(α), of a sender, S's polynomial f(x), whilst keeping both α and f(x) private. This protocol has attracted a lot of attention recently, as it has wide ranging applications in the field of cryptography. In this article we review some of these applications and, additionally, take an in-depth look at the special case of information theoretic OPE. Specifically, we provide a current and critical review of the existing information theoretic OPE protocols in the literature. We divide these protocols into two distinct cases (three-party and distributed OPE) allowing for the easy distinction and classification of future information theoretic OPE protocols. In addition to this work, we also develop several modifications and extensions to existing schemes, resulting in increased security, flexibility and efficiency. Lastly, we also identify a security flaw in a previously published OPE scheme.     

Derandomizing Polynomial Identity Tests Means Proving Circuit Lower Bounds

We show that derandomizing Polynomial Identity Testing is essentially equivalent to proving arithmetic circuit lower bounds for NEXP. More precisely, we prove that if one can test in polynomial time (or even nondeterministic subexponential time, infinitely often) whether a given arithmetic circuit over integers computes an identically zero polynomial, then either (i) (ii) Permanent is not computable by polynomial-size arithmetic circuits. We also prove a (partial) converse: If Permanent requires superpolynomial-size arithmetic circuits, then one can test in subexponential time whether a given arithmetic circuit of polynomially bounded degree computes an identically zero polynomial.Since Polynomial Identity Testing is a coRP problem, we obtain the following corollary: If then NEXP is not computable by polynomial-size arithmetic circuits. Thus establishing that RP = coRP or BPP = P would require proving superpolynomial lower bounds for Boolean or arithmetic circuits. We also show that any derandomization of RNC would yield new circuit lower bounds for a language in NEXP.We also prove unconditionally that NEXP^RP does not have polynomial-size Boolean or arithmetic circuits. Finally, we show that if both BPP = P and low-degree testing is in P; here low-degree testing is the problem of checking whether a given Boolean circuit computes a function that is close to some low-degree polynomial over a finite field.     

基于ASP的在线考试测评及成绩管理系统

系统从根本上解决了传统考试效率低、反馈时间长、资源浪费等问题。介绍了系统的数据库设计、主要功能模块及实现中的几个关键技术,并给出了具体的实现代码。     

一个新的多项式光滑支持向量机

光滑支持向量机(SSVM)是支持向量机(SVM)的快速求解模型,拥有更快的求解速度和训练效果。基于光滑的分段多项式函数和插值思想推导出一个新的光滑函数,从而可以更好地逼近正号函数。通过所得到的新光滑函数改进多项式光滑支持向量机模型(PSSVM),得到了更新的光滑支持向量机模型。还给出了新光滑函数的逼近性能和精度分析以及新模型的收敛性证明和最优解的逼近上限。数值实验表明,所提出的新光滑支持向量机模型性能优于PSSVM模型。     

New Algorithms for Polynomial J-Spectral Factorization

In this paper new algorithms are developed for J-spectral factorization of polynomial matrices. These algorithms are based on the calculus of two-variable polynomial matrices and associated quadratic differential forms, and share the common feature that the problem is lifted from the original one-variable polynomial context to a two-variable polynomial context. The problem of polynomial J-spectral factorization is thus reduced to a problem of factoring a constant matrix obtained from the coefficient matrices of the polynomial matrix to be factored. In the second part of the paper, we specifically address the problem of computing polynomial J-spectral factors in the context of H _∞ control. For this, we propose an algorithm that uses the notion of a Pick matrix associated with a given two-variable polynomial matrix. Date received: January 1, 1998. Date revised: October 15, 1998.     

A New Polynomial Factorization Algorithm and its Implementation

We consider the problem of factoring univariate polynomials over a finite field. We demonstrate that the new baby step/giant step factoring method, recently developed by Kaltofen and Shoup, can be made into a very practical algorithm. We describe an implementation of this algorithm, and present the results of empirical tests comparing this new algorithm with others. When factoring polynomials modulo large primes, the algorithm allows much larger polynomials to be factored using a reasonable amount of time and space than was previously possible. For example, this new software has been used to factor a "generic" polynomial of degree 2048 modulo a 2048-bit prime in under 12 days on a Sun SPARC-station 10, using 68 MB of main memory.     

New Results on Path Approximation

In this paper we give bounds on the complexity of some algorithms for approximating 2-D and 3-D polygonal paths with the infinite beam measure of error. While the time/space complexities of the algorithms known for other error measures are well understood, path approximation with infinite beam measure seems to be harder due to the complexity of some geometric structures that arise in the known approaches. Our results answer some open problems left in previous work. We also present a more careful analysis of the time complexity of the general iterative algorithm for infinite beam measure and show that it could perform much better in practice than in the worst case. We then propose a new approach for path approximation that consists of a breadth first traversal of the path approximation graph, without constructing the graph. This approach can be applied to any error criterion in any constant dimension. The running time of the new algorithm depends on the size of the output: if the optimal approximating path has m vertices, the algorithm performs F(m) iterations rather than n–1 in the previous approaches, where F(m) \le n–1 is the number of vertices of the path approximation graph that can be reached with at most m–2 edges. This is the first output sensitive path approximation algorithm.     

New Results on Path Approximation

In this paper we give bounds on the complexity of some algorithms for approximating 2-D and 3-D polygonal paths with the infinite beam measure of error. While the time/space complexities of the algorithms known for other error measures are well understood, path approximation with infinite beam measure seems to be harder due to the complexity of some geometric structures that arise in the known approaches. Our results answer some open problems left in previous work. We also present a more careful analysis of the time complexity of the general iterative algorithm for infinite beam measure and show that it could perform much better in practice than in the worst case. We then propose a new approach for path approximation that consists of a breadth first traversal of the path approximation graph, without constructing the graph. This approach can be applied to any error criterion in any constant dimension. The running time of the new algorithm depends on the size of the output: if the optimal approximating path has m vertices, the algorithm performs F(m) iterations rather than n–1 in the previous approaches, where F(m) \le n–1 is the number of vertices of the path approximation graph that can be reached with at most m–2 edges. This is the first output sensitive path approximation algorithm.     

Hurwitz多项式的新结果

本文给出了判别Hurwitz多项式的简便而又宽裕的方法,并推导了Routh判据的等价形式.     

Family of Ternary Arithmetic Polynomial Expansions Based on New Transforms

New classes of Linearly Independent Ternary Arithmetic (LITA) transforms being the bases of ternary arithmetic polynomial expansions are introduced here. Recursive equations defining the LITA transforms and the corresponding butterfly diagrams are shown. Various properties and relations between introduced classes of new transforms are discussed. Computational costs to calculate LITA transforms and applications of corresponding polynomial expansions in logic design are also discussed.     

Statistical Estimation and Classification on Commutative Covariance Structures

Statistical inference is investigated under the following constraints on the covariance structure for the observation vector: covariance matrices belong to some commutative matrix algebra. Commutative approximation of arbitrary covariance structures and statistical estimation of the parameters of a given commutative structure are studied. The results are applied to statistical classification of Gaussian vectors having commutative covariance structure.     

基于奇异值分解的同态可交换脆弱零水印研究

现有的水印和加密方案大多难以确保水印和加密过程的交换性以及受保护图像的视觉质量,这些方案的水印嵌入和加密过程固定且对受保护图像的内容进行了或多或少的修改,很少有方案在不影响受保护图像内容质量的前提下完成水印和加密过程的交换.因此,提出了一种基于奇异值分解的同态可交换脆弱零水印方案.在发送端,内容所有者采取同态加密对原始...     

Commutative encryption and watermarking based on orthogonal decomposition

Commutative Encryption and Watermarking (CEW) is a technology combined with encryption and watermarking, which is used for providing comprehensive security protection for multimedia information. This paper presents a novel CEW scheme based on orthogonal decomposition. Different from current CEW, the proposed CEW not only achieves the integration of encryption and watermarking in the final data but also has no specific restrictions in selecting encryption and watermarking algorithms. Therefore, the proposed CEW possesses higher security and applicability. Experimental results demonstrate that the proposed CEW can keep the performances of selected encryption and watermarking algorithm and show more robustness than other current CEW schemes.     

A Case of Depth-3 Identity Testing, Sparse Factorization and Duality

The polynomial identity testing (PIT) problem is known to be challenging even for constant depth arithmetic circuits. In this work, we study the complexity of two special but natural cases of identity testing—first is a case of depth-3 PIT, the other of depth-4 PIT. Our first problem is a vast generalization of verifying whether a bounded top fan-in depth-3 circuit equals a sparse polynomial (given as a sum of monomial terms). Formally, given a depth-3 circuit C, having constant many general product gates and arbitrarily many semidiagonal product gates test whether the output of C is identically zero. A semidiagonal product gate in C computes a product of the form ${m \cdot \prod^b_{i=1}l^{e_i}_i}$ , where m is a monomial, l _i is a linear polynomial, and b is a constant. We give a deterministic polynomial time test, along with the computation of leading monomials of semidiagonal circuits over local rings. The second problem is on verifying a given sparse polynomial factorization, which is a classical question (von zur Gathen, FOCS 1983): Given multivariate sparse polynomials f, g1, . . . , gt explicitly check whether ${f = \prod^t_{i=1} {g_i}}$ . For the special case when every gi is a sum of univariate polynomials, we give a deterministic polynomial time test. We characterize the factors of such g _i ’s and even show how to test the divisibility of f by the powers of such polynomials. The common tools used are Chinese remaindering and dual representation. The dual representation of polynomials (Saxena, ICALP 2008) is a technique to express a product-of-sums of univariates as a sum-ofproducts of univariates. We generalize this technique by combining it with a generalized Chinese remaindering to solve these two problems (over any field).