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Li Z.,Austrian Academy of Sciences | Li Z.,Key Laboratory of Mathematics Mechanization | Zhi L.,Key Laboratory of Mathematics Mechanization
Theoretical Computer Science | Year: 2013

In this paper, we derive explicit expressions for the nearest singular polynomials with given root multiplicities and its distance to the given polynomial. These expressions can be computed recursively. These results extend previous results of Zhi et al. (2004) [10] and Zhi and Wu (1998) [11]. © 2012 Elsevier B.V. All rights reserved. Source


El Din M.S.,CNRS Laboratory for Informatics | Zhi L.,Key Laboratory of Mathematics Mechanization
SIAM Journal on Optimization | Year: 2010

Let P = {h1, . . . , hs} ⊂ Z[Y1, . . . ,Yk], D ≥ deg(hi) for 1 ≤ i ≤ s, ς bounding the bit length of the coefficients of the hi's, and let ? be a quantifier-free P-formula defining a convex semialgebraic set. We design an algorithm returning a rational point in S if and only if S∩Q ≠ ∅. It requires ςO(1)DO(k3) bit operations. If a rational point is outputted, its coordinates have bit length dominated by ςDO(k3). Using this result, we obtain a procedure for deciding whether a polynomial f ∈ Z[X1, . . . ,Xn] is a sum of squares of polynomials in Q[X1, . . . ,Xn]. Denote by d the degree of f, ? the maximum bit length of the coefficients in f, D = (n+d/n), and k ≤ D(D + 1) ?(n+d/n), This procedure requires τO(1)DO(k3) bit operations, and the coefficients of the outputted polynomials have bit length dominated by τDO(k3). © 2010 Society for Industrial and Applied Mathematics. Source


Li F.,University of Delaware | Li Z.,Key Laboratory of Mathematics Mechanization | Saunders D.,University of Delaware | Yu J.,University of Delaware
Proceedings of the IEEE International Conference on Computer Vision | Year: 2011

We present a new Coprime Blurred Pair (CBP) theory that may benefit a number of computer vision applications. A CBP is constructed by blurring the same latent image with two unknown kernels, where the two kernels are co-prime when mapped to bivariate polynomials under the z-transform. We first show that the blurred contents in a CBP are difficult to restore using conventional blind deconvolution methods based on sparsity priors. We therefore introduce a new coprime prior for recovering the latent image in a CBP. Our solution maps the CBP to bivariate polynomials and sample them on the unit circle in both dimension. We show that coprimality can be derived in terms of the rank of the Bézout Matrix [2] formed by the sampled polynomials and we present an efficient algorithm to factor the Bézout Matrix for recovering the latent image. Finally, we discuss applications of the CBP theory in privacy-preserving surveillance and motion deblurring, as well as physical implementations of CBPs using flutter shutter cameras. © 2011 IEEE. Source


Wang H.-Z.,Wuhan University | Wang H.-Z.,Key Laboratory of Mathematics Mechanization | Zhang H.-G.,Wuhan University | Guan H.-M.,Chinese Electronic Equipment System Corporation | Wu Q.-H.,Wuhan University
Beijing Gongye Daxue Xuebao/Journal of Beijing University of Technology | Year: 2010

The theory of the MQ problem solving is presented in this paper. Then several major MQ public-key cryptography and the corresponding attacks are described in detail, and the basic design principles to be followed of the MQ cryptography are proposed; After the MQ problem in other cryptography applications (e.g., stream cipher) are analyzed briefly, the direction and the issues for future research are summarized in the end. Source


Wang H.Z.,Wuhan University | Wang H.Z.,Key Laboratory of Mathematics Mechanization | Zhang H.G.,Wuhan University | Wu Q.H.,Wuhan University | And 3 more authors.
Science in China, Series F: Information Sciences | Year: 2010

This paper proposes a novel hash algorithm whose security is based on the multivariate nonlinear polynomial equations of NP-hard problem over a finite field and combines with HAIFA iterative framework. Over the current widely used hash algorithms, the new algorithm has the following advantages: its security is based on a recognized difficult mathematical problem; the hash length can be changed freely; its design can be automated such that users may construct specific hash function meeting the actual needs. Furthermore, we discuss the security, efficiency and performance of the new algorithm. Under some related difficult mathematical assumptions and theoretical analysis, the new algorithm is proven practical by the experiment results, and capable of achieving security of an ideal hash function by choosing suitable parameters. In addition, it can also be used as a pseudo-random number generator for the good randomness of its output. © 2010 Science China Press and Springer-Verlag Berlin Heidelberg. Source

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