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Computing Sparse GCD of Multivariate Polynomials via Polynomial Interpolation

Journal of Systems Science & Complexity 2018年第2期31卷 552-568页

作者： TANG Min LI Bingyu ZENG Zhenbing School of Mathematics and Computing Science Guangxi Key Laboratory of Cryptography and Information Security Guilin University of Electronic Technology Guilin 541004 China Shanghai Key Laboratory of Trustworthy Computing East China Normal University Shanghai 200062 China. Email: ***@***. School of Mathematics and Statistics Northeast Normal University Changchun 130024 China. Department of Mathematics Shanghai University Shanghai 200444 China

The problem of computing the greatest common divisor（GCD） of multivariate polynomials, as one of the most important tasks of computer algebra and symbolic computation in more general scope, has been studied extensively since the beginning of the interdisciplinary of mathematics with computer science. For many real applications such as digital image restoration and enhancement,robust control theory of nonlinear systems, L1-norm convex optimization in compressed sensing techniques, as well as algebraic decoding of Reed-Solomon and BCH codes, the concept of sparse GCD plays a core role where only the greatest common divisors with much fewer terms than the original polynomials are of interest due to the nature of problems or data structures. This paper presents two methods via multivariate polynomial interpolation which are based on the variation of Zippel＇s method and ben-or/tiwari algorithm, respectively. To reduce computational complexity, probabilistic techniques and randomization are employed to deal with univariate GCD computation and univariate polynomial interpolation. The authors demonstrate the practical performance of our algorithms on a significant body of examples. The implemented experiment illustrates that our algorithms are efficient for a quite wide range of input.

关键词： ben-or/tiwari algorithm multivariate polynomial interpolation sparse GCD Zippel＇s algorithm.

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