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 extensiv...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.展开更多
This paper presents an improved early termination algorithm for sparse black box multivariate polynomials, which reduces the interpolation problem into several sub-interpolation problems with less variables and fewer ...This paper presents an improved early termination algorithm for sparse black box multivariate polynomials, which reduces the interpolation problem into several sub-interpolation problems with less variables and fewer terms. Actually, all interpolations are eventually reduced to the interpolation of a list of polynomials with less terms than that of the original polynomial. Extensive experiments show that the new algorithm is much faster than the original algorithm.展开更多
基金supported by the National Natural Science Foundation of China under Grant Nos.11471209,11561015,and 11301066Guangxi Key Laboratory of Cryptography and Information Security under Grant No.GCIS201615
文摘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.
基金supported by the National Natural Science Foundation of China under Grant No.11688101
文摘This paper presents an improved early termination algorithm for sparse black box multivariate polynomials, which reduces the interpolation problem into several sub-interpolation problems with less variables and fewer terms. Actually, all interpolations are eventually reduced to the interpolation of a list of polynomials with less terms than that of the original polynomial. Extensive experiments show that the new algorithm is much faster than the original algorithm.
