Solving Multivariate Polynomial Matrix Diophantine Equations with Gröbner Basis Method

2021 ◽  
Author(s):  
Fanghui Xiao ◽  
Dong Lu ◽  
Dingkang Wang
Keyword(s):  
Gröbner Basis ◽  
Diophantine Equations ◽  
Polynomial Matrix ◽  
Grobner Basis ◽  
Multivariate Polynomial ◽  
Basis Method

scholarly journals A note on multivariate polynomial division and Gröbner bases

2015 ◽  
Vol 97 (111) ◽  
pp. 43-48
Author(s):  
Aleksandar Lipkovski ◽  
Samira Zeada
Keyword(s):  
Gröbner Basis ◽  
Gröbner Bases ◽  
Heuristic Approach ◽  
Grobner Bases ◽  
Multivariate Polynomials ◽  
Grobner Basis ◽  
Multivariate Polynomial ◽  
Leading Term ◽  
Monomial Ordering

We first present purely combinatorial proofs of two facts: the well-known fact that a monomial ordering must be a well ordering, and the fact (obtained earlier by Buchberger, but not widely known) that the division procedure in the ring of multivariate polynomials over a field terminates even if the division term is not the leading term, but is freely chosen. The latter is then used to introduce a previously unnoted, seemingly weaker, criterion for an ideal basis to be Grobner, and to suggest a new heuristic approach to Grobner basis computations.

scholarly journals Algebraic Cryptanalysis Scheme of AES-256 Using Gröbner Basis

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Kaixin Zhao ◽  
Jie Cui ◽  
Zhiqiang Xie
Keyword(s):  
Gröbner Basis ◽  
Variable Order ◽  
Polynomial Equations ◽  
Construction Process ◽  
Grobner Basis ◽  
Multivariate Polynomial ◽  
Algebraic Cryptanalysis ◽  
Term Order ◽  
Basis Construction ◽  
Depth Study

The zero-dimensional Gröbner basis construction is a crucial step in Gröbner basis cryptanalysis on AES-256. In this paper, after performing an in-depth study on the linear transformation and the system of multivariate polynomial equations of AES-256, the zero-dimensional Gröbner basis construction method is proposed by choosing suitable term order and variable order. After giving a detailed construction process of the zero-dimensional Gröbner basis, the necessary theoretical proof is presented. Based on this, an algebraic cryptanalysis scheme of AES-256 using Gröbner basis is proposed. Analysis shows that the complexity of our scheme is lower than that of the exhaustive attack.

An Analysis of Inverse Kinematics of Robot Manipulators using Grobner Basis

1997 ◽  
Vol 9 (5) ◽  
pp. 324-331
Author(s):  
Toshimi Shimizu ◽  
◽  
Haruhisa Kawasaki
Keyword(s):  
Computer Algebra ◽  
Inverse Kinematics ◽  
Gröbner Basis ◽  
Robot Manipulators ◽  
Symbolic Analysis ◽  
New Method ◽  
Polynomial Equations ◽  
Trigonometric Functions ◽  
Grobner Basis ◽  
Multivariate Polynomial

This paper presents a new method for solving the inverse kinematics of robot manipulators symbolically using computer algebra. The kinematics equations, including the trigonometric functions of joint displacements, are expressed as multivariate polynomial equations by transforming these functions into variables. The multivariate polynomial equations can be solved by evaluating their reduced Grobner basis. The properties for efficient evaluation of the reduced Grobner basis and the inverse kinematics of a robot, whose last three joint axes intersect at a point, are shown. This procedure is implemented using Maple V and built into ROSAM (Robot Symbolic Analysis, by Maple) that is a robot analysis library made by our group. An analysis example of a structurechanged PUMA type robot is given.

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