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.

scholarly journals Reduction of Feynman integrals in the parametric representation II: reduction of tensor integrals

2021 ◽  
Vol 81 (3) ◽  
Author(s):  
Wen Chen
Keyword(s):  
Gröbner Basis ◽  
Parametric Representation ◽  
Polynomial Equations ◽  
Grobner Basis ◽  
Spacetime Dimension ◽  
Integration By Parts ◽  
Feynman Integrals ◽  
Parametric Integrals ◽  
Parametric Integration

AbstractIn a recent paper by the author (Chen in JHEP 02:115, 2020), the reduction of Feynman integrals in the parametric representation was considered. Tensor integrals were directly parametrized by using a generator method. The resulting parametric integrals were reduced by constructing and solving parametric integration-by-parts (IBP) identities. In this paper, we furthermore show that polynomial equations for the operators that generate tensor integrals can be derived. Based on these equations, two methods to reduce tensor integrals are developed. In the first method, by introducing some auxiliary parameters, tensor integrals are parametrized without shifting the spacetime dimension. The resulting parametric integrals can be reduced by using the standard IBP method. In the second method, tensor integrals are (partially) reduced by using the technique of Gröbner basis combined with the application of symbolic rules. The unreduced integrals can further be reduced by solving parametric IBP identities.

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 Divisors on graphs, Connected flags, and Syzygies

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Fatemeh Mohammadi ◽  
Farbod Shokrieh
Keyword(s):  
Gröbner Basis ◽  
Betti Numbers ◽  
Point Of View ◽  
Grobner Basis ◽  
Generating Set ◽  
Term Order ◽  
Syzygy Module ◽  
International Audience ◽  
The Given ◽  
Minimal Generating Set

International audience We study the binomial and monomial ideals arising from linear equivalence of divisors on graphs from the point of view of Gröbner theory. We give an explicit description of a minimal Gröbner basis for each higher syzygy module. In each case the given minimal Gröbner basis is also a minimal generating set. The Betti numbers of $I_G$ and its initial ideal (with respect to a natural term order) coincide and they correspond to the number of ``connected flags'' in $G$. Moreover, the Betti numbers are independent of the characteristic of the base field.

Parallel algorithms for Gröbner-basis construction

2007 ◽  
Vol 142 (4) ◽  
pp. 2248-2266 ◽  
Author(s):  
V. A. Mityunin ◽  
E. V. Pankratiev
Keyword(s):  
Parallel Algorithms ◽  
Gröbner Basis ◽  
Grobner Basis ◽  
Basis Construction

scholarly journals Grobner Basis Over Ideals of Multivariate Polynomial Ring: قواعد جروبنر فوق مثاليَّات الحدوديَّات بأكثر من متغير

2019 ◽  
Vol 3 (2) ◽  
Author(s):  
Khaled Suleiman Al-Akla
Keyword(s):  
Polynomial Ring ◽  
Gröbner Basis ◽  
Main Axis ◽  
Structural Approach ◽  
Algebraic Equations ◽  
Grobner Basis ◽  
Multivariate Polynomial ◽  
Multiple Variables ◽  
Definition Of

Grobner basis are considered one of the modern mathematical tools which has become of interest for the researchers in all fields of mathematics. Grobner basis are generally polynomials with multiple variables that has certain characteristics. it's includes two main axis:                                                                            1- The first axis we have presented the definition of Grobner basis and their properties. 2- The second axis we have studied some applications of Grobner basis, and we give some examples about its. The goal of these paper is to identify Grobner basis and some algorithms related to how to find them and talked about the most important applications, including: the issue of belonging and the issue of containment, and to reach our goal to follow the analytical and structural approach, we defined these basis and we have many results, The Grosvenor we obtained is not alone in general and to be single, some additional conditions must be set on these basis, and we conclude that Grobner basis have many applications in the solutions of algebraic equations in more than one transformer and in many fields.

scholarly journals An Algorithm to Compute the H-Bases for Ideals of Subalgebras

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Rabia ◽  
Muhammad Ahsan Binyamin ◽  
Nazia Jabeen ◽  
Adnan Aslam ◽  
Kraidi Anoh Yannick
Keyword(s):  
Polynomial Ring ◽  
Gröbner Basis ◽  
Polynomial Rings ◽  
Grobner Basis ◽  
Finitely Generated ◽  
Computational Algebra ◽  
Important Ingredient ◽  
Multivariate Polynomial

The concept of H-bases, introduced long ago by Macauly, has become an important ingredient for the treatment of various problems in computational algebra. The concept of H-bases is for ideals in polynomial rings, which allows an investigation of multivariate polynomial spaces degree by degree. Similarly, we have the analogue of H-bases for subalgebras, termed as SH-bases. In this paper, we present an analogue of H-bases for finitely generated ideals in a given subalgebra of a polynomial ring, and we call them “HSG-bases.” We present their connection to the SAGBI-Gröbner basis concept, characterize HSG-basis, and show how to construct them.

scholarly journals A novel architecture with scalable security having expandable computational complexity for stream ciphers

2017 ◽  
Vol 30 (4) ◽  
pp. 459-475
Author(s):  
Prathap Siddavaatam ◽  
Reza Sedaghat
Keyword(s):  
Computational Complexity ◽  
Nonlinear System ◽  
Linear Algebra ◽  
Stream Cipher ◽  
Algebraic Degree ◽  
Stream Ciphers ◽  
Polynomial Equations ◽  
Hardware Complexity ◽  
Multivariate Polynomial ◽  
Algebraic Cryptanalysis

Stream cipher designs are difficult to implement since they are prone to weaknesses based on usage, with properties being similar to one-time pad besides keystream is subjected to very strict requirements. Contemporary stream cipher designs are highly vulnerable to algebraic cryptanalysis based on linear algebra, in which the inputs and outputs are formulated as multivariate polynomial equations. Solving a nonlinear system of multivariate equations will reduce the complexity, which in turn yields the targeted secret information. Recently, Addition Modulo has been suggested over logic XOR as a mixing operator to guard against such attacks. However, it has been observed that the complexity of Modulo Addition can be drastically decreased with the appropriate formulation of polynomial equations and probabilistic conditions. A new design for Addition Modulo is proposed. The framework for the new design is characterized by user-defined expandable security for stronger encryption and does not impose changes in existing layout for any stream cipher such as SNOW 2.0, SOSEMANUK, CryptMT, Grain Family, etc. The structure of the proposed design is highly scalable, which boosts the algebraic degree and thwarts the probabilistic conditions by maintaining the original hardware complexity without changing the integrity of the Addition Modulo.

Resolution of a system of fuzzy polynomial equations using the Gröbner basis

2013 ◽  
Vol 220 ◽  
pp. 541-558 ◽  
Author(s):  
Ali Abbasi Molai ◽  
Abdolali Basiri ◽  
Sajjad Rahmany
Keyword(s):  
Gröbner Basis ◽  
Polynomial Equations ◽  
Grobner Basis
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