Flexible Partial Enlargement to Accelerate Gröbner Basis Computation over $\mathbb{F}_2$

2010 ◽  
pp. 69-81 ◽  
Author(s):  
Johannes Buchmann ◽  
Daniel Cabarcas ◽  
Jintai Ding ◽  
Mohamed Saied Emam Mohamed
Keyword(s):  
Gröbner Basis ◽  
Grobner Basis ◽  
Basis Computation

scholarly journals On the robust hardness of Gröbner basis computation

2019 ◽  
Vol 223 (5) ◽  
pp. 2080-2100
Author(s):  
David Rolnick ◽  
Gwen Spencer
Keyword(s):  
Gröbner Basis ◽  
Grobner Basis ◽  
Basis Computation

scholarly journals Constructions of Maximum Few-Distance Sets in Euclidean Spaces

10.37236/8565 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Ferenc Szöllősi ◽  
Patric R.J. Östergård
Keyword(s):  
Euclidean Space ◽  
Gröbner Basis ◽  
Dimensional Euclidean Space ◽  
Combined Approach ◽  
Euclidean Spaces ◽  
Grobner Basis ◽  
Finite Set ◽  
Distance Set ◽  
Distance Sets ◽  
Basis Computation

A finite set of vectors $\mathcal{X}$ in the $d$-dimensional Euclidean space $\mathbb{R}^d$ is called an $s$-distance set if the set of mutual distances between distinct elements of $\mathcal{X}$ has cardinality exactly $s$. In this paper we present a combined approach of isomorph-free exhaustive generation of graphs and Gröbner basis computation to classify the largest $3$-distance sets in $\mathbb{R}^4$, the largest $4$-distance sets in $\mathbb{R}^3$, and the largest $6$-distance sets in $\mathbb{R}^2$. We also construct new examples of large $s$-distance sets in $\mathbb{R}^d$ for $d\leq 8$ and $s\leq 6$, and independently verify several earlier results from the literature.

An Efficient Implementation of Boolean Gröbner Basis Computation

2017 ◽  
pp. 116-130
Author(s):  
Rodrigo Alexander Castro Campos ◽  
Feliú Davino Sagols Troncoso ◽  
Francisco Javier Zaragoza Martínez
Keyword(s):  
Gröbner Basis ◽  
Efficient Implementation ◽  
Grobner Basis ◽  
Basis Computation

scholarly journals Complexity bounds on Semaev’s naive index calculus method for ECDLP

2020 ◽  
Vol 14 (1) ◽  
pp. 460-485
Author(s):  
Kazuhiro Yokoyama ◽  
Masaya Yasuda ◽  
Yasushi Takahashi ◽  
Jun Kogure
Keyword(s):  
Lower Bound ◽  
Gröbner Basis ◽  
Discrete Logarithm ◽  
Polynomial Systems ◽  
Grobner Basis ◽  
Step Method ◽  
Complexity Bounds ◽  
Index Calculus ◽  
Basis Computation ◽  
The Ideal

AbstractSince Semaev introduced summation polynomials in 2004, a number of studies have been devoted to improving the index calculus method for solving the elliptic curve discrete logarithm problem (ECDLP) with better complexity than generic methods such as Pollard’s rho method and the baby-step and giant-step method (BSGS). In this paper, we provide a deep analysis of Gröbner basis computation for solving polynomial systems appearing in the point decomposition problem (PDP) in Semaev’s naive index calculus method. Our analysis relies on linear algebra under simple statistical assumptions on summation polynomials. We show that the ideal derived from PDP has a special structure and Gröbner basis computation for the ideal is regarded as an extension of the extended Euclidean algorithm. This enables us to obtain a lower bound on the cost of Gröbner basis computation. With the lower bound, we prove that the naive index calculus method cannot be more efficient than generic methods.

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