Invariant Sets and Output Dead Beat Controllability for Odd Polynomial Systems: The Gröbner Basis Method

1996 ◽  
Vol 29 (1) ◽  
pp. 2102-2107
Author(s):  
Dragan Nešić ◽  
Iven M.Y. Mareels
Keyword(s):  
Gröbner Basis ◽  
Polynomial Systems ◽  
Invariant Sets ◽  
Grobner Basis ◽  
Basis Method

scholarly journals On the first fall degree of summation polynomials

2019 ◽  
Vol 13 (3-4) ◽  
pp. 229-237
Author(s):  
Stavros Kousidis ◽  
Andreas Wiemers
Keyword(s):  
Elliptic Curve ◽  
Gröbner Basis ◽  
Discrete Logarithm ◽  
Polynomial Systems ◽  
Discrete Logarithm Problem ◽  
Grobner Basis ◽  
Weil Descent ◽  
Degree Bound

Abstract We improve on the first fall degree bound of polynomial systems that arise from a Weil descent along Semaev’s summation polynomials relevant to the solution of the Elliptic Curve Discrete Logarithm Problem via Gröbner basis algorithms.

scholarly journals An extension of Buchberger’s criteria for Gröbner basis decision

2010 ◽  
Vol 13 ◽  
pp. 111-129
Author(s):  
John Perry
Keyword(s):  
Gröbner Basis ◽  
Gröbner Bases ◽  
Characterization Theorem ◽  
Polynomial Systems ◽  
Polynomial Ideal ◽  
Grobner Bases ◽  
Grobner Basis ◽  
Case Scenario ◽  
Worst Case ◽  
New Criterion

AbstractTwo fundamental questions in the theory of Gröbner bases are decision (‘Is a basisGof a polynomial ideal a Gröbner basis?’) and transformation (‘If it is not, how do we transform it into a Gröbner basis?’) This paper considers the first question. It is well known thatGis a Gröbner basis if and only if a certain set of polynomials (theS-polynomials) satisfy a certain property. In general there arem(m−1)/2 of these, wheremis the number of polynomials inG, but criteria due to Buchberger and others often allow one to consider a smaller number. This paper presents two original results. The first is a new characterization theorem for Gröbner bases that makes use of a new criterion that extends Buchberger’s criteria. The second is the identification of a class of polynomial systemsGfor which the new criterion has dramatic impact, reducing the worst-case scenario fromm(m−1)/2 S-polynomials tom−1.

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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