Development of New Meta-Heuristic For a Bivariate Polynomial

2021 ◽  
Vol 44 (2) ◽  
pp. 58-65
Author(s):  
Sung-Ho Chang ◽  
Moonsoo Kwon ◽  
Geuntae Kim ◽  
Jonghwan Lee
Keyword(s):  
Bivariate Polynomial

scholarly journals Can we Beat the Square Root Bound for ECDLP over 𝔽p2 via Representation?

2020 ◽  
Vol 14 (1) ◽  
pp. 293-306
Author(s):  
Claire Delaplace ◽  
Alexander May
Keyword(s):  
Elliptic Curve ◽  
Quadratic Field ◽  
Discrete Logarithm ◽  
Field Size ◽  
Square Root ◽  
Multivariate Polynomial ◽  
Bivariate Polynomial ◽  
Testing Problem ◽  
Polynomial Zero ◽  
Representation Technique

AbstractWe give a 4-list algorithm for solving the Elliptic Curve Discrete Logarithm (ECDLP) over some quadratic field 𝔽p2. Using the representation technique, we reduce ECDLP to a multivariate polynomial zero testing problem. Our solution of this problem using bivariate polynomial multi-evaluation yields a p1.314-algorithm for ECDLP. While this is inferior to Pollard’s Rho algorithm with square root (in the field size) complexity 𝓞(p), it still has the potential to open a path to an o(p)-algorithm for ECDLP, since all involved lists are of size as small as $\begin{array}{} p^{\frac 3 4}, \end{array}$ only their computation is yet too costly.

scholarly journals Definite determinantal representations of multivariate polynomials

2019 ◽  
Vol 19 (07) ◽  
pp. 2050129 ◽  
Author(s):  
Papri Dey
Keyword(s):  
Semidefinite Programming ◽  
Matrix Polynomial ◽  
Heuristic Method ◽  
Linear Matrix ◽  
Multivariate Polynomials ◽  
Multivariate Polynomial ◽  
Bivariate Polynomial ◽  
Determinantal Representation ◽  
Representation Problem ◽  
Feasible Sets

In this paper, we consider the problem of representing a multivariate polynomial as the determinant of a definite (monic) symmetric/Hermitian linear matrix polynomial (LMP). Such a polynomial is known as determinantal polynomial. Determinantal polynomials can characterize the feasible sets of semidefinite programming (SDP) problems that motivates us to deal with this problem. We introduce the notion of generalized mixed discriminant (GMD) of matrices which translates the determinantal representation problem into computing a point of a real variety of a specified ideal. We develop an algorithm to determine such a determinantal representation of a bivariate polynomial of degree [Formula: see text]. Then we propose a heuristic method to obtain a monic symmetric determinantal representation (MSDR) of a multivariate polynomial of degree [Formula: see text].

Cheating identifiable secret sharing scheme using symmetric bivariate polynomial

2018 ◽  
Vol 453 ◽  
pp. 21-29 ◽  
Author(s):  
Yanxiao Liu ◽  
Chingnung Yang ◽  
Yichuan Wang ◽  
Lei Zhu ◽  
Wenjiang Ji
Keyword(s):  
Secret Sharing ◽  
Secret Sharing Scheme ◽  
Bivariate Polynomial ◽  
Sharing Scheme
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