Fast Bit-Parallel Shifted Polynomial Basis Multiplier Using Weakly Dual Basis Over $GF(2^{m})$

2011 ◽  
Vol 19 (12) ◽  
pp. 2317-2321 ◽  
Author(s):  
Sun-Mi Park ◽  
Ku-Young Chang
Keyword(s):  
Polynomial Basis ◽  
Dual Basis

scholarly journals The expansion of Hall-Littlewood functions in the dual Grothendieck polynomial basis

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Jason Bandlow ◽  
Jennifer Morse
Keyword(s):  
Symmetric Functions ◽  
Polynomial Basis ◽  
Dual Basis ◽  
Nous Obtenons ◽  
Plane Partitions ◽  
Generalize Charge ◽  
Littlewood Polynomials ◽  
International Audience ◽  
Reverse Plane ◽  
Grothendieck Polynomials

International audience A combinatorial expansion of the Hall-Littlewood functions into the Schur basis of symmetric functions was first given by Lascoux and Schützenberger, with their discovery of the charge statistic. A combinatorial expansion of stable Grassmannian Grothendieck polynomials into monomials was first given by Buch, using set-valued tableaux. The dual basis of the stable Grothendieck polynomials was given a combinatorial expansion into monomials by Lam and Pylyavskyy using reverse plane partitions. We generalize charge to set-valued tableaux and use all of these combinatorial ideas to give a nice expansion of Hall-Littlewood polynomials into the dual Grothendieck basis. \par En associant une charge à un tableau, une formule combinatoire donnant le développement des polynômes de Hall-Littlewood en termes des fonctions de Schur a été obtenue par Lascoux et Schützenberger. Une formule combinatoire donnant le développement des polynômes de Grothendieck Grassmanniens stables en termes des fonctions monomiales a quant à elle été obtenue par Buch à l'aide de tableaux à valeurs sur des ensembles. Finalement, une formule faisant intervenir des partitions planaires inverses a été obtenue par Lam et Pylyavskyy pour donner le développement de la base duale aux polynômes de Grothendieck stables en termes de monômes. Nous généralisons le concept de charge aux tableaux à valeurs sur des ensembles et, en nous servant de toutes ces notions combinatoires, nous obtenons une formule élégante donnant le développement des polynômes de Hall-Littlewood en termes de la base de Grothendieck duale.

Dual basis systolic multipliers for GF(2m)

1997 ◽  
Vol 144 (1) ◽  
pp. 43 ◽  
Author(s):  
S.T.J. Fenn ◽  
M. Benaissa ◽  
D. Taylor
Keyword(s):  
Dual Basis

On the Bases of Wrench Spaces for the Kinematic and Dynamic Analysis of Mechanisms

2003 ◽  
Vol 125 (3) ◽  
pp. 552-556 ◽  
Author(s):  
Koichi Sugimoto
Keyword(s):  
Dynamic Analysis ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Closed Loop ◽  
Computational Procedure ◽  
Analysis Procedure ◽  
Dual Basis ◽  
Passive Joint

The aim of this paper is to find out a computational procedure for the kinematic and dynamic analysis of a mechanism with multiple loops having motion spaces of a Lie algebra or Lie algebras. The basis of a motion space of the loop is determined such that it consists of passive joints axes in a loop, and a basis of a wrench space is determined to be its dual basis. The analysis of a closed loop mechanism can be done by selecting loop-cut-joints and computing values of wrenches acting on these joints from the condition that virtual works of passive joints are zero. By expressing these wrenches in the coordinate vectors on the dual bases, the concise analysis procedure can be obtained. Because a formulation for the analysis is developed based on the bases consisting of passive joint axes and their dual bases, the computational procedure can be applied to a mechanism with any Lie algebras.

scholarly journals Efficient Nonrecursive Bit-Parallel Karatsuba Multiplier for a Special Class of Trinomials

VLSI Design ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-7 ◽  
Author(s):  
Yin Li ◽  
Yu Zhang ◽  
Xiaoli Guo
Keyword(s):  
Time Delay ◽  
Special Class ◽  
Time Complexity ◽  
Space Complexity ◽  
Polynomial Basis ◽  
Gate Delay ◽  
Complexity Bound ◽  
Parallel Multiplier ◽  
Highly Efficient ◽  
First Time

Recently, we present a novel Mastrovito form of nonrecursive Karatsuba multiplier for all trinomials. Specifically, we found that related Mastrovito matrix is very simple for equally spaced trinomial (EST) combined with classic Karatsuba algorithm (KA), which leads to a highly efficient Karatsuba multiplier. In this paper, we consider a new special class of irreducible trinomial, namely, xm+xm/3+1. Based on a three-term KA and shifted polynomial basis (SPB), a novel bit-parallel multiplier is derived with better space and time complexity. As a main contribution, the proposed multiplier costs about 2/3 circuit gates of the fastest multipliers, while its time delay matches our former result. To the best of our knowledge, this is the first time that the space complexity bound is reached without increasing the gate delay.

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