scholarly journals The MOR Cryptosystem in Classical Groups with a Gaussian Elimination Algorithm for Symplectic and Orthogonal Groups

2019 ◽  
Author(s):  
Sushil Bhunia ◽  
Ayan Mahalanobis ◽  
Pralhad Shinde ◽  
Anupam Singh
Keyword(s):  
Gaussian Elimination ◽  
Classical Groups ◽  
Orthogonal Groups ◽  
Elimination Algorithm

scholarly journals Gaussian Elimination versus Greedy Methods for the Synthesis of Linear Reversible Circuits

2021 ◽  
Vol 2 (3) ◽  
pp. 1-26
Author(s):  
Timothée Goubault De Brugière ◽  
Marc Baboulin ◽  
Benoît Valiron ◽  
Simon Martiel ◽  
Cyril Allouche
Keyword(s):  
Quantum Computing ◽  
State Of The Art ◽  
Gaussian Elimination ◽  
Random Operators ◽  
Lu Factorization ◽  
Large Problem ◽  
Computational Costs ◽  
Elimination Algorithm ◽  
Reversible Circuits ◽  
Greedy Methods

Linear reversible circuits represent a subclass of reversible circuits with many applications in quantum computing. These circuits can be efficiently simulated by classical computers and their size is polynomially bounded by the number of qubits, making them a good candidate to deploy efficient methods to reduce computational costs. We propose a new algorithm for synthesizing any linear reversible operator by using an optimized version of the Gaussian elimination algorithm coupled with a tuned LU factorization. We also improve the scalability of purely greedy methods. Overall, on random operators, our algorithms improve the state-of-the-art methods for specific ranges of problem sizes: The custom Gaussian elimination algorithm provides the best results for large problem sizes (n > 150), while the purely greedy methods provide quasi optimal results when n < 30. On a benchmark of reversible functions, we manage to significantly reduce the CNOT count and the depth of the circuit while keeping other metrics of importance (T-count, T-depth) as low as possible.

scholarly journals TILING BRANCHING MULTIPLICITY SPACES WITH GL PATTERN BLOCKS

2013 ◽  
Vol 94 (3) ◽  
pp. 362-374 ◽  
Author(s):  
SANGJIB KIM
Keyword(s):  
Classical Groups ◽  
Orthogonal Groups ◽  
Branching Rules

AbstractWe study branching multiplicity spaces of complex classical groups in terms of ${\mathrm{GL} }_{2} $ representations. In particular, we show how combinatorics of ${\mathrm{GL} }_{2} $ representations are intertwined to make branching rules under the restriction of ${\mathrm{GL} }_{n} $ to ${\mathrm{GL} }_{n- 2} $. We also discuss analogous results for the symplectic and orthogonal groups.

A SOLVER OF LINEAR SYSTEMS OF EQUATIONS (REBSM) FOR LARGE-SCALE ENGINEERING PROBLEMS

2012 ◽  
Vol 09 (01) ◽  
pp. 1240011 ◽  
Author(s):  
XIAO-WEI GAO ◽  
LINGJIE LI
Keyword(s):  
Large Scale ◽  
Solution Method ◽  
Gaussian Elimination ◽  
Systems Of Equations ◽  
The Finite Element Method ◽  
Elimination Algorithm ◽  
Engineering Problems ◽  
Sparse Systems ◽  
Linear Systems Of Equations ◽  
Element Method

In this paper, a novel linear equation solution method is proposed based on a row elimination back-substitution method (REBSM). The elimination and back-substitution procedures are carried out on individual row levels. The advantage of the proposed method is that it is much faster and requires less storage than the Gaussian elimination algorithm and, therefore, is capable of solving larger systems of equations. The method is particularly efficient for solving band diagonal sparse systems with symmetric or nonsymmetric coefficient matrices, and can be easily applied to popular numerical methods, such as the finite element method and the boundary element method. Detailed Fortran codes and examples are given to demonstrate the robustness and efficiency of the proposed method.

Benchmarking a Sparse Elimination Routine on the Cyber 205 and the Cray 1-S

1983 ◽  
Vol 23 (05) ◽  
pp. 743-745 ◽  
Author(s):  
P.T. Woo ◽  
John M. Levesque
Keyword(s):  
Gaussian Elimination ◽  
Assembly Language ◽  
Triangular Matrix ◽  
Floating Point ◽  
Full Potential ◽  
Vector Computer ◽  
Solution Vector ◽  
Elimination Algorithm ◽  
Lower Triangular ◽  
Vector Computers

Abstract An existing general sparse-elimination algorithm was benchmarked on the Cyber 205 and the Cray 1-S. Computation rates on the Cyber 205 reached 16. 7 million floating point operations per second (MFLOPS) and 9. 8 MFLOPS on the Cray 1-S. It is concluded that the existing algorithm does not exploit the full potential of the vector computer. Several schemes are under investigation to improve the performance of the algorithm. Introduction General-purpose sparse-elimination techniques have been used successfully in solving systems of linear equations in reservoir simulation. These techniques allow Gaussian elimination to be performed efficiently with any given matrix ordering. The implementation of these techniques on a scalar computer such as the IBM 370/158 and on an array processor such as the Floating Point Systems AP120B are described in Refs. 1 and 2. respectively. The purpose of this paper is to report the results of a benchmark test to determine the performance of a known sparse-elimination algorithm on the vector computers Cyber 205 and Cray 1-S. The ground rules for the benchmark were as follows.The sparse-elimination algorithm was to remain unchanged.The use of vectorizable FORTRAN or vector syntax FORTRAN was allowed.The use of assembly-language coding was not allowed unless the assembly-language code was part of a systems or mathematics library callable by FORTRAN.The alternate diagonal grid ordering was to be used. Implementation on the Vector Computer The algorithm NNF in the Yale sparse matrix packages was used for benchmarking. The matrix equation to be solved is A is factored into the lower triangular matrix, L, and the upper triangular matrix, U. The elements of A, L, and U are stored in a compressed storage mode as described by Gustavson. The solution vector x is obtained by forward substitution by using L, and backward substitution by using U. The Yale algorithm uses the row operations approach, which consists of two steps:multiply one equation by a nonzero number, andadd (or subtract) the equation from above to (or from) another equation. Step 2 reduces the nonzeros in the lower triangular portion of A to zeros as factorization proceeds. portion of A to zeros as factorization proceeds. SPEJ P. 743

scholarly journals A theorem on power series with applications to classical groups over finite fields

2001 ◽  
Vol 64 (1) ◽  
pp. 121-129
Author(s):  
Andrew J. Spencer
Keyword(s):  
Power Series ◽  
Finite Fields ◽  
Generating Functions ◽  
General Linear ◽  
Classical Groups ◽  
Orthogonal Groups

For some of the classical groups over finite fields it is possible to express the proportion of eigenvalue-free matrices in terms of generating functions. We prove a theorem on the monotonicity of the coefficients of powers of power series and apply this to the generating functions of the general linear, symplectic and orthogonal groups. This proves a conjecture on the monotonicity of the proportions of eigenvalue-free elements in these groups.

scholarly journals Factorial Characters and Tokuyama's Identity for Classical Groups

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Angèle M. Hamel ◽  
Ronald C. King
Keyword(s):  
Recurrence Relations ◽  
Function Theory ◽  
Lattice Paths ◽  
Schur Function ◽  
Classical Groups ◽  
Orthogonal Groups ◽  
Schubert Polynomial ◽  
International Audience ◽  
Symplectic Case ◽  
Polynomial Theory

International audience In this paper we introduce factorial characters for the classical groups and derive a number of central results. Classically, the factorial Schur function plays a fundamental role in traditional symmetric function theory and also in Schubert polynomial theory. Here we develop a parallel theory for the classical groups, offering combinatorial definitions of the factorial characters for the symplectic and orthogonal groups, and further establish flagged factorial Jacobi-Trudi identities and factorial Tokuyama identities, providing proofs in the symplectic case. These identities are established by manipulating determinants through the use of certain recurrence relations and by using lattice paths.

scholarly journals Parallelization of the Gaussian Elimination Algorithm on Systolic Arrays

1996 ◽  
Vol 33 (1) ◽  
pp. 69-75 ◽  
Author(s):  
J-C. Bermond ◽  
C. Peyrat ◽  
I. Sakho ◽  
M. Tchuenté
Keyword(s):  
Gaussian Elimination ◽  
Systolic Arrays ◽  
Elimination Algorithm

scholarly journals A reciprocal branching problem for automorphic representations and global Vogan packets

2020 ◽  
Vol 2020 (765) ◽  
pp. 249-277 ◽  
Author(s):  
Dihua Jiang ◽  
Baiying Liu ◽  
Bin Xu
Keyword(s):  
Irreducible Representation ◽  
Symmetry Breaking ◽  
Descent Method ◽  
Classical Groups ◽  
Orthogonal Groups ◽  
Automorphic Representations ◽  
Branching Rule ◽  
Branching Problem

AbstractLet G be a group and let H be a subgroup of G. The classical branching rule (or symmetry breaking) asks: For an irreducible representation π of G, determine the occurrence of an irreducible representation σ of H in the restriction of π to H. The reciprocal branching problem of this classical branching problem is to ask: For an irreducible representation σ of H, find an irreducible representation π of G such that σ occurs in the restriction of π to H. For automorphic representations of classical groups, the branching problem has been addressed by the well-known global Gan–Gross–Prasad conjecture. In this paper, we investigate the reciprocal branching problem for automorphic representations of special orthogonal groups using the twisted automorphic descent method as developed in [13]. The method may be applied to other classical groups as well.

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