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LINBOX: A GENERIC LIBRARY FOR EXACT LINEAR ALGEBRA
The design of this generic library LinBox is described, its current range of capabilities are sketched, its guiding design principle of re-usability is given, and several examples of its use are given.
Dense Linear Algebra over Word-Size Prime Fields: the FFLAS and FFPACK Packages
This article studies high performance implementations of basic linear algebra routines over word size prime fields: especially matrix multiplication, with the goal being to provide an exact alternate to the numerical BLAS library.
On Newton–Raphson Iteration for Multiplicative Inverses Modulo Prime Powers
- J. Dumas
- MathematicsIEEE Transactions on Computers
- 28 September 2012
A hybrid combination of the explicit formula for the inverse of Newton-Raphson iteration over p-adic numbers and the best asymptotic variants is proposed, which yields then a constant factor improvement, also for large exponents.
Computing Simplicial Homology Based on Efficient Smith Normal Form Algorithms
- J. Dumas, F. Heckenbach, B. D. Saunders, V. Welker
- MathematicsAlgebra, Geometry, and Software Systems
Alternative approaches to the calculation of simplicial homology are described and motivating examples and actual experiments with the GAP package that was implemented by the authors are described.
FFPACK: finite field linear algebra package
This paper shows that those optimal complexity algorithms, such as LSP factorization, rank determinant and inverse computation can also be the most efficient.
Efficient computation of the characteristic polynomial
A probabilistic approach, based on integer minimal polynomial and Hensel factorization, is particularly well suited to sparse and/or structured matrices.
Essentially optimal interactive certificates in linear algebra
All the authors' certificates are based on interactive verification protocols with the interaction removed by a Fiat-Shamir identification heuristic, and the validity of the verification procedure is subject to standard computational hardness assumptions from cryptography.
Finite field linear algebra subroutines
Very efficient implementations of finite field dot products, matrix-vector products and matrix-matrix products (namely the symbolic equivalent of level 1, 2 and 3 BLAS) are presented.
Physical Zero-Knowledge Proofs for Akari, Takuzu, Kakuro and KenKen
Physical algorithms to realize zero-knowledge proofs for these logic games which allow a player to show that he knows a solution without revealing it are given and can be realized with simple office material as they only rely on cards and envelopes.
On Efficient Sparse Integer Matrix Smith Normal Form Computations
We present a new algorithm to compute the Integer Smith normal form of large sparse matrices. We reduce the computation of the Smith form to independent, and therefore parallel, computations modulo…