Solving sparse linear equations over finite fields
IEEE Transactions on Information Theory
The inverses of block Hankel and block Toeplitz matrices
SIAM Journal on Computing
On fast multiplication of polynomials over arbitrary algebras
Acta Informatica
Solving homogeneous linear equations over GF(2) via block Wiedemann algorithm
Mathematics of Computation
A Uniform Approach for the Fast Computation of Matrix-Type Pade Approximants
SIAM Journal on Matrix Analysis and Applications
Mathematics of Computation
On randomized Lanczos algorithms
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
Efficient parallel solution of sparse systems of linear diophantine equations
PASCO '97 Proceedings of the second international symposium on Parallel symbolic computation
Modern computer algebra
Diophantine linear system solving
ISSAC '99 Proceedings of the 1999 international symposium on Symbolic and algebraic computation
On Wiedemann's Method of Solving Sparse Linear Systems
AAECC-9 Proceedings of the 9th International Symposium, on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
Approximate algorithms to derive exact solutions to systems of linear equations
EUROSAM '79 Proceedings of the International Symposiumon on Symbolic and Algebraic Computation
Finite field linear algebra subroutines
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
On computing the determinant and Smith form of an integer matrix
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
A p-adic algorithm for univariate partial fractions
SYMSAC '81 Proceedings of the fourth ACM symposium on Symbolic and algebraic computation
On the complexity of polynomial matrix computations
ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
Black box linear algebra with the linbox library
Black box linear algebra with the linbox library
Smith normal form of dense integer matrices fast algorithms into practice
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
A BLAS based C library for exact linear algebra on integer matrices
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
The shifted number system for fast linear algebra on integer matrices
Journal of Complexity - Festschrift for the 70th birthday of Arnold Schönhage
A block Wiedemann rank algorithm
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
Solving toeplitz- and vandermonde-like linear systems with large displacement rank
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Faster inversion and other black box matrix computations using efficient block projections
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Parallel computation of the rank of large sparse matrices from algebraic K-theory
Proceedings of the 2007 international workshop on Parallel symbolic computation
Solving structured linear systems with large displacement rank
Theoretical Computer Science
Fraction-free computation of simultaneous padé approximants
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
On finding multiplicities of characteristic polynomial factors of black-box matrices
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
When e-th roots become easier than factoring
ASIACRYPT'07 Proceedings of the Advances in Crypotology 13th international conference on Theory and application of cryptology and information security
Solving Very Sparse Rational Systems of Equations
ACM Transactions on Mathematical Software (TOMS)
Exact solutions to linear systems of equations using output sensitive lifting
ACM Communications in Computer Algebra
Numeric-symbolic exact rational linear system solver
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Matrix sparsification and nested dissection over arbitrary fields
Journal of the ACM (JACM)
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We propose a new algorithm to find a rational solution to a sparse system of linear equations over the integers. This algorithm is based on a p-adic lifting technique combined with the use of block matrices with structured blocks. It achieves a sub-cubic complexity in terms of machine operations subject to a conjecture on the effectiveness of certain sparse projections. A LinBox-based implementation of this algorithm is demonstrated, and emphasizes the practical benefits of this new method over the previous state of the art.