Journal ArticleDOI
Solving sparse linear equations over finite fields
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A "coordinate recurrence" method for solving sparse systems of linear equations over finite fields is described and a probabilistic algorithm is shown to exist for finding the determinant of a square matrix.Abstract:
A "coordinate recurrence" method for solving sparse systems of linear equations over finite fields is described. The algorithms discussed all require O(n_{1}(\omega + n_{1})\log^{k}n_{1}) field operations, where n_{1} is the maximum dimension of the coefficient matrix, \omega is approximately the number of field operations required to apply the matrix to a test vector, and the value of k depends on the algorithm. A probabilistic algorithm is shown to exist for finding the determinant of a square matrix. Also, probabilistic algorithms are shown to exist for finding the minimum polynomial and rank with some arbitrarily small possibility of error.read more
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References
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Book
The Theory of Error-Correcting Codes
TL;DR: This book presents an introduction to BCH Codes and Finite Fields, and methods for Combining Codes, and discusses self-dual Codes and Invariant Theory, as well as nonlinear Codes, Hadamard Matrices, Designs and the Golay Code.
Journal ArticleDOI
Shift-register synthesis and BCH decoding
TL;DR: It is shown in this paper that the iterative algorithm introduced by Berlekamp for decoding BCH codes actually provides a general solution to the problem of synthesizing the shortest linear feedback shift register capable of generating a prescribed finite sequence of digits.
Journal ArticleDOI
Fast Probabilistic Algorithms for Verification of Polynomial Identities
TL;DR: Vanous fast probabdlsttc algonthms, with probability of correctness guaranteed a prion, are presented for testing polynomial ldentmes and propemes of systems of polynomials and ancdlary fast algorithms for calculating resultants and Sturm sequences are given.
Journal ArticleDOI
Fast solution of toeplitz systems of equations and computation of Padé approximants
TL;DR: It is proved that entries in the Pade table can be computed by the Extended Euclidean Algorithm, and an algorithm EMGCD (Extended Middle Greatest Common Divisor) is described which is faster than the algorithm HGCD of Aho, Hopcroft and Ullman, although both require time O(n log2 n).
Journal ArticleDOI
Fast evaluation of logarithms in fields of characteristic two
TL;DR: The ideas give a dramatic improvement even for moderate-sized fields such as GF (2^{127}) , and make (barely) possible computations in fields of size around 2^{400} .