Solving sparse linear equations over finite fields
IEEE Transactions on Information Theory
Representations and parallel computations for rational functions
SIAM Journal on Computing
A deterministic algorithm for sparse multivariate polynomial interpolation
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Modular rational sparse multivariate polynomial interpolation
ISSAC '90 Proceedings of the international symposium on Symbolic and algebraic computation
Parallel least-squares solution of general and Toeplitz systems
SPAA '90 Proceedings of the second annual ACM symposium on Parallel algorithms and architectures
Processor efficient parallel solution of linear systems over an abstract field
SPAA '91 Proceedings of the third annual ACM symposium on Parallel algorithms and architectures
On fast multiplication of polynomials over arbitrary algebras
Acta Informatica
Polynomial and matrix computations (vol. 1): fundamental algorithms
Polynomial and matrix computations (vol. 1): fundamental algorithms
Mathematics of Computation
Parallel computation of polynomial GCD and some related parallel computations over abstract fields
Theoretical Computer Science
Faster solution of the key equation for decoding BCH error-correcting codes
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Fast Probabilistic Algorithms for Verification of Polynomial Identities
Journal of the ACM (JACM)
ICALP '93 Proceedings of the 20th International Colloquium on Automata, Languages and Programming
Probabilistic algorithms for sparse polynomials
EUROSAM '79 Proceedings of the International Symposiumon on Symbolic and Algebraic Computation
STOC '73 Proceedings of the fifth annual ACM symposium on Theory of computing
On square-free decomposition algorithms
SYMSAC '76 Proceedings of the third ACM symposium on Symbolic and algebraic computation
Shift-register synthesis and BCH decoding
IEEE Transactions on Information Theory
The algebraic decoding of Goppa codes
IEEE Transactions on Information Theory
A method for decoding of generalized Goppa codes (Corresp.)
IEEE Transactions on Information Theory
On the complexity of decoding Goppa codes (Corresp.)
IEEE Transactions on Information Theory
Symbolic and numeric methods for exploiting structure in constructing resultant matrices
Journal of Symbolic Computation
Multivariate power series multiplication
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
Applications of FFT and structured matrices
Algorithms and theory of computation handbook
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The n coefficients of a fixed linear recurrence can be expressed through its m@?2n terms or, equivalently, the coefficients of a polynomial of a degree n can be expressed via the power sums of its zeros-by means of a polynomial equation known as the key equation for decoding the BCH error-correcting codes. The same problem arises in sparse multivariate polynomial interpolation and in various fundamental computations with sparse matrices in finite fields. Berlekamp's algorithm of 1968 solves the key equation by using order of n^2 operations in a fixed field. Several algorithms of 1975-1980 rely on the extended Euclidean algorithm and computing Pade approximation, which yields a solution in O(n(log n)^2 log log n) operations, though a considerable overhead constant is hidden in the ''O'' notation. We show algorithms (depending on the characteristic c of the ground field of the allowed constants) that simplify the solution and lead to further improvements of the latter bound, by factors ranging from order of log n, for c=0 and cn (in which case the overhead constant drops dramatically), to order of min (c, log n), for 2@?c@?n; the algorithms use Las Vegas type randomization in the case of 2