A Sorting Algorithm for Polynomial Multiplication
Journal of the ACM (JACM)
The Altran system for rational function manipulation — a survey
Communications of the ACM
On computing the fast Fourier transform
Communications of the ACM
The Design and Analysis of Computer Algorithms
The Design and Analysis of Computer Algorithms
Chinese remainder and interpolation algorithms
SYMSAC '71 Proceedings of the second ACM symposium on Symbolic and algebraic manipulation
Studies in fast algebraic algorithms.
Studies in fast algebraic algorithms.
A fast, low-space algorithm for multiplying dense multivariate polynomials
ACM Transactions on Mathematical Software (TOMS)
Functional decomposition ofpolynomials: the tame case
Journal of Symbolic Computation - Special issue on computational algebraic complexity
Parallel multiplication and powering of polynomials
Journal of Symbolic Computation
Parallel univariate p-adic lifting on shared-memory multiprocessors
ISSAC '92 Papers from the international symposium on Symbolic and algebraic computation
A case study in interlanguage communication: Fast LISP polynomial operations written in 'C'
SYMSAC '81 Proceedings of the fourth ACM symposium on Symbolic and algebraic computation
Efficient polynomial substitutions of a sparse argument
ACM SIGSAM Bulletin
Implementation techniques for fast polynomial arithmetic in a high-level programming environment
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
Exact and approximate bandwidth
Theoretical Computer Science
An alternative class of irreducible polynomials for optimal extension fields
Designs, Codes and Cryptography
Attribute and object selection queries on objects with probabilistic attributes
ACM Transactions on Database Systems (TODS)
Impact of Intel's new instruction sets on software implementation of GF(2)[x] multiplication
Information Processing Letters
On the bit-complexity of sparse polynomial and series multiplication
Journal of Symbolic Computation
Practical lattice-based cryptography: a signature scheme for embedded systems
CHES'12 Proceedings of the 14th international conference on Cryptographic Hardware and Embedded Systems
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The “fast” polynomial multiplication algorithms for dense univariate polynomials are those which are asymptotically faster than the classical O(N2) method. These “fast” algorithms suffer from a common defect that the size of the problem at which they start to be better than the classical method is quite large; so large, in fact that it is impractical to use them in an algebraic manipulation system. A number of techniques are discussed here for improving these fast algorithms. The combination of the best of these improvements results in a Hybrid Mixed Basis FFT multiplication algorithm which has a cross-over point at degree 25 and is generally faster than a basic FFT algorithm, while retaining the desirable O(N log N) timing function of an FFT approach. The application of these methods to multivariate polynomials is also discussed. The use is advocated of the Kronecker Trick to speed up a fast algorithm. This results in a method which has a cross-over point at degree 5 for bivariate polynomials. Both theoretical and empirical computing times are presented for all algorithms discussed.