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 Diskr. Mat., 2000, Volume 12, Issue 3, Pages 124–153 (Mi dm340)

Remarks on the fast multiplication of polynomials, and Fourier and Hartley transforms

S. B. Gashkov

Abstract: We give a fast algorithm for multiplication of polynomials with real-valued coefficients without resort to complex numbers and the fast Fourier transformation. The efficiency of this algorithm is compared with the multiplication algorithm based on the discrete Hartley transformation. We demonstrate that the complexity of the Hartley transformation coincides, to within a linear term, with the complexity of the Fourier transformation, but the use of the Hartley transformation leads to a more efficient multiplication algorithm.
We give analogues of these results for finite fields. In some cases, the multiplicative constants in bounds for the complexity of multiplication of polynomials and of the Fourier and Hartley transformations over finite fields are smaller than those in the case of the field of real numbers.
This research was supported by the Russian Foundation for Basic Research, grant 99–01–01175, and the Federal Special Program ‘Integration’, grant 473.

DOI: https://doi.org/10.4213/dm340

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English version:
Discrete Mathematics and Applications, 2000, 10:5, 499–528

Bibliographic databases:

UDC: 519.7

Citation: S. B. Gashkov, “Remarks on the fast multiplication of polynomials, and Fourier and Hartley transforms”, Diskr. Mat., 12:3 (2000), 124–153; Discrete Math. Appl., 10:5 (2000), 499–528

Citation in format AMSBIB
\Bibitem{Gas00} \by S.~B.~Gashkov \paper Remarks on the fast multiplication of polynomials, and Fourier and Hartley transforms \jour Diskr. Mat. \yr 2000 \vol 12 \issue 3 \pages 124--153 \mathnet{http://mi.mathnet.ru/dm340} \crossref{https://doi.org/10.4213/dm340} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1810959} \zmath{https://zbmath.org/?q=an:1001.11060} \transl \jour Discrete Math. Appl. \yr 2000 \vol 10 \issue 5 \pages 499--528 

• http://mi.mathnet.ru/eng/dm340
• https://doi.org/10.4213/dm340
• http://mi.mathnet.ru/eng/dm/v12/i3/p124

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This publication is cited in the following articles:
1. Gashkov S.B., Gashkov I.B., “Some Remarks on Testing Irreducibility of Polynomials and Normality of Bases in Finite Fields”, Fundamenta Informaticae, 104:3 (2010), 227–238
2. S. B. Gashkov, I. S. Sergeev, “Complexity of computation in finite fields”, J. Math. Sci., 191:5 (2013), 661–685
3. Burtsev A.A., Gashkov S.B., “O skhemakh dlya arifmetiki v konechnykh polyakh”, Trudy moskovskogo fiziko-tekhnicheskogo instituta, 2012, 15–22
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