Academic Journal
A Lower Bound on the Complexity of Polynomial Multiplication over Finite Fields: A lower bound on the complexity of polynomial multiplication over finite fields
| Title: | A Lower Bound on the Complexity of Polynomial Multiplication over Finite Fields: A lower bound on the complexity of polynomial multiplication over finite fields |
|---|---|
| Authors: | Michael Kaminski |
| Source: | Lecture Notes in Computer Science ISBN: 9783540249986 |
| Publisher Information: | Society for Industrial & Applied Mathematics (SIAM), 2005. |
| Publication Year: | 2005 |
| Subject Terms: | polynomial multiplication, 4. Education, 0102 computer and information sciences, 02 engineering and technology, Symbolic computation and algebraic computation, 01 natural sciences, Polynomials over finite fields, 0202 electrical engineering, electronic engineering, information engineering, Computational aspects of field theory and polynomials, Recurrences, quadratic algorithms, finite fields, Number-theoretic algorithms, complexity, Polynomials in number theory |
| Description: | In infinite fields, it is possible to compute the coefficients of the product of two degree \(n\) polynomials in \(2n+1\) nonscalar operations. In finite fields, where the number of elements can be smaller than \(2n\), this lower bound is not achieved. In this paper, the author proves that any quadratic algorithm computing the coefficients of the product of two degree \(n\) polynomials over a field of \(q\) elements requires at least \((3 + \frac{(q-1)^2}{q^5+(q-1)^3})n- o(n)\) multiplications. The technique used by the author to prove the lower bound is a combination of Hankel matrices representation and a counting argument from coding theory. |
| Document Type: | Article Part of book or chapter of book |
| File Description: | application/xml |
| Language: | English |
| ISSN: | 1095-7111 0097-5397 |
| DOI: | 10.1137/s0097539704442118 |
| DOI: | 10.1007/978-3-540-31856-9_40 |
| Access URL: | https://zbmath.org/2205880 https://doi.org/10.1137/s0097539704442118 https://link.springer.com/chapter/10.1007/978-3-540-31856-9_40 https://dblp.uni-trier.de/db/conf/stacs/stacs2005.html#Kaminski05 https://rd.springer.com/chapter/10.1007/978-3-540-31856-9_40 https://dblp.uni-trier.de/db/journals/siamcomp/siamcomp34.html#Kaminski05 https://locus.siam.org/doi/abs/10.1137/S0097539704442118 https://epubs.siam.org/doi/abs/10.1137/S0097539704442118 https://doi.org/10.1137/S0097539704442118 |
| Rights: | Springer TDM |
| Accession Number: | edsair.doi.dedup.....4947a28f953fe4460f7c20704c2a8834 |
| Database: | OpenAIRE |
| ISSN: | 10957111 00975397 |
|---|---|
| DOI: | 10.1137/s0097539704442118 |