Fast multipoint evaluation and interpolation of polynomials in the LCH-basis over FPr
Pages 344 - 351
Abstract
Lin, Chung and Han introduced in 2014 the LCH-basis in order to derive FFT-based multipoint evaluation and interpolation algorithms with respect to this polynomial basis. Considering an affine space of n = 2j points, their algorithms require O(n · log2 n) operations in F2r. The LCH-basis has then been extended over finite fields of characteristic p by Lin et al. in 2016 and an n-point evaluation algorithm has been derived for n = pj with complexity O(n · logp n · p). However, the problem of interpolating polynomials represented in such a basis over FPr has not been addressed.
In this paper, we fill this gap and also derive a faster algorithm for evaluating polynomials in the LCH-basis at multiple points over FPr. We follow a different approach where we represent the multipoint evaluation and interpolation maps by well-defined matrices. We present factorizations of such matrices into the product of sparse matrices which can be evaluated efficiently. These factorizations lead to fast algorithms for both the multipoint evaluation and the interpolation of polynomials represented in the LCH-basis at n = pj points with optimized complexity O(n · log2 n · log2 p · log2 log2 p).
A particular attention is paid to provide in-place algorithms with high memory-locality. Our implementations written in C confirm that our approach improves the original transforms.
References
[1]
Elwyn R. Berlekamp. Algebraic coding theory. McGraw-Hill series in systems science. McGraw-Hill, 1968.
[2]
Ming-Shing Chen, Chen-Mou Cheng, Po-Chun Kuo, Wen-Ding Li, and Bo-Yin Yang. Multiplying boolean polynomials with frobenius partitions in additive fast fourier transform. CoRR, abs/1803.11301, 2018.
[3]
James W. Cooley and John W. Tukey. An algorithm for the machine calculation of complex Fourier series. Math. Comput., 19:297--301, 1965.
[4]
M. Davio. Kronecker products and shuffle algebra. "IEEE Transactions on Computers", 30(2):116--125, 1981.
[5]
Joachim Von Zur Gathen and Jurgen Gerhard. Modern Computer Algebra. Cambridge University Press, New York, NY, USA, 2 edition, 2003.
[6]
Pascal Giorgi, Bruno Grenet, and Daniel S. Roche. Fast in-place algorithms for polynomial operations: division, evaluation, interpolation, 2020.
[7]
W. Hart, F. Johansson, and S. Pancratz. FLINT: Fast Library for Number Theory, 2015. Version 2.5.2, http://flintlib.org.
[8]
Leslie Hogben. Handbook of linear algebra; 2nd ed. Discrete Mathematics and Its Applications. Taylor and Francis, Hoboken, NJ, 2013.
[9]
Runzhou Li, Qin Huang, and Zulin Wang. Encoding of non-binary quasi-cyclic codes by lin-chung-han transform. In IEEE Information Theory Workshop, ITW 2018, Guangzhou, China, November 25--29, 2018, pages 1--5, 2018.
[10]
Sian-Jheng Lin, Tareq Y. Al-Naffouri, and Yunghsiang S. Han. Efficient frequency-domain decoding algorithms for reed-solomon codes. CoRR, abs/1503.05761, 2015.
[11]
Sian-Jheng Lin, Tareq Y. Al-Naffouri, and Yunghsiang S. Han. FFT algorithm for binary extension finite fields and its application to reed-solomon codes. IEEE Trans. Information Theory, 62(10):5343--5358, 2016.
[12]
Sian-Jheng Lin, Tareq Y. Al-Naffouri, Yunghsiang S. Han, and Wei-Ho Chung. Novel polynomial basis with fast fourier transform and its application to reed-solomon erasure codes. IEEE Trans. Information Theory, 62(11):6284--6299, 2016.
[13]
Sian-Jheng Lin, Wei-Ho Chung, and Yunghsiang S. Han. Novel polynomial basis and its application to reed-solomon erasure codes. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18--21, 2014, pages 316--325, 2014.
[14]
Joris van der Hoeven and Robin Larrieu. The frobenius FFT. In Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation, ISSAC 2017, Kaiserslautern, Germany, July 25--28, 2017, pages 437--444, 2017.
[15]
Qingxuan Yang, John Ellis, Khalegh Mamakani, and Frank Ruskey. In-place permuting and perfect shuffling using involutions. Information Processing Letters, 113(10):386--391, 2013.
Index Terms
- Fast multipoint evaluation and interpolation of polynomials in the LCH-basis over FPr
Recommendations
Fast, algebraic multivariate multipoint evaluation in small characteristic and applications
STOC 2022: Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of ComputingMultipoint evaluation is the computational task of evaluating a polynomial given as a list of coefficients at a given set of inputs. Besides being a natural and fundamental question in computer algebra on its own, fast algorithms for this problem are ...
Fast in-place algorithms for polynomial operations: division, evaluation, interpolation
ISSAC '20: Proceedings of the 45th International Symposium on Symbolic and Algebraic ComputationWe consider space-saving versions of several important operations on univariate polynomials, namely power series inversion and division, division with remainder, multi-point evaluation, and interpolation. Now-classical results show that such problems ...
Fast polynomial factorization and modular composition in small characteristic
STOC '08: Proceedings of the fortieth annual ACM symposium on Theory of computingWe obtain randomized algorithms for factoring degree n univariate polynomials over F_q that use O(n1.5 + o(1) + n1 + o(1)log q) field operations, when the characteristic is at most no(1). When log q < n, this is asymptotically faster than the best ...
Comments
Information & Contributors
Information
Published In
July 2020
480 pages
ISBN:9781450371001
DOI:10.1145/3373207
- General Chairs:
- Ioannis Z. Emiris,
- Lihong Zhi,
- Publications Chair:
- Angelos Mantzaflaris
Copyright © 2020 ACM.
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]
In-Cooperation
Publisher
Association for Computing Machinery
New York, NY, United States
Publication History
Published: 27 July 2020
Check for updates
Author Tags
Qualifiers
- Research-article
Conference
ISSAC '20
ISSAC '20: International Symposium on Symbolic and Algebraic Computation
July 20 - 23, 2020
Kalamata, Greece
Acceptance Rates
ISSAC '20 Paper Acceptance Rate 58 of 102 submissions, 57%;
Overall Acceptance Rate 395 of 838 submissions, 47%
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 79Total Downloads
- Downloads (Last 12 months)24
- Downloads (Last 6 weeks)5
Reflects downloads up to 03 Mar 2025
Other Metrics
Citations
View Options
Login options
Check if you have access through your login credentials or your institution to get full access on this article.
Sign in