research-article

Sparse multiplication for skew polynomials

Authors:

Mark Giesbrecht,

Qiao-Long Huang,

Éric SchostAuthors Info & Claims

ISSAC '20: Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation

Pages 194 - 201

https://doi.org/10.1145/3373207.3404023

Published: 27 July 2020 Publication History

Abstract

Consider the skew polynomial ring L[x; σ], where L is a field and σ is an automorphism of L of order r. We present two randomized algorithms for the multiplication of sparse skew polynomials in L[x;σ].

The first algorithm is Las Vegas; it relies on evaluation and interpolation on a normal basis, at successive powers of a normal element. For inputs A, B ∈ L[x; σ] of degrees at most d, its expected runtime is O^~(max(d,r)rR^ω^-2) operations in K, where K = L^σ is the fixed field of σ in L and R ≤ r is the size of the Minkowski sum supp(A) + supp(B) taken modulo r; here, the supports supp(A), supp(B) are the exponents of non-zero terms in A and B.

The second algorithm is Monte Carlo; it is "super-sparse", in the sense that its expected runtime is O^~(log(d)Sr^ω), where S is the size of supp(A) + supp(B). Using a suitable form of Kronecker substitution, we extend this second algorithm to handle multivariate polynomials, for certain families of extensions.

References

[1]

A. Arnold, M. Giesbrecht, and D. Roche. 2013. Faster sparse interpolation of straight-line programs. In International Workshop on Computer Algebra in Scientific Computing. Springer, 61--74.

[2]

A. Arnold and D. Roche. 2015. Output-sensitive algorithms for sumset and sparse polynomial multiplication. In ISSAC'15. ACM Press, 29--36.

[3]

D. Boucher, P. Gaborit, W. Geiselmann, O. Ruatta, and F. Ulmer. 2010. Key exchange and encryption schemes based on non-commutative skew polynomials. In International Workshop on Post-Quantum Cryptography. Springer, 126--141.

[4]

D. Boucher, W. Geiselmann, and F. Ulmer. 2007. Skew-cyclic codes. Applicable Algebra in Engineering, Communication and Computing 18, 4 (2007), 379--389.

[5]

D. Boucher and F. Ulmer. 2009. Coding with skew polynomial rings. Journal of Symbolic Computation 44, 12 (2009), 1644--1656.

Digital Library

[6]

X. Caruso and J. Le Borgne. 2017. Fast multiplication for skew polynomials. In ISSAC'17. ACM, 77--84.

[7]

D. Coppersmith and S. Winograd. 1990. Matrix multiplication via arithmetic progressions. J. Symb. Comput. 9, 3 (1990), 251--280.

Digital Library

[8]

J.-M. Couveignes and R. Lercier. 2009. Elliptic periods for finite fields. Finite Fields Their Appl. 15, 1 (2009), 1--22.

Digital Library

[9]

E. Gabidulin. 1985. Theory of codes with maximum rank distance. Problemy Peredachi Informatsii 21, 1 (1985), 3--16.

[10]

F. Le Gall. 2014. Powers of tensors and fast matrix multiplication. In ISSAC'14. ACM Press, 296--303.

Digital Library

[11]

J. von zur Gathen and M. Giesbrecht. 1990. Constructing normal bases in finite fields. J. Symb. Comput 10 (1990), 547--570.

Digital Library

[12]

J. von zur Gathen and V. Shoup. 1992. Computing Frobenius maps and factoring polynomials. Computational Complexity 2, 3 (1992), 187--224.

[13]

W. Geiselmann and F. Ulmer. 2019. Skew Reed-Muller codes. Contemporary mathematics (2019), 107--116.

[14]

M. Giesbrecht. 1998. Factoring in skew-polynomial rings over finite fields. Journal of Symbolic Computation 26, 4 (1998), 463--486.

Digital Library

[15]

M. Giesbrecht, A. Jamshidpey, and É Schost. 2019. Quadratic-Time Algorithms for Normal Elements. In ISSAC'19. ACM Press, 179--186.

[16]

K. Girstmair. 1999. An algorithm for the construction of a normal basis. Journal of Number Theory 78, 1 (1999), 36--45.

[17]

D. Goss. 1996. Basic Structures of Function Field Arithmetic. Springer Berlin Heidelberg.

[18]

S. Johnson. 1974. Sparse polynomial arithmetic. ACM SIGSAM Bulletin 8, 3 (1974), 63--71.

Digital Library

[19]

E. Kaltofen and V. Shoup. 1998. Subquadratic-time factoring of polynomials over finite fields. Math. Comp. 67, 223 (1998), 1179--1197.

Digital Library

[20]

U. Martínez-Penas. 2019. Classification of multivariate skew polynomial rings over finite fields via affine transformations of variables. arXiv: 1908.06833 (2019).

[21]

U. Martínez-Penas and F. R. Kschischang. 2019. Evaluation and interpolation over multivariate skew polynomial rings. Journal of Algebra 525 (2019), 111--139.

[22]

M. Monagan and R. Pearce. 2009. Parallel Sparse Polynomial Multiplication Using Heaps. In ISSAC'09. 263--269.

[23]

M. Monagan and R. Pearce. 2011. Sparse Polynomial Pseudo Division Using a Heap. J. Symb. Comp. 46, 7 (2011), 807--822.

Digital Library

[24]

O. Ore. 1933. Theory of non-commutative polynomials. Annals of Mathematics (1933), 480--508.

[25]

S. Puchinger and A. Wachter-Zeh. 2016. Sub-quadratic decoding of Gabidulin codes. In 2016 IEEE International Symposium on Information Theory (ISIT). IEEE, 2554--2558.

[26]

S. Puchinger and A. Wachter-Zeh. 2018. Fast operations on linearized polynomials and their applications in coding theory. Journal of Symbolic Computation 89 (2018), 194--215.

Digital Library

[27]

F. Kschischang R. Koetter. 2008. Coding for errors and erasures in random network coding. IEEE Transactions on Information Theory 54, 8 (2008), 3579--3591.

Digital Library

[28]

D. Roche. 2018. What can (and can't) we do with sparse polynomials?. In ISSAC'18. 25--30.

Digital Library

[29]

V. Shoup. 1994. Fast construction of irreducible polynomials over finite fields. Journal of Symbolic Computation 17, 5 (1994), 371--391.

Digital Library

[30]

J. van der Hoeven and G. Lecerf. 2012. On the Complexity of Multivariate Blockwise Polynomial Multiplication. In ISSAC'12. 211--218.

[31]

J. van der Hoeven and G. Lecerf. 2013. On the bit-complexity of sparse polynomial and series multiplication. J. Symbolic Computation 50 (2013), 227--254.

Digital Library

[32]

Y. Zhang. 2010. A secret sharing scheme via skew polynomials. In 2010 International Conference on Computational Science and Its Applications. IEEE, 33--38.

Digital Library

Cited By

Rasheed RSadiq AKaiwartya O(2024)Elimination Algorithms for Skew Polynomials with Applications in CybersecurityMathematics10.3390/math1220325812:20(3258)Online publication date: 17-Oct-2024
https://doi.org/10.3390/math12203258
Ding XWang DXiao FZheng X(2024)An Algorithm for Computing Greatest Common Right Divisors of Parametric Ore PolynomialsProceedings of the 2024 International Symposium on Symbolic and Algebraic Computation10.1145/3666000.3669694(226-233)Online publication date: 16-Jul-2024
https://dl.acm.org/doi/10.1145/3666000.3669694
Chen QYe K(2024)A quasi-optimal lower bound for skew polynomial multiplicationProceedings of the 2024 International Symposium on Symbolic and Algebraic Computation10.1145/3666000.3669677(74-81)Online publication date: 16-Jul-2024
https://dl.acm.org/doi/10.1145/3666000.3669677
Show More Cited By

Sparse multiplication for skew polynomials
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms

Recommendations

A near-optimal algorithm for computing real roots of sparse polynomials
ISSAC '14: Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation

Let p ∈ Z[x] be an arbitrary polynomial of degree n with k non-zero integer coefficients of absolute value less than 2^τ. In this paper, we answer the open question whether the real roots of p can be computed with a number of arithmetic operations over ...
Finding small degree factors of multivariate supersparse (lacunary) polynomials over algebraic number fields
ISSAC '06: Proceedings of the 2006 international symposium on Symbolic and algebraic computation

We present algorithms that compute all irreducible factors of degree ≤ d of supersparse (lacunary) multivariate polynomials in n variables over an algebraic number field in deterministic polynomial-time in (l+d)ⁿ, where l is the size of the input ...
Sparse shifts for univariate polynomials

Letf(x) be a polynomial of degreed with rational coefficients and lett be a positive integer ? deg(f). We consider the problem of finding at-sparse shift forf(x). The problem is to find an a, if one exists (in some algebraic extension of the rationals), ...

Comments

Information & Contributors

Information

Published In

cover image ACM Other conferences

ISSAC '20: Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation

July 2020

480 pages

ISBN:9781450371001

DOI:10.1145/3373207

General Chairs:
Ioannis Z. Emiris
National and Kapodistrian University of Athens, ATHENA Research and Innovation Center, Greece
,
Lihong Zhi
Academia Sinica, China
,
Publications Chair:
Angelos Mantzaflaris
Inria, France

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

SIGSAM: ACM Special Interest Group on Symbolic and Algebraic Manipulation

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 27 July 2020

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article

Funding Sources

Natural Sciences and Engineering Research Council of Canada

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

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

5
Total Citations
View Citations
93
Total Downloads

Downloads (Last 12 months)4
Downloads (Last 6 weeks)2

Reflects downloads up to 08 Feb 2025

Other Metrics

View Author Metrics

Citations

Cited By

Rasheed RSadiq AKaiwartya O(2024)Elimination Algorithms for Skew Polynomials with Applications in CybersecurityMathematics10.3390/math1220325812:20(3258)Online publication date: 17-Oct-2024
https://doi.org/10.3390/math12203258
Ding XWang DXiao FZheng X(2024)An Algorithm for Computing Greatest Common Right Divisors of Parametric Ore PolynomialsProceedings of the 2024 International Symposium on Symbolic and Algebraic Computation10.1145/3666000.3669694(226-233)Online publication date: 16-Jul-2024
https://dl.acm.org/doi/10.1145/3666000.3669694
Chen QYe K(2024)A quasi-optimal lower bound for skew polynomial multiplicationProceedings of the 2024 International Symposium on Symbolic and Algebraic Computation10.1145/3666000.3669677(74-81)Online publication date: 16-Jul-2024
https://dl.acm.org/doi/10.1145/3666000.3669677
Giesbrecht MHuang QSchost ÉChyzak FLabahn G(2021)Sparse Multiplication of Multivariate Linear Differential OperatorsProceedings of the 2021 International Symposium on Symbolic and Algebraic Computation10.1145/3452143.3465527(155-162)Online publication date: 18-Jul-2021
https://dl.acm.org/doi/10.1145/3452143.3465527
Rasheed R(2021)Resultant-based Elimination for Skew Polynomials2021 23rd International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC)10.1109/SYNASC54541.2021.00014(11-18)Online publication date: Dec-2021
https://doi.org/10.1109/SYNASC54541.2021.00014

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Figures

Tables

Media

View Table of Conten