Abstract
This paper concerns an exhaustive search for normal bases with minimum complexity in finite fields \(\mathbb {F}_{2^n}\) over \(\mathbb {F}_2\) for \(n \le 46\). This is a followup paper to [11], which appeared one decade ago in 2008 and completed the cases \(n \le 39\). We extend the results in [11] by taking advantage of a combination of algorithmic improvements, more efficient implementations and massive parallelism.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Arnault, F., Pickett, E.J., Vinatier, S.: Construction of self-dual normal bases and their complexity. Finite Fields Appl. 18, 458–472 (2012)
Compute Canada: 2018 Final Allocations for Publication. https://www.computecanada.ca/research-portal/accessing-resources/resource-allocation-competitions/rac-2018-results/. Accessed 17 May 2018
Blake, I.F., Gao, S., Mullin, R.C.: Specific irreducible polynomials with linearly independent roots over finite fields. Linear Algebr. Appl. 253, 227–249 (1997)
Gao, S., Lenstra, H.W.: Optimal normal bases. Des. Codes Cryptogr. 2, 315–323 (1992)
Gao, S., Panario, D.: Density of normal elements. Finite Fields Appl. 3, 141–150 (1997)
von zur Gathen, J., Gerhard, J.: Modern Computer Algebra. Cambridge University Press, Cambridge (2013)
gperftools: Authors unlisted \(\copyright \) Google Inc. (2005). www.github.io/gperftools. Accessed 27 Mar 2018
Jungnickel, D.: Finite Fields: Structure and Arithmetics, Bibliographisches Institut, Mannheim GE (1993)
Kreher, D.L., Stinson, D.R.: Combinatorial Algorithms: Generation Enumeration and Search. CRC Press, Boca Raton (1999)
Lidl, R., Niederreiter, H.: Finite Fields. Cambridge University Press, Oxford (1997)
Masuda, A., Moura, L., Panario, D., Thomson, D.: Low complexity normal elements over finite fields of characteristic two. IEEE Trans. Comput. 57, 990–1001 (2008)
Mullin, R.C., Onyszchuk, I.M., Vanstone, S.A., Wilson, R.M.: Optimal normal bases in \(GF(p^n)\), Discrete Appl. Math. 22, 149–161 (1988/1989)
Mullen, G.L., Panario, D.: Handbook of Finite Fields. CRC Press, Boca Raton (2013)
National Institute for Standards in Technology: Digital Signature Standard FIPS PUB 186–4 (2013)
SageMath: the Sage Mathematics Software System (Version 7.1.0). The Sage Developers (2018). http://www.sagemath.org
Shoup, V.: NTL: number theory library. www.shoup.net/ntl. Accessed 3 Mar 2018
Silva, D., Kschischang, F.: Fast encoding and decoding of Gabidulin codes. IEEE Int. Symp. Inf. Theory 2009, 2858–2862 (2009)
Thomson, D.: Exhaustive search for normal basis, Version 1.0.0. www.github.com/dgthoms/exh_normal. Accessed 26 May 2018
Acknowledgement
We would like to thank the three reviewers for their helpful suggestions, which greatly improved the presentation of this paper.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this paper
Cite this paper
Moura, L., Panario, D., Thomson, D. (2018). Normal Basis Exhaustive Search: 10 Years Later. In: Budaghyan, L., Rodríguez-Henríquez, F. (eds) Arithmetic of Finite Fields. WAIFI 2018. Lecture Notes in Computer Science(), vol 11321. Springer, Cham. https://doi.org/10.1007/978-3-030-05153-2_10
Download citation
DOI: https://doi.org/10.1007/978-3-030-05153-2_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-05152-5
Online ISBN: 978-3-030-05153-2
eBook Packages: Computer ScienceComputer Science (R0)