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Title: Asymptotic bound for multiplication complexity in the extensions of small finite fields
Authors: Cascudo, Ignacio
Cramer, Ronald
Xing, Chaoping
Yang, An
Issue Date: 2011
Source: Cascudo, I., Cramer, R., Xing, C., & Yang, A. (2012). Asymptotic Bound for Multiplication Complexity in the Extensions of Small Finite Fields. IEEE Transactions on Information Theory, 58(7), 4930-4935.
Series/Report no.: IEEE transactions on information theory
Abstract: In 1986, D. V. Chudnovsky and G. V. Chudnovsky first employed algebraic curves over finite fields to construct bilinear multiplication algorithms implicitly through supercodes introduced by Shparlinski-Tsfasman-Vladuţ, or equivalently, multiplication-friendly codes that we will introduce in this paper. This idea was further developed by Shparlinski-Tsfasman-Vladuţ in order to study the asymptotic behavior of multiplication complexity in extension fields. Later on, Ballet et al. further investigated the method and obtained some improvements. Recently, Ballet and Pieltant made use of curves over an extension field of to obtain an improvement on the complexity of multiplications in extensions of the binary field. In this paper, we develop the multiplication-friendly splitting technique and then apply this technique to study asymptotic behavior of multiplications in extension fields. By combining this with the idea of using algebraic function fields, we are able to improve further the asymptotic results of multiplication complexity. In particular, the improvement for small fields such as the binary and ternary fields is substantial.
ISSN: 0018-9448
DOI: 10.1109/TIT.2011.2180696
Rights: © 2011 IEEE.
Fulltext Permission: none
Fulltext Availability: No Fulltext
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