On the tensor rank of the multiplication in the finite fields

Abstract

First, we prove the existence of certain types of non-special divisors of degree g − 1 in the algebraic function fields of genus g defined over F q . Then, it enables us to obtain upper bounds of the tensor rank of the multiplication in any extension of quadratic finite fields F q by using Shimura and modular curves defined over F q . From the preceding results, we obtain upper bounds of the tensor rank of the multiplication in any extension of certain non-quadratic finite fields F q , notably in the case of F 2 . These upper bounds attain the best asymptotic upper bounds of Shparlinski-Tsfasman-Vladut [I.E. Shparlinski, M.A. Tsfasman, S.G. Vladut, Curves with many points and multiplication in finite fields, in:

On multiplication in finite fields

“…We first define the multiplications m i , for 1 ≤ i ≤ 26 and then we give the formula for obtaining the coefficients of γ using these multiplications. m 1 = a 9 b 9 m 2 = (2a 2 + 2a 3 + 2a 4 + a 7 + 2a 8 + a 9 )(2b 2 + 2b 3 + 2b 4 + b 7 + 2b 8 + b 9 ) m 3 = (a 2 + 2a 8 + 2a 4 + 2a 7 + a 9 )(b 2 + 2b 8 + 2b 4 + 2b 7 + b 9 ) m 4 = (a 8 + 2 a 4 + a 9 + 2 a 3 )(b 8 …”

“…Arnaud [1] presented a method using local expansions with multiplicity 2 and places of degree one and two. In [8], new upper bounds of the bilinear complexity of multiplication in F q n over F q are obtained by proving the existence of certain types of non-special divisors of degree g −1 in the algebraic function fields of genus g defined over F q . Moreover, concerning the use of places of degree greater than one, Ballet and Rolland use places of degree one, two and four to improve the asymptotic bilinear complexity of multiplication in the extensions of F 2 in [10].…”

“…Here many times refers to using first u i > 1 coefficients instead of the first (u i = 1) coefficient in the local expansion of a place P i (see the map ϕ in Section 3). The proposed method is a generalization of the methods introduced in [14,23,22,[2][3][4][5][6][7]1,8,9]. In the papers cited above, mostly degree one places are used only.…”

“…Among these papers, only [4][5][6][7]1,8,9] use places of degree greater than one. In these papers, [4][5][6][7][8][9] use only places of degree one and degree two but always with u i = 1, i.e., using places only once. In [1], only places of degree one and degree two are used and all of such places are used at most 2 times.…”

Efficient multiplications in F55n and Cenk1, Özbudak2 2011Journal of Computational and Applied Mathematics0000View full textAdd to dashboardCite

“…For example, recently it has been shown that one can obtain the explicit formula for multiplication in F 5 5n with µ 5 n (5) ≤ 11 in [4,5]. In this paper, by using the recent methods for multiplication in F q m (see, [6][7][8][9][10]) improved the values for the multiplications in F 5 5n and F 7 7n are obtained. The explicit formulas having µ 5 n (5) ≤ 10 and µ 7 n (7) ≤ 15, which also improve the corresponding result in [4,5] are given.…”