Efficient multiplications in ${\mathbb{F}}_{5^{5n}}$ and ${\mathbb{F}}_{7^{7n}}$

Abstract

Efficient multiplications in finite fields of characteristics 5 and 7 are used for computing the Eta pairing over divisor class groups of the hyperelliptic curves Lee et al. (2008) [1]. In this paper, using the recent methods for multiplication in finite fields, the explicit formulas for multiplication in ${\mathbb{F}}_{5^{5n}}$ and ${\mathbb{F}}_{7^{7n}}$ are obtained with 10 multiplications in ${\mathbb{F}}_{5^n}$ for ${\mathbb{F}}_{5^{5n}}$ and 15 multiplications in ${\mathbb{F}}_{7^n}$ for ${\mathbb{F}}_{7^{7n}}$ improving the results in Cenk and Özbudak (2008) [4], Cenk et al. (2009) [5], Lee et al. (2008) [1] and Montgomery (2005) [12]. The timing results of implementations of the Karatsuba type formulas and the proposed formula for multiplication in ${\mathbb{F}}_{5^{5\cdot 89}}$ are given.