2016 年 E99.A 巻 11 号 p. 2060-2074
We propose subquadratic space complexity multipliers for any finite field $\mathbb{F}_{q^n}$ over the base field $\mathbb{F}_q$ using the Dickson basis, where q is a prime power. It is shown that a field multiplication in $\mathbb{F}_{q^n}$ based on the Dickson basis results in computations of Toeplitz matrix vector products (TMVPs). Therefore, an efficient computation of a TMVP yields an efficient multiplier. In order to derive efficient $\mathbb{F}_{q^n}$ multipliers, we develop computational schemes for a TMVP over $\mathbb{F}_{q}$. As a result, the $\mathbb{F}_{2^n}$ multipliers, as special cases of the proposed $\mathbb{F}_{q^n}$ multipliers, have lower time complexities as well as space complexities compared with existing results. For example, in the case that n is a power of 3, the proposed $\mathbb{F}_{2^n}$ multiplier for an irreducible Dickson trinomial has about 14% reduced space complexity and lower time complexity compared with the best known results.
