Digit-Level Serial-In Parallel-Out Multiplier Using Redundant Representation for a Class of Finite Fields

Abstract

Two digit-level finite field multipliers in $\mathbb {F}_{2^{m}}$ using redundant representation are presented. Embedding $\mathbb {F}_{2^{m}}$ in cyclotomic field $\mathbb {F}_{2}^{(n)}$ causes a certain amount of redundancy and consequently performing field multiplication using redundant representation would require more hardware resources. Based on a specific feature of redundant representation in a class of finite fields, two new multiplication algorithms along with their pertaining architectures are proposed to alleviate this problem. Considering area-delay product as a measure of evaluation, it has been shown that both the proposed architectures considerably outperform existing digit-level multipliers using the same basis. It is also shown that for a subset of the fields, the proposed multipliers are of higher performance in terms of area-delay complexities among several recently proposed optimal normal basis multipliers. The main characteristics of the postplace&route application specific integrated circuit implementation of the proposed multipliers for three practical digit sizes are also reported.