Low Complexity Implementation of Unified Systolic Multipliers for NIST Pentanomials and Trinomials Over <inline-formula> <tex-math notation="LaTeX">$\textit{GF}(2^{m})$ </tex-math> </inline-formula>

Abstract

Systolic finite field multiplier over $\textit {GF}(2^{m})$ based on the National Institute of Standards and Technology (NIST) recommended pentanomials or trinomials can be used as a critical component in many cryptosystems. In this paper, for the first time, we propose a novel low-complexity unified (hybrid field size) systolic multiplier for NIST pentanomials and trinomials over $\textit {GF}(2^{m})$ . We have proposed a computation-core-based design strategy to obtain the desired low-complexity unified multiplier for both NIST pentanomials and trinomials. The proposed multiplier can swift between pentanomial-based and trinomial-based multipliers through a control signal. First of all, a novel strategy is briefly introduced to implement a certain matrix-vector multiplication, which can be packed as a standard computation core (or computation core like). Then, based on the computation-core concept, a novel unified multiplication algorithm is derived that it can realize both the pentanomial-based and trinomial-based multiplications. After that, an efficient systolic structure is presented that it can fully employ the introduced computation core. A detailed example of the proposed unified multiplier (for $\textit {GF}(2^{163})$ and $\textit {GF}(2^{233})$ ) is also presented. Both the theoretical and field-programmable gate array implementation results show that the proposed design has efficient performance in area-time-power complexities, e.g., the proposed design (the one performs $\textit {GF}(2^{163})$ and $\textit {GF}(2^{233})$ multiplications) is found to have at least 14.2% and 13.3% less area-delay product and power-delay product than the combination of the existing individual $\textit {GF}(2^{163})$ and $\textit {GF}(2^{233})$ multipliers (best among all competing designs), respectively. Because of its structural regularity and functional flexibility, the proposed unified multiplier can be used as an intellectual property core for various cryptosystems.