Low Complexity Implementation of Unified Systolic Multipliers for NIST Pentanomials and Trinomials Over GF(2m)

Systolic finite field multiplier over 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 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 GF(2(163)) and 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 GF(2(163)) and 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 GF(2(163)) and 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.