An area-efficient composite field inverter for elliptic curve cryptosystems

This chapter presents a new area-efficient composite field inverter of the form GF(q¹) with q=2^n.m suitable for the hardware realization of an elliptic curve (EC) cryptosystem. Considering both the security aspect and the hardware cost required, the authors propose the utilization of the composite field GF(((2²)²)⁴¹) for EC cryptosystem. For efficient implementation, they have derived a compact inversion circuit over GF(2¹⁶⁴)=GF(((2²)²)⁴¹) to achieve an optimal saving in the hardware cost required. Furthermore, the authors have also developed a composite field digit serial Sunar-Koc multiplier for the multiplication in the extension field. All of the arithmetic operations in the subfield GF(2⁴) are performed in its isomorphic composite field, GF((2²)²), leading to a full combinatorial implementation without resorting to the conventional look-up table approach. To summarize the work, the final hardware implementation and the complexity analysis of the inversion is reported towards the end of this chapter.