Fast Inversion in ${schmi{GF(2^m)}}$ with Normal Basis Using Hybrid-Double Multipliers

Fast inversion in finite fields is crucial for high-performance cryptography and codes. We present techniques to exploit the recently proposed hybrid-double multipliers for fast inversions in binary fields GF(2^m) with normal bases. A hybrid-double multiplier computes a double multiplication, the product of three elements in GF(2^m), with a latency comparable to the latency of single multiplication of two elements. Traditional approaches, such as Itoh-Tsujii, cannot utilize hybrid-double multipliers. We devise a new inversion algorithm based on ternary representations that exploits their potential. The algorithm reduces the latency of inversion significantly for the fields recommended by NIST if hybrid-double multipliers are employed. For example, the algorithm computes an inversion in GF(2¹⁶³) with only five double multiplications whereas the Itoh-Tsujii algorithm requires nine single or double multiplications. We propose a new inverter architecture using this new algorithm and a hybrid-double multiplier. We show that it is faster than the existing techniques by providing ASIC synthesis results using 65-nm CMOS technology. For example, our inverter for GF(2¹⁶³) achieves about 34 percent shorter computation time than an inverter using the Itoh-Tsujii algorithm and a single multiplier.