High-Performance Architectures for Finite Field Inversion Over GF(2163)

Abstract

Inversion is the most computationally expensive finite field operation in public-key cryptographic such as elliptic curve cryptography (ECC). This paper presents highperformance architectures for performing the finite field inversion using Gaussian Normal Bases (GNB) and a digit-level serial-in parallel-out multiplier (DL-SIPO) over GF(2163). We propose three architectures to carry out the inversion operation. The first one is based on classic Itoh-Tsujji Algorithm (ITA), the second one carries out the inversion operation according to the NIST binary fields over GF $(2^{163})$ and finally, the last one is based on Fermat's Little Theorem (FLT). The architectures were designed using VHDL description, synthesized on the Stratix IV FPGA using Quartus Prime 17.0, and verified in ModelSim and Matlab. The synthesis results show that the designed architectures present a very good performance using low area. In this case, the processing time and area resources to compute the inversion operation were 114.2, 115.9 and 114.5 ns using 11624, 11558 and 11690 LUTs, respectively.