Error Detection Constructions for ITA Finite Field Inversions Over $\text{GF}(2^{m})$ on FPGA Using CRC and Hamming Codes

Abstract

Finite field arithmetic operations over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\text{GF}(2^{m})$</tex-math></inline-formula> are widely used in critical applications, such as cryptography, coding theory, error-correcting codes, and digital signal processing. Finite field inversions are the most time-consuming operations among other widely-used ones and require a large footprint as well as power/energy to be performed. To reduce such complexity, the Itoh–Tsujii algorithm (ITA) has received prominent attention in the literature; however, implementations using ITA are still vulnerable to natural very-large-scale integration defects. To overcome the challenge of detecting naturally-induced faults in such constructions, for the first time, we propose error detection schemes based on Hamming codes for architectures performing finite field inversions using the ITA algorithm over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\text{GF}(2^{m})$</tex-math></inline-formula> with polynomial basis. Additionally, CRC-oriented error detection schemes for inversions in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\text{GF}(2^{m})$</tex-math></inline-formula> with normal basis are also studied and new approaches to protect them are presented. In this article, general formulations are provided along with different case studies to show the feasibility of our schemes with any finite field size. Moreover, field-programmable gate array (FPGA) implementations are performed on two Xilinx FPGA families, i.e., Kintex UltraScale+ and Xilinx Virtex-7 UltraScale+, to verify that the overheads added by the error detection architectures to provide reliability are suitable for deeply-constrained embedded systems.