CRC-Based Error Detection Constructions for FLT and ITA Finite Field Inversions Over GF(2m)

Binary extension finite fields <inline-formula> <tex-math notation="LaTeX">${\mathrm{ GF}}(2^{m})$ </tex-math></inline-formula> have received prominent attention in the literature due to their application in many modern public-key cryptosystems and error-correcting codes. In particular, the inversion over <inline-formula> <tex-math notation="LaTeX">${\mathrm{ GF}}(2^{m})$ </tex-math></inline-formula> is crucial for current and postquantum cryptographic applications. Schemes such as Fermat’s little theorem (FLT) and the Itoh–Tsujii algorithm (ITA) have been studied to achieve better performance; however, this arithmetic operation is a complex, expensive, and time-consuming task that may require thousands of gates, increasing its vulnerability chance to natural defects. In this work, we propose efficient hardware architectures based on cyclic redundancy check (CRC) as error detection schemes for state-of-the-art finite field inversion over <inline-formula> <tex-math notation="LaTeX">${\mathrm{ GF}}(2^{m})$ </tex-math></inline-formula> for a polynomial basis. To verify the derivations of the formulations, software implementations are performed. Likewise, hardware implementations of the original finite field inversions with the proposed error detection schemes are performed over Xilinx field-programmable gate array (FPGA) verifying that the proposed schemes achieve high error coverage with acceptable overhead.