Efficient Error Detection Cryptographic Architectures Benchmarked on FPGAs for Montgomery Ladder
Kasra Ahmadi, Saeed Aghapour, Mehran Mozaffari Kermani, Reza Azarderakhsh
Abstract
Elliptic curve scalar multiplication (ECSM) is a fundamental element of public key cryptography. The ECSM implementations on deeply embedded architectures and Internet-of-nano-Things have been vulnerable to both permanent and transient errors, as well as fault attacks. Consequently, error detection is crucial. In this work, we present a novel algorithm-level error detection scheme on Montgomery Ladder often used for a number of elliptic curves featuring highly efficient point arithmetic, known as Montgomery curves. Our error detection simulations achieve high error coverage on loop abort and scalar bit flipping fault model using binary tree data structure. Assuming n is the size of the private key, the overhead of our error detection scheme is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(n)$ </tex-math></inline-formula>. Finally, we conduct a benchmark of our proposed error detection scheme on both ARMv8 and field-programmable gate array (FPGA) platforms to illustrate the implementation and resource utilization. Deployed on Cortex-A72 processors, our proposed error detection scheme maintains a clock cycle overhead of less than 5.2%. In addition, integrating our error detection approach into FPGAs, including AMD/Xilinx Zynq Ultrascale+ and Artix Ultrascale+, results in a comparable throughput and less than 2% increase in area compared with the original hardware implementation. We note that we envision using adoptions of the proposed architectures in the postquantum cryptography (PQC) based on elliptic curves.