High-Performance Accelerator for Constant-Time Cross-Domain Integer and Montgomery Inversion on FPGA

Modular Inversion (MI) is one of the fundamental arithmetic operations in the finite field, which plays an essential role in various cryptographic applications and requires high performance and security. Unfortunately, the simple MI algorithm is vulnerable to side-channel attacks, such as the timing attack, which can compromise the cryptographic system by analyzing the time taken to execute cryptographic algorithms. Attackers may recover the initial data since the time can differ based on the input. Besides, the low complexity and low resource consumption of hardware implementations in MI are also challenging. In this article, we propose two novel modular inversion algorithms, named Constant-Time Integer Modular Inversion (CT-IMI) and Constant-Time Complementary Montgomery Modular Inversion (CT-CMMI). They both consist of constant iteration rounds to resist the timing attack. CT-IMI processes the data in the integer field, which is designed for common scenarios. CT-CMMI is suitable for the cross-domain case, which can directly use data in the Montgomery domain and avoid the conversion steps for some specific applications, e.g., scalar multiplication in Elliptic Curve Cryptography (ECC). In software simulations, we measure the average clock cycles for a single inversion and illustrate the relationship between various bit lengths and the latency. The significant differences between constant and non-constant algorithms demonstrate the vulnerability of modular inversion to timing attacks. In addition, we design two efficient hardware architectures on FPGA. Experimental results show that our CT-IMI can finish a single inversion in 2.56 \(\mu\) s with 4.2k LUTs, 1.8k FFs, and our CT-CMMI requires 2.45 \(\mu\) s with 2.7k LUTs, 1.6k FFs. The product of area and latency of our CT-IMI and CT-CMMI can reach 10.50 and 6.62, respectively, which shows optimal performance compared with all the results in the existing literature.