FELIX: FPGA-Based Scalable and Lightweight Accelerator for Large Integer Extended GCD

Abstract

The extended greatest common divisor (XGCD) computation is a critical component in various cryptographic applications and algorithms, including both pre- and postquantum cryptosystems. In addition to computing the greatest common divisor (GCD) of two integers, the XGCD also produces Bézout coefficients $b_{a}$ and $b_{b}$ which satisfy $\mathrm {GCD}(a,b) = a\times b_{a} + b\times b_{b}$ . In particular, computing the XGCD for large integers is of significant interest. Most recently, XGCD computation between 6479-bit integers is required for solving Nth-degree truncated polynomial ring unit (NTRU) trapdoors in Falcon, a National Institute of Standards and Technology (NIST)-selected postquantum digital signature scheme. To this point, existing literature has primarily focused on exploring software-based implementations for XGCD. The few existing high-performance hardware architectures require significant hardware resources and may not be desirable for practical usage, and the lightweight architectures suffer from poor performance. To fill the research gap, this work proposes a novel FPGA-based scalable and lightweight accelerator for large integer XGCD (FELIX). First, a new algorithm suitable for scalable and lightweight computation of XGCD is proposed. Next, a hardware accelerator (FELIX) is presented, including both constant- and variable-time versions. Finally, a thorough evaluation is carried out to showcase the efficiency of the proposed FELIX. In certain configurations, FELIX involves 81% less equivalent area-time product (eATP) than the state-of-the-art design for 1024-bit integers, and achieves a 95% reduction in latency over the software for 6479-bit integers (Falcon parameter set) with reasonable resource usage. Overall, the proposed FELIX is highly efficient, scalable, lightweight, and suitable for very large integer computation, making it the first such XGCD accelerator in the literature (to the best of our knowledge).