Improved Frobenius FFT for Code-Based Cryptography on Cortex-M4

Polynomial multiplication over finite fields is one of the most significant operations in code-based cryptography, including hamming quasi-cyclic (HQC), which has been selected as a standardized algorithm in the NIST post-quantum cryptography (PQC) round 4 process. In the standardization process, the performance of an algorithm is important not only in general-purpose (GP) systems but also in embedded systems. In particular, NIST has recommended the ARM Cortex-M4 as the benchmark platform for embedded systems. In CHES2021, Chen et al. optimized bit flipping key encapsulation (BIKE) on the ARM Cortex-M4, using the Frobenius Additive FFT as the polynomial multiplication algorithm. However, although HQC was finally selected as a standard algorithm in March 2025, an efficient implementation for the ARM Cortex-M4 platform, which NIST recommends as the benchmark for embedded systems, has not yet been reported. In this article, we propose an optimized implementation of the Frobenius Additive FFT to accelerate polynomial multiplication in BIKE and HQC on the ARM Cortex-M4 platform. Our approach exploits the fact that one operand of field multiplications in the Frobenius Additive FFT is fixed, allowing the transformation of these operations into binary matrix-vector products. We then apply xor-efficient linear layer techniques combined with a register scheduling strategy specifically designed for the constraints of the Cortex-M4 architecture. We evaluate our implementation on the Nucleo-L4R5ZI evaluation board and compare it against existing state-of-the-art implementations. Our results demonstrate an 8% improvement in polynomial multiplication for BIKE, leading to up to 6% performance gains in key generation, encapsulation, and decapsulation. Moreover, we achieve an 80%–92% speedup in polynomial multiplication for HQC compared to the PQClean implementation, resulting in overall performance gains of up to 84% in key generation, encapsulation, and decapsulation.