In this work, we propose an optimized method for polynomial multiplication over \(\text {GF}(2)[x]\) , targeting SIMD environments such as AVX2 and Neon. Polynomial multiplication is a core operation in code-based cryptographic schemes, including the recently standardized HQC. Our method provides a systematic, branch-free approach that selects efficient Toom-Cook/Karatsuba decomposition for \(\text {GF}(2)[x]\) polynomial multiplication. We systematically explore Toom-Cook/Karatsuba decomposition trees and generate specialized implementations without table look-ups or conditional branches, enabling precise selection for a given parameter set. Unlike general-purpose libraries such as gf2x, which use table look-ups and conditional branches to support arbitrary polynomial degrees, our parameter-specific, branch-free design avoids this runtime selection overhead and enables precise algorithm selection for each parameter set. We further implement architecture-specific base multiplications using VPCLMULQDQ for AVX2 and PMULL for Neon. To evaluate our method, we compare against the gf2x library and state-of-the-art implementations of HQC and BIKE in both AVX2 and Neon environments. In particular, for HQC-1 and HQC-3 on AVX2 with PCLMULQDQ, our method reduces polynomial multiplication cycles by 17-32% compared to the official HQC implementation. For HQC-5, we observe that FFT-based multiplication can be preferable. For BIKE in the same environment, we observe a 5-11% improvement in polynomial multiplication cycles compared to the method of Chen et al. (CHES 2021).
