The recent advance in the post-quantum cryptography (PQC) field has gradually shifted from the theory to the implementation of the cryptosystem, especially on the hardware platforms. Following this trend, in this paper, we aim to present efficient implementations of the finite field arithmetic (key component) for the binary Ring-Learning-with-Errors (Ring-LWE) PQC through a novel lookup-table (LUT)-like method. In total, we have carried out four stages of interdependent efforts: (i) an algorithm-hardware co-design driven derivation of the proposed LUT-like method is provided detailedly for the key arithmetic of the BRLWE scheme; (ii) the proposed hardware architecture is then presented along with the internal structural description; (iii) we have also presented a novel hybrid size structure suitable for flexible operation, which is the first report in the literature; (iv) the final implementation and comparison processes have also been given, demonstrating that our proposed structures deliver significant improved performance over the state-of-the-art solutions. The proposed designs are highly efficient and are expected to be employed in many emerging applications.