Polynomial multiplication algorithms such as Toom-Cook and the Number Theoretic Transform are fundamental building blocks for lattice-based post-quantum cryptography. In this work we present correlation power-analysis-based side-channel analysis methodologies targeting every polynomial multiplication strategy for all lattice-based post-quantum key encapsulation mechanisms in the final round of the NIST post-quantum standardization procedure. We perform practical experiments on real side-channel measurements, demonstrating that our method allows to extract the secret key from all lattice-based post-quantum key encapsulation mechanisms. Our analysis shows that the used polynomial multiplication strategy can significantly impact the time complexity of the attack.