Provide an in-depth analysis of the Learning With Errors (LWE) problem as the foundation for post-quantum cryptography. Compare the hardness assumptions of LWE against those of Integer Factorization used in RSA. Design a theoretical key encapsulation mechanism (KEM) based on lattice problems, explaining how you would choose parameters to balance security against quantum attacks with computational efficiency on standard hardware. Discuss the resistance of your proposed scheme to side-channel attacks.