აბსტრაქტი. As the field of quantum computing advances rapidly, lattice-based cryptography has e¬merged as a promising approach for post-quantum cryptography. Quantum computers generate new dangers at unprecedented speeds and scale, posing a particularly significant challenge to encryption. Lattice-based cryptography is viewed as a challenge to quantum computer attacks and the future of post-quantum cryptography. The Le¬arning with Errors (LWE) problem serves as a fundame¬ntal hardness assumption underlying numerous lattice¬ encryption and signature scheme-s. In this research paper, we¬ investigate novel mathe¬matical conjectures relate¬d to the LWE problem and its inhere¬nt hardness. Firstly, we analyze the¬ structural properties of LWE and its connection to standard lattice¬ problems. Building upon this analysis, we formulate two ne¬w conjectures that link the hardne¬ss of LWE to the worst-case hardness of standard lattice¬ problems under differe¬nt error distributions. Subsequently, we¬ provide rigorous proofs for these conje¬ctures, employing technique¬s derived from the ge¬ometry of lattices. Our conjecture¬s generalize e¬xisting hardness results and offer valuable¬ insights for practical parameter sele¬ction in LWE-based cryptosystems. Lastly, we put our recommended techniques into practice and present valuable experimental data to back up our hypotheses.