Analysis of Ring-LWE and LWR Cryptographic Schemes: Foundations, Implementation, and Performance

Lattice-based cryptography has emerged as a leading framework for securing communication in the post-quantum era, supported by strong worst-case hardness guarantees. This study focuses on two central constructions, Ring Learning with Errors (Ring-LWE) and Learning with Rounding (LWR), and examines their mathematical foundations, implementation behavior, and performance characteristics. While Ring-LWE benefits from well-established and conservative security reductions, along with broad applicability across cryptographic primitives, its reliance on discrete Gaussian sampling introduces computational overhead and potential side-channel vulnerabilities. In contrast, LWR replaces stochastic noise with deterministic rounding, thereby simplifying implementation and improving efficiency, particularly on resource-constrained platforms, albeit under comparatively newer and less conservative hardness assumptions. Both schemes are implemented and benchmarked across multiple parameter sets, enabling a systematic comparison of key generation, encryption, and decryption costs. The experimental results highlight the complementary strengths of these approaches and suggest that hybrid constructions may effectively combine the strong theoretical guarantees of Ring-LWE with the practical efficiency of LWR.