The Algebraic Way to Solve LWE

本文研究了基于错误学习(LWE)问题的密码学构造，重点探讨了通过Kannan嵌入方法将其转化为代数问题的理论框架。我们系统分析了Kannan嵌入在LWE实例中的应用，证明了该转化过程的严谨性及其在密码分析中的有效性。具体而言，本文展示了如何将LWE的搜索问题嵌入到格结构中，并利用代数工具(如最短向量问题，SVP)求解，同时严格论证了转化后的代数形式与原问题的等价性。此外，我们讨论了该方法在参数选择、计算复杂度及安全性证明中的优势，为基于LWE的密码方案设计提供了理论支撑。实验结果进一步验证了该方法的可行性与效率。This paper studies cryptographic constructions based on the Learning With Errors (LWE) problem, with a focus on exploring the theoretical framework of transforming it into an algebraic problem through Kannan’s embedding method. We systematically analyze the application of Kannan’s embedding in LWE instances and prove the rigor of this transformation process and its effectiveness in cryptographic analysis. Specifically, this paper demonstrates how to embed the search problem of LWE into a lattice structure and solve it using algebraic tools (such as the Shortest Vector Problem, SVP), while strictly proving the equivalence between the transformed algebraic form and the original problem. Additionally, we discuss the advantages of this method in parameter selection, computational complexity, and security proof, providing theoretical support for the design of LWE-based cryptographic schemes. Experimental results further validate the feasibility and efficiency of this approach.