# Learning with Errors: A Lattice-Based Keystone of Post-Quantum Cryptography
## 6 Conclusions and Future Research  
Lattice theory, originating from diverse mathematical inquiries into geometry, number theory and abstract algebra, has evolved into a foundational concept in modern cryptography, particularly in the context of post-quantum cryptography. Recent advances in quantum computing necessitate the evolution of cryptographic practices that anticipate the emergence of powerful quantum computers and the potential threat they pose to existing cryptographic algorithms. LWE is pivotal in post-quantum cryptography due to its fundamental role in creating cryptographic schemes resilient to attacks from both classical and quantum computers. For the science of cryptography and computer science, the importance of LWE is why it has been a constant subject of study, research and analysis since its inception. In this paper, we introduced a variant of the LWE cryptographic scheme, adding to the key generation step of the protocol a transformation that changes the elements of the product matrix of the secret key \(S\) and the uniformly random chosen matrix \(A\). Avoiding interference with the rest of the structure of the LWE algorithm, this additional step maintains the efficiency and security of the protocol without scaling up the complexity, as the operations added into the protocol are additions and multiplications of matrices. We discussed certain instances of our variant, choosing a particular transformation mapping \(f\), and presented discrete implementations in small dimensions. The main advantage of our variant is the addition of a transformation in the key generation algorithm, which has both an algebraic and a geometric interpretation into a lattice. Moreover, our variant is regarded as a primitive for applications that employ robust encryption, and our future work aims to implement the protocol both generally and to certain applications. Due to its potential security features, LWE has attracted a lot of attention lately, especially in light of the concerns posed by quantum computing. Even though LWE and its variants are widely studied and performed and have been advanced significantly, there are still a number of open issues and creative uses that merit further investigation. The computational expense of lattice-based encryption in comparison to more conventional cryptographic techniques is one of its primary drawbacks. The goal of ongoing research is to make lattice-based primitives more efficient in terms of computing complexity and usefulness across a range of platforms. Blockchain technology and cryptocurrencies may benefit from the use of lattice-based cryptography, which can improve privacy, scalability and resilience to quantum attacks. Additionally, there are still open questions about the best approaches for standardization. Investigating lattice-based consensus methods, signature protocols and privacy-boosting strategies for blockchain networks may result in novel blockchain systems with improved security features. This covers factors like compatibility across various implementations and the proper parameter selection for security and performance trade-offs. Although our LWE-based cryptographic scheme is reasonably flexible, efficient and secure, improvements and optimizations can be made. Optimizing LWE cryptosystems requires a multidisciplinary approach that combines mathematical research, algorithmic advances, hardware breakthroughs and a thorough understanding of both theoretical and practical aspects of cryptography. Ongoing research focuses on the implementation and comparison of the proposed variant, the key size of cryptosystems to improve their complexity and security, and the investigation of potential vulnerabilities to strengthen them against possible attacks. Conceptualization, M.E.S., methodology, M.E.S. ; validation, M.E.S., I.K.S. and G.G. ; writing--original draft preparation, M.E.S. ; writing, M.E.S.--review and editing, M.E.S., I.K.S. and G.G. All authors have read and agreed to the published version of the manuscript..