Post-Quantum Cryptography Utilizing Lattices

The advent of quantum computing poses a significant threat to classical cryptographic systems, necessitating the development of post-quantum cryptography because it can withstand quantum assaults, lattice-based encryption has become a viable option. Lattice-based cryptography is a fundamental component of secure system design and is resistant to quantum attacks because it makes use of the geometric structure of high-dimensional lattices. The main lattice-based issues that form the foundation of several post-quantum cryptography systems, such as Learning With Errors (LWE) and Short Integer Solution (SIS), are the subject of this research. We look at the challenges' theoretical underpinnings, real-world applications in public-key cryptosystems, and significance for maintaining security in a world allowed by quantum computing. Lattice-based methods are particularly versatile beyond simple encryption and signatures, as evidenced by their capacity to adapt to sophisticated cryptographic capabilities like secure multi-party computing and homomorphic encryption. A thorough examination of Regev's LWE-based cryptosystem is given, showing how its intrinsic hardness provides robust security guarantees. This includes an in-depth analysis of parameter selection, error distribution, and the effects these factors have on computing efficiency and security in real-world applications. Moreover, the SIS problem is investigated about its potential applications in digital signatures and collision-resistant hash functions. The study also looks at the trade-offs between security assurances and computational complexity, with an emphasis on how feasible these systems are in practice with limited resources. This research examines the present level of post-quantum cryptography optimisation, highlights remaining issues, and suggests future paths by examining the complex links between LWE, SIS, and other lattice problems. The objective of this effort is to connect theoretical developments with real-world applications by tackling the computational difficulties of key generation, encryption, and decryption. In the final analysis, this study advances safe cryptography frameworks for the quantum age.