Exploring the Underlying Assumptions of Lattice Constructions:

Abstract

Owing to its adaptability in cryptographic protocols and possible defence against quantum attacks, lattice-based cryptography has become a very attractive topic. This survey explores the fundamental hard problems in lattice theory, such as the Shortest Vector Problem (SVP), the Closest Vector Problem (CVP), and the Learning With Errors (LWE) problem, which form the cornerstone of latticebased cryptosystems. We explore the intricate mathematical structures and specifics of all of these problems, highlighting their computational difficulty and importance. In addition, we look at the idea of “crypto dark matter,” which refers to cryptographic structures and protocols that function outside of the accepted frameworks for cryptographic analysis and application. Our aim is to gain knowledge regarding the incorporation of lattice-based hard problems into the crypto dark matter framework through a review of the literature and uncover new dimensions of security and functionality that challenge traditional approaches. This analysis emphasises the application of current developments in latticebased cryptography in building secure cryptographic primitives while o↵ering a thorough overview of the field. In the era of quantum computing, our studies highlight the importance of lattice-based hard problems as a frontier for innovative cryptography research as well as a solid foundation for strong cryptographic systems. The aim of this study is to help researchers and practitioners better understand how advanced cryptographic applications interact with lattice theory, which will ultimately lead to the development of cryptographic solutions that are more e↵ective and secure.