International Journal on Science and Technology

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A Number Theoretic Investigation Of The Hardness Assumptions Underlying The Shortest Vector Problem In Lattice Cryptography

Author(s) Mr. Jitendra Kumar Sharma
Country India
Abstract The lattice-based encryption paradigm (LBE) represents a new security model emphasizing flexible development, high levels of security, and efficient computations. It is now becoming more commonly seen as an alternative to other systems of post-quantum cryptography due to its foundations for both security (i.e., there is a high degree of confidence that lattice-based cryptography is secure) and efficiency with respect to computation. The theoretical basis for this report is related to the difficulty of solving the short vector problem (SVP) and other cryptographic hard problems concerning lattices. The report focuses on the hardness of lattice-based cryptography relative to three sub-components of the theoretical work: the relationship between geometric number theory and lattice-based systems, the application of AI to help determine parameters and the efficiencies of finding new lattice reduction algorithms, and finally estimating the difficulty to solve lattice problems. This report reviews the inherent difficulty of short vector problems (SVPs) and other kinds of SVPs, such as the closest vector problem. The report analyzes how to reduce both the time and space complexities for classical and quantum computing solutions through the discovery of new techniques based upon number-theoretic concepts. The report also discusses a number of examples, including digital signature products, encryption systems and homomorphic encryption systems as primitive constructions of cryptography based on lattices or through the properties of the lattice structures. Finally, based on number-theoretical relationships among complex numbers, the report provides a new theoretical approach to the construction of lattice-based systems and their strength relative to other types of cryptographic systems via the provision of theoretical bounds related to the lengths of lattice basis vectors.As a result of this research into bridging the divide between theoretical and practical cryptography, the results show that lattice-based systems have the capability of providing the strength to withstand quantum attacks. Through this research study, there is an improvement in how we understand mathematically the difficulty of lattices and how to create post-quantum secure cryptographic techniques through the use of lattices...
Keywords Keywords: Lattice-based cryptography, Shortest Vector Problem (SVP), Number theory, post-quantum, cryptography, Homomorphic encryption, AI, Hardness assumptions, Geometry of numbers
Field Mathematics
Published In Volume 17, Issue 1, January-March 2026
Published On 2026-03-31
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