A SHEAF-THEORETIC AND ETALÉ SPACE APPROACHTO THE SHORTEST VECTOR PROBLEM: ORTHOGONALIZATION, COBOUNDARY MAPS,AND MEMORY-EFFICIENT SIEVING

The Shortest Vector Problem (SVP) is a fundamental challenge in computational number theory and lattice-based cryptography, with critical implications for cryptanalysis, coding theory, and computational geometry. Classical algorithms such as enumeration, sieving, and basis reduction techniques either suffer from exponential time complexity or prohibitive memory demands. This work presents a novel SVP-solving framework rooted in sheaf theory and the geometry of Etalé spaces, providing a categorical and topological interpretation of vector localization, difference computation, and sieving procedures. By embedding the lattice problem into a manifold structure and defining fractional coordinate spaces via exact sequences, we develop a generalized coboundary map formulation that subsumes classical vector-difference methods used in sieving. Orthogonalization through algorithms such as LLL reduces search complexity, while the Etalé-space-based sieving allows for localized, memory-aware updates. Complexity analysis shows that while the algorithm retains exponential dependence on lattice dimension in the worst case, careful selection of subset coverings and coarse-graining strategies can reduce memory requirements from exponential to polynomial scaling. The proposed approach bridges the gap between geometric group theory, algebraic topology, and algorithmic lattice reduction, offering a mathematically rigorous and computationally viable method for SVP in moderate dimensions.