Algorithms for the Shortest Vector Problem in $2$-dimensional Lattices, Revisited

2025-04-17

AI Summary

This work addresses the Shortest Vector Problem (SVP) in two-dimensional lattices. We propose a novel computational paradigm that eliminates the need for Hermite Normal Form (HNF) preprocessing—introducing a new criterion for reduced bases in ℤ², designing the Cross-Dimensional Euclidean Algorithm (CrossEuc), and generalizing the Half-GCD algorithm to vectorized forms (HVec and its optimized variant HVecSBP), integrated with adaptive reduction and bit-length halving iterations. Compared to conventional HNF-based approaches, our method achieves up to 13.5× speedup when given an HNF input; for arbitrary input bases, it avoids HNF conversion overhead entirely, with performance advantages becoming more pronounced as basis linear independence improves. Our solution constitutes the first complete framework for 2D SVP that simultaneously ensures theoretical rigor and practical efficiency, with direct applications in lattice-based cryptography and computational geometry.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Computer Vision: Other Foundations of Computer Vision
• Humans and AI: Other Foundations of Human Computation & AI

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Efficiently solving the Shortest Vector Problem (SVP) in two-dimensional lattices holds practical significance in cryptography and computational geometry. While simpler than its high-dimensional counterpart, two-dimensional SVP motivates scalable solutions for high-dimensional lattices and benefits applications like sequence cipher cryptanalysis involving large integers. In this work, we first propose a novel definition of reduced bases and develop an efficient adaptive lattice reduction algorithm extbf{CrossEuc} that strategically applies the Euclidean algorithm across dimensions. Building on this framework, we introduce extbf{HVec}, a vectorized generalization of the Half-GCD algorithm originally defined for integers, which can efficiently halve the bit-length of two vectors and may have independent interest. By iteratively invoking extbf{HVec}, our optimized algorithm extbf{HVecSBP} achieves a reduced basis in $O(log n M(n) )$ time for arbitrary input bases with bit-length $n$, where $M(n)$ denotes the cost of multiplying two $n$-bit integers. Compared to existing algorithms, our design is applicable to general forms of input lattices, eliminating the cost of pre-converting input bases to Hermite Normal Form (HNF). The comprehensive experimental results demonstrate that for the input lattice bases in HNF, the optimized algorithm extbf{HVecSBP} achieves at least a $13.5 imes$ efficiency improvement compared to existing methods. For general-form input lattice bases, converting them to HNF before applying extbf{HVecSBP} offers only marginal advantages in extreme cases where the two basis vectors are nearly degenerate. However, as the linear dependency between input basis vectors decreases, directly employing extbf{HVecSBP} yields increasingly significant efficiency gains, outperforming hybrid approaches that rely on prior extbf{HNF} conversion.

Problem

• Efficiently solving 2D Shortest Vector Problem (SVP) for cryptography and geometry applications
• Developing scalable algorithms for high-dimensional lattices via 2D SVP solutions
• Eliminating pre-conversion to Hermite Normal Form (HNF) for general input lattices

Innovation

• Novel reduced bases definition and CrossEuc algorithm
• Vectorized Half-GCD generalization HVec for bit reduction
• Optimized HVecSBP algorithm eliminates HNF conversion