Saved in:
| Main Authors: | , , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.12948 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915365823447040 |
|---|---|
| author | Zhao, Lihao Tian, Chengliang Bi, Jingguo Xu, Guangwu Yu, Jia |
| author_facet | Zhao, Lihao Tian, Chengliang Bi, Jingguo Xu, Guangwu Yu, Jia |
| contents | 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 \textbf{CrossEuc} that strategically applies the Euclidean algorithm across dimensions. Building on this framework, we introduce \textbf{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 \textbf{HVec}, our optimized algorithm \textbf{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 \textbf{HVecSBP} achieves at least a $13.5\times$ efficiency improvement compared to existing methods. For general-form input lattice bases, converting them to HNF before applying \textbf{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 \textbf{HVecSBP} yields increasingly significant efficiency gains, outperforming hybrid approaches that rely on prior \textbf{HNF} conversion. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_12948 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Algorithms for the Shortest Vector Problem in $2$-dimensional Lattices, Revisited Zhao, Lihao Tian, Chengliang Bi, Jingguo Xu, Guangwu Yu, Jia Computational Geometry Cryptography and Security 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 \textbf{CrossEuc} that strategically applies the Euclidean algorithm across dimensions. Building on this framework, we introduce \textbf{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 \textbf{HVec}, our optimized algorithm \textbf{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 \textbf{HVecSBP} achieves at least a $13.5\times$ efficiency improvement compared to existing methods. For general-form input lattice bases, converting them to HNF before applying \textbf{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 \textbf{HVecSBP} yields increasingly significant efficiency gains, outperforming hybrid approaches that rely on prior \textbf{HNF} conversion. |
| title | Algorithms for the Shortest Vector Problem in $2$-dimensional Lattices, Revisited |
| topic | Computational Geometry Cryptography and Security |
| url | https://arxiv.org/abs/2504.12948 |