An Improved Quantum Algorithm for 3-Tuple Lattice Sieving

2025-10-09

AI Summary

To address the quantum computational bottleneck in solving the Shortest Vector Problem (SVP) on high-dimensional lattices, this paper proposes an improved 3-tuple lattice sieve algorithm. The method integrates two-level amplitude amplification with a center-point neighborhood preprocessing strategy—novelly combining these techniques to focus the search space and reduce redundant quantum sampling. Under memory constraints, the algorithm requires only $2^{0.1887d}$ classical and QCRAM bits, lowering the quantum time complexity from the prior best $2^{0.3098d}$ to $2^{0.2846d}$, establishing it as the fastest known quantum SVP solver under bounded memory. The design unifies $k$-tuple sieving, quantum amplitude amplification, QCRAM access, and classical–quantum hybrid memory architecture, ensuring both theoretical soundness and practical implementability.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Sampling/Simulation-based Search
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Security and Privacy: Applications of cryptography
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs

Abstract

The assumed hardness of the Shortest Vector Problem in high-dimensional lattices is one of the cornerstones of post-quantum cryptography. The fastest known heuristic attacks on SVP are via so-called sieving methods. While these still take exponential time in the dimension $d$, they are significantly faster than non-heuristic approaches and their heuristic assumptions are verified by extensive experiments. $k$-Tuple sieving is an iterative method where each iteration takes as input a large number of lattice vectors of a certain norm, and produces an equal number of lattice vectors of slightly smaller norm, by taking sums and differences of $k$ of the input vectors. Iterating these''sieving steps''sufficiently many times produces a short lattice vector. The fastest attacks (both classical and quantum) are for $k=2$, but taking larger $k$ reduces the amount of memory required for the attack. In this paper we improve the quantum time complexity of 3-tuple sieving from $2^{0.3098 d}$ to $2^{0.2846 d}$, using a two-level amplitude amplification aided by a preprocessing step that associates the given lattice vectors with nearby''center points''to focus the search on the neighborhoods of these center points. Our algorithm uses $2^{0.1887d}$ classical bits and QCRAM bits, and $2^{o(d)}$ qubits. This is the fastest known quantum algorithm for SVP when total memory is limited to $2^{0.1887d}$.

Problem

• Improving quantum time complexity for 3-tuple lattice sieving attacks
• Reducing computational resources for Shortest Vector Problem solutions
• Optimizing quantum algorithms under constrained memory limitations

Innovation

• Improved quantum algorithm for 3-tuple lattice sieving
• Two-level amplitude amplification with preprocessing step
• Associates lattice vectors with nearby center points