divide-and-conquer lattice reduction

High-dimensional lattice basis reduction faces a fundamental trade-off between efficiency and output quality. This work proposes a novel hierarchical reduction method that, for the first time, integrates the divide-and-conquer principle of merge sort into the LLL framework. By performing parallel local reductions on sub-bases and combining PotLLL-style deep insertions with Gram–Schmidt orthogonalization, the algorithm achieves logarithmic parallel depth while preserving unimodular transformations. Experimental results demonstrate significant improvements over classical algorithms on subset-sum and NTRU-derived lattices, exhibiting stronger orthogonality, fewer swaps, better Hermite factors, and excellent scalability for both multi-core and distributed execution environments.

This paper investigates the Shortest Vector Problem (SVP) and the construction of dense sublattices in integer lattices. We propose the first abstract, recursion-based lattice reduction framework that is independent of any specific basis, Gram–Schmidt orthogonalization, or projection operations; instead, it achieves structured approximation via hierarchical search over low-rank sublattices and their duals. Our method unifies SVP solving and the search for arbitrary-rank ℓ-sublattices satisfying min{ℓ, n−ℓ} ≤ n−k+1, yielding improved trade-offs between oracle query complexity and approximation quality. Theoretically, we obtain a quasipolynomial-time algorithm for finding short vectors. Practically, leveraging automated algorithm search, we recover the performance of classical basis reduction algorithms and discover novel, provably superior alternatives.

This work addresses lattice basis reduction—a classical combinatorial optimization problem—by proposing the first self-supervised geometric deep learning method that requires no labeled data. Methodologically, it parameterizes the space of reduction algorithms via neural networks and directly outputs decomposable unimodular matrices to optimize basis orthogonality. It is the first to embed isometric/scaling invariance and hyperoctahedral group equivariance into a self-supervised framework, and introduces a grid-convolutional architecture enabling joint reduction of multiple lattices and computational amortization. Experiments demonstrate that the method achieves reduction quality and time complexity comparable to the LLL algorithm on standard benchmarks; its grid-convolutional variant significantly improves efficiency for multi-lattice processing and shows strong practical utility in real-world applications such as wireless communications. The core contribution lies in establishing a novel paradigm for lattice reduction: unsupervised, geometric-prior-driven, and interpretable via unimodular factorization.

This paper investigates lattice basis reduction under indefinite quadratic forms—i.e., generalizing the LLL algorithm when the inner product is replaced by an arbitrary (not necessarily positive-definite) quadratic form. The authors propose a signature-aware reduction framework: reduction difficulty is characterized by the quadratic form’s inertia index (rather than ambient dimension); a modified Gram–Schmidt orthogonalization is designed to avoid isotropic vectors; and an explicit relationship is established between approximation factors and the signature. Theoretically, the shortest-vector approximation ratio depends only on the difference between the numbers of positive and negative eigenvalues—yielding a bound substantially tighter than classical dimension-dependent ones. The algorithm retains polynomial-time complexity and, empirically, achieves superior reduction quality and shorter vector approximations on typical indefinite lattices.

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.

This work addresses the efficient computation of a Hermite normal form (HNF) basis for lattices defined by integer relations. Specifically, for the lattice consisting of all integer row vectors \( p \) satisfying \( pF \in \mathcal{L}(M) \), the paper proposes a Las Vegas randomized algorithm. The method generalizes HNF computation to arbitrary integer-relation lattices by integrating randomization techniques, lattice theory, and integer matrix arithmetic, producing a correct basis with failure probability at most \( 1/2 \). Notably, when \( M \) is square and \( F \) is the identity matrix, the algorithm achieves a bit complexity nearly matching the lower bound dictated by matrix multiplication, thereby significantly improving computational efficiency.

This work addresses the absence of a unified, verifiable catalog for small-scale fast matrix multiplication algorithms scattered across diverse domains with inconsistent formats and naming conventions. The authors construct a comprehensive algorithm repository covering all instances up to size 32×32×32, supporting multiple number fields and commutative variants. By introducing the notion of “non-overlappingness,” they clearly distinguish between discovering novel bilinear kernels and composing existing ones, thereby decoupling algorithm discovery from composition and resolving attribution disputes in the literature. Leveraging a state-of-the-art closure-search framework augmented with techniques such as axis flipping, Kronecker products, axis concatenation, random products, distributive recomposition (including output stripping and pair fusion), and downward projection, the study systematically recombines and extends known algorithms, yielding numerous new low-rank schemes—including ternary integer algorithms—and automatically generates DIS09 comparison tables categorized by number field and commutativity.

This work addresses the poor basis quality produced by the LLL algorithm in high-dimensional lattices, which often fails to meet practical requirements. The authors formulate lattice reduction as a single-agent Markov decision process and propose a novel approach that integrates AlphaZero-style self-play with adaptive-horizon Monte Carlo Tree Search (MCTS) to learn improved reduction strategies directly within the original LLL operation space. Their method employs a deep residual network enhanced with an entropy-gated expansion mechanism. Remarkably, the resulting policy, DeltaStar—trained solely on 8-dimensional q-ary lattices—requires fewer row operations than LLL and demonstrates strong zero-shot generalization to lattices with unknown moduli and dimensions up to 32, substantially enhancing both reduction efficiency and generalization capability.

本文通过提供一个从通用格版本到理想格问题的多项式时间归约，证明了在$\ell_2$范数下几个理想格问题（包括SVP和CVP）的最坏情况下的硬度。

This work addresses the inefficiency of existing submodular function minimization methods on distributive lattices, which typically require embedding the lattice into a Boolean lattice, thereby inducing an exponential blow-up in the search space. To overcome this limitation, the paper introduces the first general-purpose optimization framework that operates directly within the distributive lattice without resorting to Boolean lattice expansion. The proposed framework inherently avoids exponential space growth, remains compatible with classical algorithms designed for Boolean lattices, and achieves significantly improved scalability and computational efficiency. Both theoretical analysis and empirical evaluation demonstrate that the method substantially outperforms traditional approaches in terms of runtime while preserving correctness and generality.