Indefiniteness makes lattice reduction easier

2025-11-20

AI Summary

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.

Technology Category

Machine Learning: Learning with ManifoldsSearch and Optimization: Other Foundations of Search & OptimizationConstraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingSecurity and Privacy: Applications of cryptography

Abstract

Since the invention of the famous LLL algorithm, lattice reduction has been an extremely useful tool in computational number theory. By construction, the LLL algorithm deals with lattices living in a vector space endowed with a positive definite scalar product. However, it seems quite nature to ask about the indefinite case, where the scalar product is replaced by an arbitrary quadratic form, possibily indefinite. This question was considered independently in two lines of work. One by G{á}bor Ivanyos and {Á}gnes Sz{á}nt{ó} and one by Denis Simon. Both lead to an algorithm that generalizes LLL and whose performance is very similar to LLL, i.e. a polynomial-time algorithm that approximates the shortest vector within an approximation factor exponential in the dimension. Denis Simon achieves an approximation factor close to that of LLL under the assumption that no isotropic vectors arise during reduction. G{á}bor Ivanyos and {Á}gnes Sz{á}nt{ó} show that it is possible to avoid isotropic vectors altogether, at the cost of a somewhat worse approximation factor. In this paper, we revisit the reduction of indefinite lattices and conclude that it can lead to much better reduced representations that previously thought. We also conclude that the approximation factor depends on the signature of the indefinite lattice rather than on its dimension.

Problem

• Generalizing LLL algorithm to indefinite quadratic forms in lattice reduction
• Analyzing approximation factors under isotropic vector constraints
• Establishing dependency of approximation quality on lattice signature

Innovation

• Generalizes LLL algorithm to indefinite quadratic forms
• Avoids isotropic vectors during lattice reduction process
• Approximation factor depends on signature not dimension