Generalized Theta Series of a Lattice

2025-07-03

AI Summary

This paper addresses three fundamental problems in lattice theory: stability determination, isomorphism identification, and the secrecy gain conjecture for formally self-dual lattices. To tackle them, we introduce a novel invariant—the generalized theta series—inspired by the generalized Hamming weights of linear codes. This series captures fine-grained structural information about sublattices, significantly enhancing both stability classification and isomorphism discrimination. Crucially, we construct explicit counterexamples among formally self-dual lattices, thereby disproving the long-standing conjecture that the theta quotient attains its minimum at the symmetry point; we demonstrate that the minimum may instead occur away from this point. Integrating techniques from algebraic coding theory and lattice theory, and combining analytical derivations with numerical verification, our work provides new theoretical tools and rigorous foundations for security analysis in lattice-based cryptography.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Game Theory and Economic Paradigms: Adversarial Learning
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

Mimicking the idea of the generalized Hamming weight of linear codes, we introduce a new lattice invariant, the generalized theta series. Applications range from identifying stable lattices to the lattice isomorphism problem. Moreover, we provide counterexamples for the secrecy gain conjecture on isodual lattices, which claims that the ratio of the theta series of an isodual (and more generally, formally unimodular) lattice by the theta series of the integer lattice $mathbb{Z}^n$ is minimized at a (unique) symmetry point.

Problem

• Introduce generalized theta series for lattices
• Apply to identify stable lattices and isomorphism
• Counterexamples for secrecy gain conjecture

Innovation

• Introducing generalized theta series for lattices
• Applying to stable lattices identification
• Countering secrecy gain conjecture examples