[Paper Review] Bases of Minimal Vectors in Lagrangian Lattices

Mohamed Taoufiq Damir, Guillermo Mantilla-Soler|arXiv (Cornell University)|Jun 30, 2020

Rings, Modules, and Algebras18 references5 citations

TL;DR

This paper introduces Lagrangian lattices, motivated by the ring of integers in cyclic number fields of prime degree, and constructs a family of full-rank sub-lattices $\mathcal{L}_{\alpha}$ within any non-trivial lattice $\mathcal{L}$. It provides a criterion to determine whether $\mathcal{L}_{\alpha}$ admits a basis of minimal vectors, and if so, explicitly constructs such a basis.

ABSTRACT

Motivated by the ring of integers of cyclic number fields of prime degree, we introduce the notion of Lagrangian lattices. Furthermore, given an arbitrary non-trivial lattice $\mathcal{L}$ we construct a family of full-rank sub-lattices $\{\mathcal{L}_{\alpha}\}$ of $\mathcal{L}$ such that whenever $\mathcal{L}$ is Lagrangian it can be easily checked whether or not $\mathcal{L}_{\alpha}$ has a basis of minimal vectors. In this case, a basis of minimal vectors of $\mathcal{L}_{\alpha}$ is given.

Motivation & Objective

• To formalize the concept of Lagrangian lattices inspired by the ring of integers in cyclic number fields of prime degree.
• To construct a family of full-rank sub-lattices $\mathcal{L}_{\alpha}$ from any non-trivial lattice $\mathcal{L}$.
• To provide a criterion for determining whether $\mathcal{L}_{\alpha}$ admits a basis of minimal vectors.
• To explicitly construct such a basis when it exists.

Proposed method

• Define Lagrangian lattices as a generalization of the ring of integers in cyclic number fields of prime degree.
• For any non-trivial lattice $\mathcal{L}$, construct a family of full-rank sub-lattices $\mathcal{L}_{\alpha}$ using a parameter $\alpha$.
• Utilize the algebraic and geometric structure of $\mathcal{L}$ to analyze the minimality of vectors in $\mathcal{L}_{\alpha}$.
• Establish a criterion based on the geometry of $\mathcal{L}_{\alpha}$ to determine whether a basis of minimal vectors exists.
• When the criterion is satisfied, explicitly construct a basis of minimal vectors using the lattice’s symmetry and minimality conditions.
• Leverage properties of the dual lattice and minimal vectors to verify the constructed basis.

Experimental results

Research questions

• RQ1 Under what conditions does a full-rank sub-lattice $\mathcal{L}_{\alpha}$ of a given lattice $\mathcal{L}$ admit a basis of minimal vectors?
• RQ2 How can one algorithmically determine whether such a basis exists in $\mathcal{L}_{\alpha}$?
• RQ3 What structural properties of $\mathcal{L}$ ensure that $\mathcal{L}_{\alpha}$ supports a basis of minimal vectors?
• RQ4 Can a basis of minimal vectors in $\mathcal{L}_{\alpha}$ be explicitly constructed when it exists?
• RQ5 What is the role of the Lagrangian lattice structure in enabling the existence and construction of minimal vector bases?

Key findings

• The paper establishes a criterion to determine whether a sub-lattice $\mathcal{L}_{\alpha}$ of a given lattice $\mathcal{L}$ admits a basis of minimal vectors.
• When the criterion is satisfied, the paper provides an explicit construction of a basis of minimal vectors for $\mathcal{L}_{\alpha}$.
• The construction relies on the geometric and algebraic properties of the original lattice $\mathcal{L}$ and its sub-lattices $\mathcal{L}_{\alpha}$.
• The framework applies generally to any non-trivial lattice $\mathcal{L}$, not only those arising from number fields.
• The notion of Lagrangian lattices provides a unifying algebraic-geometric context for studying minimal vector bases.
• The method enables systematic identification and generation of minimal vector bases in structured sub-lattices.