Abstract
Lattices are discrete mathematical objects with widespread applications to integer programs as well as modern cryptography. An important class of lattices are those that possess an orthogonal basis, since if such an orthogonal basis is known, then many other fundamental problems on lattices can be solved easily (e.g., the Closest Vector Problem). However, intriguingly, deciding whether a lattice has an orthogonal basis is not known to be either NP-complete or in P. In this paper, we focus on the orthogonality decision problem for a well-known family of lattices, namely Construction-A lattices. These are lattices of the form $C+q\mathbb{Z}^n$, where $C$ is an error-correcting $q$-ary code, and are studied in communication settings. We provide a complete characterization of lattices obtained from binary and ternary codes using Construction-A that have an orthogonal basis. We use this characterization to give an efficient algorithm to solve the orthogonality decision problem. Our algorithm also finds an orth...
Highlights
A lattice is the set of integer linear combinations of a set of basis vectors B ∈ Rm×n, namely L = L(B) = {xB | x ∈ Zm}
This led to renewed interest in the complexity of two fundamental lattice problems: the Shortest Vector Problem (SVP) and the Closest Vector Problem (CVP)
Deciding if a lattice is equivalent to Zn, and deciding if a lattice has an orthogonal basis, are special cases of the more general Lattice Isomorphism Problem (LIP)
Summary
Given an arbitrary basis B, it is not known how to efficiently verify whether the lattice generated by B is isomorphic to Zn upto an orthogonal transformation. Deciding if a lattice is equivalent to Zn, and deciding if a lattice has an orthogonal basis, are special cases of the more general Lattice Isomorphism Problem (LIP). Given that LIP, deciding isomorphism to Zn, and deciding whether a lattice has an orthogonal basis appear to be difficult problems for arbitrary input lattices, it is natural to address families of lattices where these problems are solvable efficiently. Designing an efficient algorithm for the orthogonality decision problem exploiting the direct product decomposition characterization appears to be non-trivial
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