Lattices with Symmetry
This provides a solution for lattice theory and cryptography applications where symmetry is present, but it is incremental as it builds on existing methods.
The paper tackles the problem of deciding whether a given lattice has an orthonormal basis, which lacks efficient algorithms for large ranks, and presents a deterministic polynomial-time algorithm that works when the lattice has sufficient symmetry, based on prior work by Gentry and Szydlo.
For large ranks, there is no good algorithm that decides whether a given lattice has an orthonormal basis. But when the lattice is given with enough symmetry, we can construct a provably deterministic polynomial-time algorithm to accomplish this, based on the work of Gentry and Szydlo. The techniques involve algorithmic algebraic number theory, analytic number theory, commutative algebra, and lattice basis reduction.