Random Integer Lattice Generation via the Hermite Normal Form

Lattices used in cryptography are integer lattices. Defining and generating a “random integer lattice” are interesting topics. A generation algorithm for a random integer lattice can be used to serve as a random input of all the lattice algorithms. In this paper, we recall the definition of the random integer lattice given by G. Hu et al. and present an improved generation algorithm for it via the Hermite normal form. It can be proven that with probability ≥0.99, this algorithm outputs an n-dim random integer lattice within O(n2) operations.

A Note on Decomposable and Reducible Integer Matrices

We propose necessary and sufficient conditions for an integer matrix to be decomposable in terms of its Hermite normal form. Specifically, to each integer matrix, we associate a symmetric integer matrix whose reducibility can be efficiently determined by elementary linear algebra techniques, and which completely determines the decomposability of the first one.

Symmetric Wilson systems

The Wilson orthonormal basis was constructed in 1991 by Daubechies, Jaffard and Journé using combinations of elements of Gabor tight frame with redundancy 2. In 1994, Auscher gave a characterization of the atoms for which the Wilson system is an orthonormal basis. Then, Kutyniok and Strohmer generalized the notion of Wilson system to the lattices whose generator matrix is in Hermite normal form.In the present publication, we introduce Wilson systems where the time-frequency shifts are combined symmetrically with respect to the origin. Using the arguments from previous papers that work also in this case, we give a full characterization of atoms for which the obtained Wilson systems are orthonormal bases in [Formula: see text] and we generalize these results to different sign sequences and to symplectic lattices of density [Formula: see text].