Integral Matrix Gram Root and Lattice Gaussian Sampling Without Floats

Many advanced lattice based cryptosystems require to sample lattice points from Gaussian distributions. One challenge for this task is that all current algorithms resort to floating-point arithmetic (FPA) at some point, which has numerous drawbacks in practice: it requires numerical stability analysis, extra storage for high-precision, lazy/backtracking techniques for efficiency, and may suffer from weak determinism which can completely break certain schemes. In this paper, we give techniques to implement Gaussian sampling over general lattices without using FPA. To this end, we revisit the approach of Peikert, using perturbation sampling. Peikert’s approach uses continuous Gaussian sampling and some decomposition minimal amsmath wasysym amsfonts amssymb amsbsy mathrsfs upgreek -69pt document$\mathbf {\Sigma }= \mathbf {A}\mathbf {A}^t$documentΣ=AAt of the target covariance matrix minimal amsmath wasysym amsfonts amssymb amsbsy mathrsfs upgreek -69pt document$\mathbf {\Sigma }$documentΣ. The suggested decomposition, e.g. the Cholesky decomposition, gives rise to a square matrix minimal amsmath wasysym amsfonts amssymb amsbsy mathrsfs upgreek -69pt document$\mathbf {A}$documentA with real (not integer) entries. Our idea, in a nutshell, is to replace this decomposition by an integral one. While there is in general no integer solution if we restrict minimal amsmath wasysym amsfonts amssymb amsbsy mathrsfs upgreek -69pt document$\mathbf {A}$documentA to being a square matrix, we show that such a decomposition can be efficiently found by allowing minimal amsmath wasysym amsfonts amssymb amsbsy mathrsfs upgreek -69pt document$\mathbf {A}$documentA to be wider (say minimal amsmath wasysym amsfonts amssymb amsbsy mathrsfs upgreek -69pt document$n \times 9n$documentn×9n). This can be viewed as an extension of Lagrange’s four-square theorem to matrices. In addition, we adapt our integral decomposition algorithm to the ring setting: for power-of-2 cyclotomics, we can exploit the tower of rings structure for improved complexity and compactness.