Paper 2019/320

Integral Matrix Gram Root and Lattice Gaussian Sampling without Floats

Léo Ducas, Steven Galbraith, Thomas Prest, and Yang Yu


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 $\mathbf{\Sigma} = \mathbf{A} \mathbf{A}^t$ of the target covariance matrix $\mathbf{\Sigma}$. The suggested decomposition, e.g. the Cholesky decomposition, gives rise to a square matrix $\mathbf{A}$ 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 $\mathbf{A}$ to being a square matrix, we show that such a decomposition can be efficiently found by allowing $\mathbf{A}$ to be wider (say $n \times 9n$). 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.

Available format(s)
Public-key cryptography
Publication info
A minor revision of an IACR publication in EUROCRYPT 2020
Lattice based cryptographyDiscrete Gaussian samplingMatrix decomposition
Contact author(s)
ducas @ cwi nl
s galbraith @ auckland ac nz
thomas prest @ pqshield com
yang yu0986 @ gmail com
2020-05-30: last of 6 revisions
2019-03-29: received
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Creative Commons Attribution


      author = {Léo Ducas and Steven Galbraith and Thomas Prest and Yang Yu},
      title = {Integral Matrix Gram Root and Lattice Gaussian Sampling without Floats},
      howpublished = {Cryptology ePrint Archive, Paper 2019/320},
      year = {2019},
      note = {\url{}},
      url = {}
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