International Association for Cryptologic Research

International Association
for Cryptologic Research


Integral Matrix Gram Root and Lattice Gaussian Sampling without Floats

Léo Ducas , Centrum Wiskunde en Informatica
Steven D. Galbraith , University of Auckland
Thomas Prest , PQShield Ltd
Yang Yu , IRISA, Université Rennes 1, and CNRS
DOI: 10.1007/978-3-030-45724-2_21 (login may be required)
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Conference: EUROCRYPT 2020
Abstract: 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 $\BSigma = \matA \matA^t$ of the target covariance matrix $\BSigma$. The suggested decomposition, e.g. the Cholesky decomposition, gives rise to a square matrix $\matA$ 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 $\matA$ to being a square matrix, we show that such a decomposition can be efficiently found by allowing $\matA$ 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.
Video from EUROCRYPT 2020
  title={Integral Matrix Gram Root and Lattice Gaussian Sampling without Floats},
  booktitle={39th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Zagreb, Croatia, May 10–14, 2020, Proceedings},
  series={Lecture Notes in Computer Science},
  keywords={Lattice based cryptography;Discrete Gaussian sampling;Matrix decomposition},
  author={Léo Ducas and Steven D. Galbraith and Thomas Prest and Yang Yu},