Discrete Gaussian Leftover Hash Lemma over Infinite Domains

The classic Leftover Hash Lemma (LHL) is one of the most useful tools in cryptography, and is often used to argue that certain distributions arising from modular subset-sums are close to uniform over some finite domain. Though extremely useful and powerful in general, the applicability of the leftover hash lemma to lattice based cryptography is limited for two reasons. First, typically the distributions we care about in lattice-based cryptography are {\em discrete Gaussians}, not uniform. Second, the elements chosen from these discrete Gaussian distributions lie in an infinite domain: a lattice rather than a finite field. In this work we prove a ``lattice world analog of LHL over infinite domains, proving that certain ``generalized subset sum\u27\u27 distributions are statistically close to well behaved discrete Gaussian distributions, even without any modular reduction. Specifically, given many vectors $\{\vec x_i\}_{i=1}^m$ from some lattice $L\subset\R^n$, we analyze the probability distribution $\sum_{i=1}^m z_i \vec x_i$ where the integer vector $\vec z \in \Z^m$ is chosen from a discrete Gaussian distribution. We show that when the $\vec x_i$\u27s are ``random enough\u27\u27 and the Gaussian from which the $\vec z$\u27s are chosen is ``wide enough\u27\u27, then the resulting distribution is statistically close to a near-spherical discrete Gaussian over the lattice $L$. Beyond being interesting in its own right, this ``lattice-world analog of LHL has applications for the new construction of multilinear maps \cite{GGH12}, where it is used to sample Discrete Gaussians obliviously. Specifically, given encoding of the $\vec x_i$\u27s, it is used to produce an encoding of a near-spherical Gaussian distribution over the lattice. We believe that our new lemma will have other applications, and sketch some plausible ones in this work