Paper 2024/1695 Original: eprint.iacr.org/2024/1695 ↗

Discrete Gaussians Modulo Sub-Lattices: New Leftover Hash Lemmas for Discrete Gaussians

Abstract

The Leftover Hash Lemma (LHL) is a powerful tool for extracting randomness from an entropic distribution, with numerous applications in cryptography. LHLs for discrete Gaussians have been explored in both integer settings by Gentry et al. (GPV, STOC'08) and algebraic ring settings by Lyubashevsky et al. (LPR, Eurocrypt'13). However, the existing LHLs for discrete Gaussians have two main limitations: they require the Gaussian parameter to be larger than certain smoothing parameters, and they cannot handle cases where fixed and arbitrary information is leaked. In this work, we present new LHLs for discrete Gaussians in both integer and ring settings. Our results show that the Gaussian parameter can be improved by a factor of $\omega(\sqrt{\log\lambda}) $ and $ O(\sqrt{N}) $ compared to the regularity lemmas of GPV and LPR, respectively, under similar parameter choices such as the dimension and ring. Furthermore, our new LHLs can be applied to leaked discrete Gaussians, and the result can be used to establish asymptotic hardness of the extended MLWE assumptions, addressing an open question in recent works by Lyubashevsky et al. (LNP, Crypto'22). Our central techniques involve new fine-grained analyses of the min-entropy in discrete Gaussians modulo sublattices and should be of interest.