International Association
for Cryptologic Research

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.

This work conducts a comprehensive investigation on determining the entropic hardness of (Ring/Module) Learning with Rounding (LWR) under polynomial modulus. Particularly, we establish the hardness of (M)LWR for general entropic secret distributions from (Module) LWE assumptions based on a new conceptually simple framework called rounding lossiness. By combining this hardness result and a trapdoor inversion algorithm with asymptotically the most compact parameters, we obtain a compact lossy trapdoor function (LTF) with improved efficiency. Extending our LTF with other techniques, we can derive a compact all-but-many LTF and PKE scheme against selective opening and chosen ciphertext attacks, solely based on (Module) LWE assumptions within a polynomial modulus. Additionally, we show a search-to-decision reduction for RLWR with Gaussian secrets from a new Rényi divergence-based analysis.