International Association
for Cryptologic Research

This paper studies the hardness of decision Module Learning with Errors (MLWE) under linear leakage, which has been used as a foundation to derive more efficient lattice-based zero-knowledge proofs in a recent paradigm of Lyubashevsky, Nguyen, and Seiler (PKC 21). Unlike in the plain LWE setting, it was unknown whether this problem remains provably hard in the module/ring setting. This work shows a reduction from the search MLWE to decision MLWE with linear leakage. Thus, the main problem remains hard asymptotically as long as the non-leakage version of MLWE is hard. Additionally, we also refine the paradigm of Lyubashevsky, Nguyen, and Seiler (PKC 21) by showing a more fine-grained tradeoff between efficiency and leakage. This can lead to further optimizations of lattice proofs under the paradigm.

We derive the first adaptively secure \ibe~and \abe for t-CNF, and selectively secure \abe for general circuits from lattices, with $1-o(1)$ leakage rates, in the both relative leakage model and bounded retrieval model (\BRM). To achieve this, we first identify a new fine-grained security notion for \abe~-- partially adaptive/selective security, and instantiate this notion from \LWE. Then, by using this notion, we design a new key compressing mechanism for identity-based/attributed-based weak hash proof system (\ib/\ab-\whps) for various policy classes, achieving (1) succinct secret keys and (2) adaptive/selective security matching the existing non-leakage resilient lattice-based designs. Using the existing connection between weak hash proof system and leakage resilient encryption, the succinct-key \ib/\ab-\whps~can yield the desired leakage resilient \ibe/\abe schemes with the optimal leakage rates in the relative leakage model. Finally, by further improving the prior analysis of the compatible locally computable extractors, we can achieve the optimal leakage rates in the \BRM.

This work proposes a new lattice two-stage sampling technique, generalizing the prior two-stage sampling method of Gentry, Peikert, and Vaikuntanathan (STOC '08). By using our new technique as a key building block, we can significantly improve security and efficiency of the current state of the arts of simulation-based functional encryption. Particularly, our functional encryption achieves $(Q,\poly)$ simulation-based semi-adaptive security that allows arbitrary pre- and post-challenge key queries, and has succinct ciphertexts with only an additive $O(Q)$ overhead. %This significantly improves the current research frontier of simulation-based functional encryption. Additionally, our two-stage sampling technique can derive new feasibilities of indistinguishability-based adaptively-secure $\IB$-$\FE$ for inner products and semi-adaptively-secure $\AB$-$\FE$ for inner products, breaking several technical limitations of the recent work by Abdalla, Catalano, Gay, and Ursu (Asiacrypt '20).

In this work, we first present general methods to construct information rate-1 PKE that is $\KDM^{(n)}$-secure with respect to \emph{block-affine} functions for any unbounded polynomial $n$. To achieve this, we propose a new notion of extractor that satisfies \emph{reusability}, \emph{homomorphic}, and \emph{security against correlated-source attacks}, and show how to use this extractor to improve the information rate of the \KDM-secure PKE of Brakerski et al.~(Eurocrypt 18). Then, we show how to amplify \KDM~security from block-affine function class into general bounded size circuits via a variant of the technique of Applebaum (Eurocrypt 11), achieving better efficiency. Furthermore, we show how to generalize these approaches to the IBE setting. Additionally, our PKE and IBE schemes are also leakage resilient, with leakage rates $1-o(1)$ against a slightly smaller yet still general class -- block leakage functions. We can instantiate the required building blocks from $\LWE$ or $\DDH$.

Achieving tight security is a fundamental task in cryptography. While one of the most important purposes of this task is to improve the overall efficiency of a construction (by allowing smaller security parameters), many current lattice-based instantiations do not completely achieve the goal. Particularly, a super-polynomial modulus seems to be necessary in all prior work for (almost) tight schemes that allow the adversary to conduct queries, such as PRF, IBE, and Signatures. As the super-polynomial modulus would affect the noise-to-modulus ratio and thus increase the parameters, this might cancel out the advantages (in efficiency) brought from the tighter analysis. To determine the full power of tight security/analysis in lattices, it is necessary to determine whether the super-polynomial modulus restriction is inherent. In this work, we remove the super-polynomial modulus restriction for many important primitives – PRF, IBE, All-but-many Lossy Trapdoor Functions, and Signatures. The crux relies on an improvement over the framework of Boyen and Li (Asiacrypt 16), and an almost tight reduction from LWE to LWR, which improves prior work by Alwen et al. (Crypto 13), Bogdanov et al. (TCC 16), and Bai et al. (Asiacrypt 15). By combining these two advances, we are able to derive these almost tight schemes under LWE with a polynomial modulus.