We initiate a systematic study of information-theoretic cryptography with weak privacy, only requiring that the adversary cannot rule out any possible secret. For a parameter 00 . We obtain the following main results.
We revisit multi-server Private Information Retrieval (PIR), where the client interacts with S non-colluding servers. Ideally, we want a scalable family of multi-server PIR schemes where all the performance metrics of the scheme decrease as S increases. However, no prior work achieved scalability under any setting, and any hardness assumption. In this paper we construct new multi-server, information-theoretically secure scalable PIR schemes for three natural settings. First, we give a construction where all the performance metrics scale at equal rate. Second, we give a scalable construction that minimizes the per-query bandwidth. Third, we give a scalable construction that minimizes the per-query online bottleneck cost (the maximum of the bandwidth and computation). For the first two settings, our constructions are doubly efficient with only a super-constant number of servers. In comparison, the best known prior works in the information-theoretic setting required super-logarithmically many servers to achieve the doubly efficient notion. Our techniques for achieving scalable PIR also enable us to advance the state of the art in the polynomial space setting. In this setting, we show how to improve the space consumption of prior works by a polynomial factor while preserving all other metrics. Further, we show a new balancing technique that allows us to further minimize the bandwidth per query by trading off the computation and server space, thus enabling a more smooth tradeoff between the metrics and generalizing the design space.
We develop a toolbox for modular construction and analysis of succinct, non-interactive commitments and vector commitments in the random oracle model, while guaranteeing universally composable security. To demonstrate its power, we use the toolbox to construct and analyze a modular variant of the Kilian-Micali ZK-SNARK. Along the way we also propose a new UC formulation of a global random oracle, that avoids a weakness in existing formulations and also enables expressing more nuanced, session-specific abstractions. We hope that this toolbox will be useful for building secure applications in settings where both succinctness and non-interactivity are key.
In its decisional form, the module-Lattice Isomorphism Problem (decision module-LIP) has received first attention in [12]. The authors gave a polynomial-time algorithm to distinguish spinor genera within the genus of a quadratic binary 𝒪_F -lattice, assuming that 𝒪_F is a principal ideal domain. However, this algorithm would not impact cryptographic schemes based on decision module-LIP for lattices such as those employed in HAWK [5], i.e., for binary 𝒪_K -lattices equipped with a Hermitian form (with K a cyclotomic number field). Motivated by HAWK’s framework, we investigate a concept that serves as an analogue of the spinor genus for Hermitian lattices, called special genus. This notion was studied by Shimura [20] who provided a complete set of invariants for describing special genera. Building on this result, we propose an algorithm to determine whether two Hermitian lattices belong to the same special genus. Specifically for HAWK’s lattice and siblings, our algorithm runs in classical polynomial-time. Nevertheless we provide numerical evidence suggesting that the ability to distinguish special genera does not, in practice, constitute a significative advantage for solving decision module-LIP.
We give a new approach for constructing statistical ZAP arguments (a two-message public-coin statistically witness indistinguishable argument) from quasi-polynomial hardness of the learning with errors (LWE) assumption with a polynomial modulus-to-noise ratio. Previously, all ZAP arguments from lattice-based assumptions relied on correlation-intractable hash functions. In this work, we present the first construction of a ZAP from LWE via the classic hidden-bits paradigm. Our construction matches previous lattice-based schemes by being public-coin and satisfying statistical witness indistinguishability.
