Foundations of Homomorphic Secret Sharing

Homomorphic secret sharing (HSS) is the secret sharing analogue of homomorphic encryption. An HSS scheme supports a local evaluation of functions on shares of one or more secret inputs, such that the resulting shares of the output are short. Some applications require the stronger notion of additive HSS, where the shares of the output add up to the output over some finite Abelian group. While some strong positive results for HSS are known under specific cryptographic assumptions, many natural questions remain open. We initiate a systematic study of HSS, making the following contributions. - A definitional framework. We present a general framework for defining HSS schemes that unifies and extends several previous notions from the literature, and cast known results within this framework. - Limitations. We establish limitations on information-theoretic multi-input HSS with short output shares via a relation with communication complexity. We also show that additive HSS for non-trivial functions, even the AND of two input bits, implies non-interactive key exchange, and is therefore unlikely to be implied by public-key encryption or even oblivious transfer. - Applications. We present two types of applications of HSS. First, we construct 2-round protocols for secure multiparty computation from a simple constant-size instance of HSS. As a corollary, we obtain 2-round protocols with attractive asymptotic efficiency features under the Decision Diffie Hellman (DDH) assumption. Second, we use HSS to obtain nearly optimal worst-case to average-case reductions in P. This in turn has applications to fine-grained average-case hardness and verifiable computation

Ideal homogeneous access structures constructed from graphs

Starting from a new relation between graphs and secret sharing schemes introduced by Xiao, Liu and Zhang, we show a method to construct more general ideal homogeneous access structures. The method has some advantages: it efficiently gives an ideal homogeneous access structure for the desired rank, and some conditions can be imposed (such as forbidden or necessary subsets of players), even if the exact composition of the resulting access structure cannot be fully controlled. The number of homogeneous access structures that can be constructed in this way is quite limited; for example, we show that (t, l)-threshold access structures can be constructed from a graph only when t = 1, t = l - 1 or t = l.Peer ReviewedPostprint (published version

Secrecy without Perfect Randomness: Cryptography with (Bounded) Weak Sources

Cryptographic protocols are commonly designed and their security proven under the assumption that the protocol parties have access to perfect (uniform) randomness. Physical randomness sources deployed in practical implementations of these protocols often fall short in meeting this assumption, but instead provide only a steady stream of bits with certain high entropy. Trying to ground cryptographic protocols on such imperfect, weaker sources of randomness has thus far mostly given rise to a multitude of impossibility results, including the impossibility to construct provably secure encryption, commitments, secret sharing, and zero-knowledge proofs based solely on a weak source. More generally, indistinguishability-based properties break down for such weak sources. In this paper, we show that the loss of security induced by using a weak source can be meaningfully quantified if the source is bounded, e.g., for the well-studied Santha-Vazirna (SV) sources. The quantification relies on a novel relaxation of indistinguishability by a quantitative parameter. We call the resulting notion differential indistinguishability in order to reflect its structural similarity to differential privacy. More concretely, we prove that indistinguishability with uniform randomness implies differential indistinguishability with weak randomness. We show that if the amount of weak randomness is limited (e.g., by using it only to seed a PRG), all cryptographic primitives and protocols still achieve differential indistinguishability

Access Structure Hiding Secret Sharing from Novel Set Systems and Vector Families

Secret sharing provides a means to distribute shares of a secret such that any authorized subset of shares, specified by an access structure, can be pooled together to recompute the secret. The standard secret sharing model requires public access structures, which violates privacy and facilitates the adversary by revealing high-value targets. In this paper, we address this shortcoming by introducing \emph{hidden access structures}, which remain secret until some authorized subset of parties collaborate. The central piece of this work is the construction of a set-system $\mathcal{H}$ with strictly greater than $\exp\left(c \dfrac{1.5 (\log h)^2}{\log \log h}\right)$ subsets of a set of $h$ elements. Our set-system $\mathcal{H}$ is defined over $\mathbb{Z}_m$, where $m$ is a non-prime-power, such that the size of each set in $\mathcal{H}$ is divisible by $m$ but the sizes of their pairwise intersections are not divisible by $m$, unless one set is a subset of another. We derive a vector family $\mathcal{V}$ from $\mathcal{H}$ such that superset-subset relationships in $\mathcal{H}$ are represented by inner products in $\mathcal{V}$. We use $\mathcal{V}$ to "encode" the access structures and thereby develop the first \emph{access structure hiding} secret sharing scheme. For a setting with $\ell$ parties, our scheme supports $2^{2^{\ell/2 - O(\log \ell) + 1}}$ out of the $2^{2^{\ell - O(\log \ell)}}$ total monotone access structures, and its maximum share size for any access structures is $(1+ o(1)) \dfrac{2^{\ell+1}}{\sqrt{\pi \ell/2}}$. The scheme assumes semi-honest polynomial-time parties, and its security relies on the Generalized Diffie-Hellman assumption.Comment: This is the full version of the paper that appears in D. Kim et al. (Eds.): COCOON 2020 (The 26th International Computing and Combinatorics Conference), LNCS 12273, pp. 246-261. This version contains tighter bounds on the maximum share size, and the total number of access structures supporte

Secure multiparty quantum computation for summation and multiplication

As a fundamental primitive, Secure Multiparty Summation and Multiplication can be used to build complex secure protocols for other multiparty computations, specially, numerical computations. However, there is still lack of systematical and efficient quantum methods to compute Secure Multiparty Summation and Multiplication. In this paper, we present a novel and efficient quantum approach to securely compute the summation and multiplication of multiparty private inputs, respectively. Compared to classical solutions, our proposed approach can ensure the unconditional security and the perfect privacy protection based on the physical principle of quantum mechanics

Encrypted control for networked systems -- An illustrative introduction and current challenges

Cloud computing and distributed computing are becoming ubiquitous in many modern control systems such as smart grids, building automation, robot swarms or intelligent transportation systems. Compared to "isolated" control systems, the advantages of cloud-based and distributed control systems are, in particular, resource pooling and outsourcing, rapid scalability, and high performance. However, these capabilities do not come without risks. In fact, the involved communication and processing of sensitive data via public networks and on third-party platforms promote, among other cyberthreats, eavesdropping and manipulation of data. Encrypted control addresses this security gap and provides confidentiality of the processed data in the entire control loop. This paper presents a tutorial-style introduction to this young but emerging field in the framework of secure control for networked dynamical systems.Comment: The paper is a preprint of an accepted paper in the IEEE Control Systems Magazin