International Association
for Cryptologic Research

Despite active research on secret-sharing schemes for arbitrary access structures for more than 35 years, we do not understand their share size -- the best known upper bound for an arbitrary $n$-party access structure is $2^{O(n)}$, while the best known lower bound is $\Omega(n/\log(n))$. Consistent with our knowledge, the share size can be anywhere between these bounds. To better understand this question, one can study specific families of secret-sharing schemes. For example, linear secret-sharing schemes, in which the sharing and reconstruction functions are linear mappings, have been studied in many papers, e.g., it is known that they require shares of size at least $2^{0.5n}$. Secret-sharing schemes in which the sharing and/or reconstruction are computed by low-degree polynomials have been recently studied by Paskin-Cherniavsky and Radune [ITC 2020] and by Beimel, Othman, and Peter [CRYPTO 2021]. It was shown that secret-sharing schemes with sharing and reconstruction computed by polynomials of degree $2$ are more efficient than linear schemes (i.e., schemes in which the sharing and reconstruction are computed by polynomials of degree one). Prior to our work, it was not known if using polynomials of higher degree can reduce the share size. We show that this is indeed the case, i.e., we construct secret-sharing schemes for arbitrary access structures with reconstruction by degree-$d$ polynomials, where as the reconstruction degree $d$ increases, the share size decreases. As a step in our construction, we construct conditional disclosure of secrets (CDS) protocols. For example, we construct 2-server CDS protocols for functions $f:[N]\times [N] \to \{0,1\}$ with reconstruction computed by degree-$d$ polynomials with message size $N^{O(\log \log d/\log d)}$. Combining our results with a lower bound of Beimel et al.~[CRYPTO 2021], we show that increasing the degree of the reconstruction function in CDS protocols provably reduces the message size. To construct our schemes, we define \emph{sparse} matching vectors, show constructions of such vectors, and design CDS protocols and secret-sharing schemes with degree-$d$ reconstruction from sparse matching vectors.

Cache side-channel attacks allow adversaries to learn sensitive information about co-running processes by using only access latency measures and cache contention.This vulnerability has been shown to lead to several microarchitectural attacks. As a promising solution, recent work proposes Randomization-based Protected Caches (RPCs). RPCs randomize cache addresses, changing keys periodically so as to avoid long-term leakage. Unfortunately, recent attacks have called the security of state-of-the-art RPCs into question.In this work, we tackle the problem of formally defining and analyzing the security properties of RPCs. We first give security definitions against access-based cache sidechannel attacks that capture security against known attacks such as Prime+Probe and Evict+Probe. Then, using these definitions, we obtain results that allow to guarantee security by adequately choosing the rekeying period, the key generation algorithm and the cache randomizer, thus providing security proofs for RPCs under certain assumptions.

The share size of general secret-sharing schemes is poorly understood. The gap between the best known upper bound on the total share size per party of $2^{0.64n}$ (Applebaum et al., STOC 2020) and the best known lower bound of $\Omega(n/\log n)$ (Csirmaz, J. of Cryptology 1997) is huge (where $n$ is the number of parties in the scheme). To gain some understanding on this problem, we study the share size of secret-sharing schemes of almost all access structures, i.e., of almost all collections of authorized sets. This is motivated by the fact that in complexity, many times almost all objects are hardest (e.g., most Boolean functions require exponential size circuits). All previous constructions of secret-sharing schemes were for the worst access structures (i.e., all access structures) or for specific families of access structures. We prove upper bounds on the share size for almost all access structures. We combine results on almost all monotone Boolean functions (Korshunov, Probl. Kibern. 1981) and a construction of (Liu and Vaikuntanathan, STOC 2018) and conclude that almost all access structures have a secret-sharing scheme with share size $2^{\tilde{O}(\sqrt{n})}$. We also study graph secret-sharing schemes. In these schemes, the parties are vertices of a graph and a set can reconstruct the secret if and only if it contains an edge. Again, for this family there is a huge gap between the upper bounds -- $O(n/\log n)$ (Erd\"{o}s and Pyber, Discrete Mathematics 1997) -- and the lower bounds -- $\Omega(\log n)$ (van Dijk, Des. Codes Crypto. 1995). We show that for almost all graphs, the share size of each party is $n^{o(1)}$. This result is achieved by using robust 2-server conditional disclosure of secrets protocols, a new primitive introduced and constructed in (Applebaum et al., STOC 2020), and the fact that the size of the maximal independent set in a random graph is small. Finally, using robust conditional disclosure of secrets protocols, we improve the total share size for all very dense graphs.

A secret-sharing scheme allows some authorized sets of parties to reconstruct a secret; the collection of authorized sets is called the access structure. For over 30 years, it was known that any (monotone) collection of authorized sets can be realized by a secret-sharing scheme whose shares are of size $$2^{n-o(n)}$$ and until recently no better scheme was known. In a recent breakthrough, Liu and Vaikuntanathan (STOC 2018) have reduced the share size to $$O(2^{0.994n})$$. Our first contribution is improving the exponent of secret sharing down to 0.892. For the special case of linear secret-sharing schemes, we get an exponent of 0.942 (compared to 0.999 of Liu and Vaikuntanathan).Motivated by the construction of Liu and Vaikuntanathan, we study secret-sharing schemes for uniform access structures. An access structure is k-uniform if all sets of size larger than k are authorized, all sets of size smaller than k are unauthorized, and each set of size k can be either authorized or unauthorized. The construction of Liu and Vaikuntanathan starts from protocols for conditional disclosure of secrets, constructs secret-sharing schemes for uniform access structures from them, and combines these schemes in order to obtain secret-sharing schemes for general access structures. Our second contribution in this paper is constructions of secret-sharing schemes for uniform access structures. We achieve the following results:A secret-sharing scheme for k-uniform access structures for large secrets in which the share size is $$O(k^2)$$ times the size of the secret.A linear secret-sharing scheme for k-uniform access structures for a binary secret in which the share size is $$\tilde{O}(2^{h(k/n)n/2})$$ (where h is the binary entropy function). By counting arguments, this construction is optimal (up to polynomial factors).A secret-sharing scheme for k-uniform access structures for a binary secret in which the share size is $$2^{\tilde{O}(\sqrt{k \log n})}$$. Our third contribution is a construction of ad-hoc PSM protocols, i.e., PSM protocols in which only a subset of the parties will compute a function on their inputs. This result is based on ideas we used in the construction of secret-sharing schemes for k-uniform access structures for a binary secret.

Multipartite secret sharing schemes are those having a multipartite access structure, in which the set of participants is divided into several parts and all participants in the same part play an equivalent role. In this work, the characterization of ideal multipartite access structures is studied with all generality. Our results are based on the well-known connections between ideal secret sharing schemes and matroids and on the introduction of a new combinatorial tool in secret sharing, integer polymatroids .Our results can be summarized as follows. First, we present a characterization of multipartite matroid ports in terms of integer polymatroids. As a consequence of this characterization, a necessary condition for a multipartite access structure to be ideal is obtained. Second, we use representations of integer polymatroids by collections of vector subspaces to characterize the representable multipartite matroids. In this way we obtain a sufficient condition for a multipartite access structure to be ideal, and also a unified framework to study the open problems about the efficiency of the constructions of ideal multipartite secret sharing schemes. Finally, we apply our general results to obtain a complete characterization of ideal tripartite access structures, which was until now an open problem.