On publicly verifiable secret sharing schemes

Secret sharing allows a dealer to distribute shares of a secret to a set of parties such that only so-called authorised subsets of these parties can recover the secret, whilst forbidden sets gain at most some restricted amount of information. This idea has been built upon in verifiable secret sharing to allow parties to verify that their shares are valid and will therefore correctly reconstruct the same secret. This can then be further extended to publicly verifiable secret sharing by firstly considering only public channels of communication, hence imposing the need for encryption of the shares, and secondly by requiring that any party be able to verify any other parties shares from the public encryption. In this thesis we work our way up from the original secret sharing scheme by Shamir to examples of various approaches of publicly verifiable schemes. Due to the need for encryption in private communication, different cryptographic methods allow for certain interesting advantages in the schemes. We review some important existing methods and their significant properties of interest, such as being homomorphic or efficiently verifiable. We also consider recent improvements in these schemes and make a contribution by showing that an encryption scheme by Castagnos and Laguillaumie allows for a publicly verifiable secret sharing scheme to have some interesting homomorphic properties. To explore further we look at generalisations to the recently introduced idea of Abelian secret sharing, and we consider some examples of such constructions. Finally we look at some applications of secret sharing schemes, and present our own implementation of Schoenmaker’s scheme in Python, along with a voting system on which it is based

Timed Secret Sharing

Secret sharing has been a promising tool in cryptographic schemes for decades. It allows a dealer to split a secret into some pieces of shares that carry no sensitive information on their own when being treated individually but lead to the original secret when having a sufficient number of them together. Existing schemes lack considering a guaranteed delay prior to secret reconstruction and implicitly assume once the dealer shares the secret, a sufficient number of shareholders will get together and recover the secret at their wish. This, however, may lead to security breaches when a timely reconstruction of the secret matters as the early knowledge of a single revealed share is catastrophic assuming a threshold adversary. This paper presents the notion of timed secret sharing (TSS), providing lower and upper time bounds for secret reconstruction with the use of time-based cryptography. The recent advances in the literature including short-lived proofs [Asiacrypt 2022], enable us to realize an upper time bound shown to be useful in breaking public goods game, an inherent issue in secret sharing-based systems. Moreover, we establish an interesting trade-off between time and fault tolerance in a secret sharing scheme by having dealer gradually release additional shares over time, offering another approach with the same goal. We propose several constructions that offer a range of security properties while maintaining practical efficiency. Our constructions leverage a variety of techniques and state-of-the-art primitives

On Polynomial Secret Sharing Schemes

Nearly all secret sharing schemes studied so far are linear or multi-linear schemes. Although these schemes allow to implement any monotone access structure, the share complexity, $SC$, may be suboptimal -- there are access structures for which the gap between the best known lower bounds and best known multi-linear schemes is exponential. There is growing evidence in the literature, that non-linear schemes can improve share complexity for some access structures, with the work of Beimel and Ishai (CCC \u2701) being among the first to demonstrate it. This motivates further study of non linear schemes. We initiate a systematic study of polynomial secret sharing schemes (PSSS), where shares are (multi-variate) polynomials of secret and randomness vectors $\vec{s},\vec{r}$ respectively over some finite field \F_q. Our main hope is that the algebraic structure of polynomials would help obtain better lower bounds than those known for the general secret sharing. Some of the initial results we prove in this work are as follows. \textbf{On share complexity of polynomial schemes.}\\ First we study degree (at most) 1 in randomness variables $\vec{r}$ (where the degree of secret variables is unlimited). We have shown that for a large subclass of these schemes, there exist equivalent multi-linear schemes with $O(n)$ share complexity overhead. Namely, PSSS where every polynomial misses monomials of exact degree $c\geq 2$ in $\vec{s}$ and 0 in $\vec{r}$, and PSSS where all polynomials miss monomials of exact degree $\geq 1$ in $\vec{s}$ and 1 in $\vec{r}$. This translates the known lower bound of $\Omega(n^{\log(n)})$ for multi linear schemes onto a class of schemes strictly larger than multi linear schemes, to contrast with the best $\Omega(n^2/\log(n))$ bound known for general schemes, with no progress since 94\u27. An observation in the positive direction we make refers to the share complexity (per bit) of multi linear schemes (polynomial schemes of total degree 1). We observe that the scheme by Liu et. al obtaining share complexity $O(2^{0.994n})$ can be transformed into a multi-linear scheme with similar share complexity per bit, for sufficiently long secrets. % For the next natural degree to consider, 2 in $\vec{r}$, we have shown that PSSS where all share polynomials are of exact degree 2 in $\vec{r}$ (without exact degree 1 in $\vec{r}$ monomials) where \F_q has odd characteristic, can implement only trivial access structures where the minterms consist of single parties. Obtaining improved lower bounds for degree-2 in $\vec{r}$ PSSS, and even arbitrary degree-1 in $\vec{r}$ PSSS is left as an interesting open question. \textbf{On the randomness complexity of polynomial schemes.}\\ We prove that for every degree-2 polynomial secret sharing scheme, there exists an equivalent degree-2 scheme with identical share complexity with randomness complexity, $RC$, bounded by $2^{poly(SC)}$. For general PSSS, we obtain a similar bound on $RC$ (preserving $SC$ and \F_q but not degree). So far, bounds on randomness complexity were known only for multi linear schemes, demonstrating that $RC \leq SC$ is always achievable. Our bounds are not nearly as practical as those for multi-linear schemes, and should be viewed as a proof of concept. If a much better bound for some degree bound $d=O(1)$ is obtained, it would lead directly to super-polynomial counting-based lower bounds for degree-$d$ PSSS over constant-sized fields. Another application of low (say, polynomial) randomness complexity is transforming polynomial schemes with polynomial-sized (in $n$) algebraic formulas $C(\vec{s},\vec{r})$ for each share , into a degree-3 scheme with only polynomial blowup in share complexity, using standard randomizing polynomials constructions

Ideal hierarchical secret sharing schemes

Hierarchical secret sharing is among the most natural generalizations of threshold secret sharing, and it has attracted a lot of attention from the invention of secret sharing until nowadays. Several constructions of ideal hierarchical secret sharing schemes have been proposed, but it was not known what access structures admit such a scheme. We solve this problem by providing a natural definition for the family of the hierarchical access structures and, more importantly, by presenting a complete characterization of the ideal hierarchical access structures, that is, the ones admitting an ideal secret sharing scheme. Our characterization deals with the properties of the hierarchically minimal sets of the access structure, which are the minimal qualified sets whose participants are in the lowest possible levels in the hierarchy. By using our characterization, it can be efficiently checked whether any given hierarchical access structure that is defined by its hierarchically minimal sets is ideal. We use the well known connection between ideal secret sharing and matroids and, in particular, the fact that every ideal access structure is a matroid port. In addition, we use recent results on ideal multipartite access structures and the connection between multipartite matroids and integer polymatroids. We prove that every ideal hierarchical access structure is the port of a representable matroid and, more specifically, we prove that every ideal structure in this family admits ideal linear secret sharing schemes over fields of all characteristics. In addition, methods to construct such ideal schemes can be derived from the results in this paper and the aforementioned ones on ideal multipartite secret sharing. Finally, we use our results to find a new proof for the characterization of the ideal weighted threshold access structures that is simpler than the existing one.Peer ReviewedPostprint (author's final draft

Approximate Quantum Error-Correcting Codes and Secret Sharing Schemes

It is a standard result in the theory of quantum error-correcting codes that no code of length n can fix more than n/4 arbitrary errors, regardless of the dimension of the coding and encoded Hilbert spaces. However, this bound only applies to codes which recover the message exactly. Naively, one might expect that correcting errors to very high fidelity would only allow small violations of this bound. This intuition is incorrect: in this paper we describe quantum error-correcting codes capable of correcting up to (n-1)/2 arbitrary errors with fidelity exponentially close to 1, at the price of increasing the size of the registers (i.e., the coding alphabet). This demonstrates a sharp distinction between exact and approximate quantum error correction. The codes have the property that any $t$ components reveal no information about the message, and so they can also be viewed as error-tolerant secret sharing schemes. The construction has several interesting implications for cryptography and quantum information theory. First, it suggests that secret sharing is a better classical analogue to quantum error correction than is classical error correction. Second, it highlights an error in a purported proof that verifiable quantum secret sharing (VQSS) is impossible when the number of cheaters t is n/4. More generally, the construction illustrates a difference between exact and approximate requirements in quantum cryptography and (yet again) the delicacy of security proofs and impossibility results in the quantum model.Comment: 14 pages, no figure

Generic Secure Repair for Distributed Storage

This paper studies the problem of repairing secret sharing schemes, i.e., schemes that encode a message into $n$ shares, assigned to $n$ nodes, so that any $n-r$ nodes can decode the message but any colluding $z$ nodes cannot infer any information about the message. In the event of node failures so that shares held by the failed nodes are lost, the system needs to be repaired by reconstructing and reassigning the lost shares to the failed (or replacement) nodes. This can be achieved trivially by a trustworthy third-party that receives the shares of the available nodes, recompute and reassign the lost shares. The interesting question, studied in the paper, is how to repair without a trustworthy third-party. The main issue that arises is repair security: how to maintain the requirement that any colluding $z$ nodes, including the failed nodes, cannot learn any information about the message, during and after the repair process? We solve this secure repair problem from the perspective of secure multi-party computation. Specifically, we design generic repair schemes that can securely repair any (scalar or vector) linear secret sharing schemes. We prove a lower bound on the repair bandwidth of secure repair schemes and show that the proposed secure repair schemes achieve the optimal repair bandwidth up to a small constant factor when $n$ dominates $z$, or when the secret sharing scheme being repaired has optimal rate. We adopt a formal information-theoretic approach in our analysis and bounds. A main idea in our schemes is to allow a more flexible repair model than the straightforward one-round repair model implicitly assumed by existing secure regenerating codes. Particularly, the proposed secure repair schemes are simple and efficient two-round protocols

On the optimization of bipartite secret sharing schemes

Optimizing the ratio between the maximum length of the shares and the length of the secret value in secret sharing schemes for general access structures is an extremely difficult and long-standing open problem. In this paper, we study it for bipartite access structures, in which the set of participants is divided in two parts, and all participants in each part play an equivalent role. We focus on the search of lower bounds by using a special class of polymatroids that is introduced here, the tripartite ones. We present a method based on linear programming to compute, for every given bipartite access structure, the best lower bound that can be obtained by this combinatorial method. In addition, we obtain some general lower bounds that improve the previously known ones, and we construct optimal secret sharing schemes for a family of bipartite access structures.Peer ReviewedPostprint (author's final draft