Optimal non-perfect uniform secret sharing schemes

A secret sharing scheme is non-perfect if some subsets of participants that cannot recover the secret value have partial information about it. The information ratio of a secret sharing scheme is the ratio between the maximum length of the shares and the length of the secret. This work is dedicated to the search of bounds on the information ratio of non-perfect secret sharing schemes. To this end, we extend the known connections between polymatroids and perfect secret sharing schemes to the non-perfect case. In order to study non-perfect secret sharing schemes in all generality, we describe their structure through their access function, a real function that measures the amount of information that every subset of participants obtains about the secret value. We prove that there exists a secret sharing scheme for every access function. Uniform access functions, that is, the ones whose values depend only on the number of participants, generalize the threshold access structures. Our main result is to determine the optimal information ratio of the uniform access functions. Moreover, we present a construction of linear secret sharing schemes with optimal information ratio for the rational uniform access functions.Peer ReviewedPostprint (author's final draft

Mixed-mode oscillations and interspike interval statistics in the stochastic FitzHugh-Nagumo model

We study the stochastic FitzHugh-Nagumo equations, modelling the dynamics of neuronal action potentials, in parameter regimes characterised by mixed-mode oscillations. The interspike time interval is related to the random number of small-amplitude oscillations separating consecutive spikes. We prove that this number has an asymptotically geometric distribution, whose parameter is related to the principal eigenvalue of a substochastic Markov chain. We provide rigorous bounds on this eigenvalue in the small-noise regime, and derive an approximation of its dependence on the system's parameters for a large range of noise intensities. This yields a precise description of the probability distribution of observed mixed-mode patterns and interspike intervals.Comment: 36 page

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

Conditional Disclosure of Secrets: Amplification, Closure, Amortization, Lower-bounds, and Separations

In the \emph{conditional disclosure of secrets} problem (Gertner et al., J. Comput. Syst. Sci., 2000) Alice and Bob, who hold inputs $x$ and $y$ respectively, wish to release a common secret $s$ to Carol (who knows both $x$ and $y$) if only if the input $(x,y)$ satisfies some predefined predicate $f$. Alice and Bob are allowed to send a single message to Carol which may depend on their inputs and some joint randomness and the goal is to minimize the communication complexity while providing information-theoretic security. Following Gay, Kerenidis, and Wee (Crypto 2015), we study the communication complexity of CDS protocols and derive the following positive and negative results. 1. *Closure* A CDS for $f$ can be turned into a CDS for its complement $\bar{f}$ with only a minor blow-up in complexity. More generally, for a (possibly non-monotone) predicate $h$, we obtain a CDS for $h(f_1,\ldots,f_m)$ whose cost is essentially linear in the formula size of $h$ and polynomial in the CDS complexity of $f_i$. 2. *Amplification* It is possible to reduce the privacy and correctness error of a CDS from constant to $2^{-k}$ with a multiplicative overhead of $O(k)$. Moreover, this overhead can be amortized over $k$-bit secrets. 3. *Amortization* Every predicate $f$ over $n$-bit inputs admits a CDS for multi-bit secrets whose amortized communication complexity per secret bit grows linearly with the input length $n$ for sufficiently long secrets. In contrast, the best known upper-bound for single-bit secrets is exponential in $n$. 4. *Lower-bounds* There exists a (non-explicit) predicate $f$ over $n$-bit inputs for which any perfect (single-bit) CDS requires communication of at least $\Omega(n)$. This is an exponential improvement over the previously known $\Omega(\log n)$ lower-bound. 5. *Separations* There exists an (explicit) predicate whose CDS complexity is exponentially smaller than its randomized communication complexity. This matches a lower-bound of Gay et. al., and, combined with another result of theirs, yields an exponential separation between the communication complexity of linear CDS and non-linear CDS. This is the first provable gap between the communication complexity of linear CDS (which captures most known protocols) and non-linear CDS

On the Power of Amortization in Secret Sharing: dd-Uniform Secret Sharing and CDS with Constant Information Rate

Consider the following secret-sharing problem. Your goal is to distribute a long file $s$ between $n$ servers such that $(d-1)$-subsets cannot recover the file, $(d+1)$-subsets can recover the file, and $d$-subsets should be able to recover $s$ if and only if they appear in some predefined list $L$. How small can the information ratio (i.e., the number of bits stored on a server per each bit of the secret) be? We initiate the study of such $d$-uniform access structures, and view them as a useful scaled-down version of general access structures. Our main result shows that, for constant $d$, any $d$-uniform access structure admits a secret sharing scheme with a *constant* asymptotic information ratio of $c_d$ that does not grow with the number of servers $n$. This result is based on a new construction of $d$-party Conditional Disclosure of Secrets (Gertner et al., JCSS \u2700) for arbitrary predicates over $n$-size domain in which each party communicates at most four bits per secret bit. In both settings, previous results achieved non-constant information ratio which grows asymptotically with $n$ even for the simpler (and widely studied) special case of $d=2$. Moreover, our results provide a unique example for a natural class of access structures $F$ that can be realized with information rate smaller than its bit-representation length $\log |F|$ (i.e., $\Omega( d \log n)$ for $d$-uniform access structures) showing that amortization can beat the representation size barrier. Our main result applies to exponentially long secrets, and so it should be mainly viewed as a barrier against amortizable lower-bound techniques. We also show that in some natural simple cases (e.g., low-degree predicates), amortization kicks in even for quasi-polynomially long secrets. Finally, we prove some limited lower-bounds, point out some limitations of existing lower-bound techniques, and describe some applications to the setting of private simultaneous messages

The Communication Complexity of Private Simultaneous Messages, Revisited

Private Simultaneous Message (PSM) protocols were introduced by Feige, Kilian and Naor (STOC \u2794) as a minimal non-interactive model for information-theoretic three-party secure computation. While it is known that every function $f:\{0,1\}^k\times \{0,1\}^k \rightarrow \{0,1\}$ admits a PSM protocol with exponential communication of $2^{k/2}$ (Beimel et al., TCC \u2714), the best known (non-explicit) lower-bound is $3k-O(1)$ bits. To prove this lower-bound, FKN identified a set of simple requirements, showed that any function that satisfies these requirements is subject to the $3k-O(1)$ lower-bound, and proved that a random function is likely to satisfy the requirements. We revisit the FKN lower-bound and prove the following results: (Counterexample) We construct a function that satisfies the FKN requirements but has a PSM protocol with communication of $2k+O(1)$ bits, revealing a gap in the FKN proof. (PSM lower-bounds) We show that, by imposing additional requirements, the FKN argument can be fixed leading to a $3k-O(\log k)$ lower-bound for a random function. We also get a similar lower-bound for a function that can be computed by a polynomial-size circuit (or even polynomial-time Turing machine under standard complexity-theoretic assumptions). This yields the first non-trivial lower-bound for an explicit Boolean function partially resolving an open problem of Data, Prabhakaran and Prabhakaran (Crypto \u2714, IEEE Information Theory \u2716). We further extend these results to the setting of imperfect PSM protocols which may have small correctness or privacy error. (CDS lower-bounds) We show that the original FKN argument applies (as is) to some weak form of PSM protocols which are strongly related to the setting of Conditional Disclosure of Secrets (CDS). This connection yields a simple combinatorial criterion for establishing linear $\Omega(k)$-bit CDS lower-bounds. As a corollary, we settle the complexity of the Inner Product predicate resolving an open problem of Gay, Kerenidis, and Wee (Crypto \u2715)

Creating a multi-center rare disease consortium - the Consortium of Eosinophilic Gastrointestinal Disease Researchers (CEGIR).

 Eosinophilic gastrointestinal disorders (EGIDs) affect various segments of the gastrointestinal tract. Since these disorders are rare, collaboration is essential to enroll subjects in clinical studies and study the broader population. The Rare Diseases Clinical Research Network (RDCRN), a program of the National Center for Advancing Translational Sciences (NCATS), funded the Consortium of Eosinophilic Gastrointestinal Disease Researchers (CEGIR) in 2014 to advance the field of EGIDs. CEGIR facilitates collaboration among various centers, subspecialties, patients, professional organizations and patient-advocacy groups and includes 14 clinical sites. It has successfully initiated two large multi-center clinical studies looking to refine EGID diagnoses and management. Several pilot studies are underway that focus on various aspects of EGIDs including novel therapeutic interventions, diagnostic and monitoring methods, and the role of the microbiome in pathogenesis. CEGIR currently nurtures five physician-scholars through a career training development program and has published more than 40 manuscripts since its inception. This review focuses on CEGIR's operating model and progress and how it facilitates a framework for exchange of ideas and stimulates research and innovation. This consortium provides a model for progress on other potential clinical areas

A review of combined advanced oxidation technologies for the removal of organic pollutants from water

Water pollution through natural and anthropogenic activities has become a global problem causing short-and long-term impact on human and ecosystems. Substantial quantity of individual or mixtures of organic pollutants enter the surface water via point and nonpoint sources and thus affect the quality of freshwater. These pollutants are known to be toxic and difficult to remove by mere biological treatment. To date, most researches on the removal of organic pollutants from wastewater were based on the exploitation of individual treatment process. This single-treatment technology has inherent challenges and shortcomings with respect to efficiency and economics. Thus, application of two advanced treatment technologies characterized with high efficiency with respect to removal of primary and disinfection by-products in wastewater is desirable. This review article focuses on the application of integrated technologies such as electrohydraulic discharge with heterogeneous photocatalysts or sonophotocatalysis to remove target pollutants. The information gathered from more than 100 published articles, mostly laboratories studies, shows that process integration effectively remove and degrade recalcitrant toxic contaminants in wastewater better than single-technology processing. This review recommends an improvement on this technology (integrated electrohydraulic discharge with heterogeneous photocatalysts) viz-a-vis cost reduction in order to make it accessible and available in the rural and semi-urban settlement. Further recommendation includes development of an economic model to establish the cost implications of the combined technology. Proper monitoring, enforcement of the existing environmental regulations, and upgrading of current wastewater treatment plants with additional treatment steps such as photocatalysis and ozonation will greatly assist in the removal of environmental toxicants