In this paper, we study the problem of distributed multi-user secret sharing, including a trusted master node, N ∈ N storage nodes, and K users, where each user has access to the contents of a subset of storage nodes. Each user has an independent secret message with certain rate, defined as the size of the message normalized by the size of a storage node. Having access to the secret messages, the trusted master node places encoded shares in the storage nodes, such that (i) each user can recover its own message from the content of the storage nodes that it has access to, (ii) each user cannot gain any information about the message of any other user. We characterize the capacity region of the distributed multi-user secret sharing, defined as the set of all achievable rate tuples, subject to the correctness and privacy constraints. In the achievable scheme, for each user, the master node forms a polynomial with the degree equal to the number of its accessible storage nodes minus one, where the value of this polynomial at certain points are stored as the encoded shares. The message of that user is embedded in some of the coefficients of the polynomial. The remaining coefficients are determined such that the content of each storage node serves as the encoded shares for all users that have access to that storage node.

In order to investigate the effect of nano iron chelate and compare to EDDHSA chelate on saffron, a factorial experiment with six treatments and three replications was conducted in saffron research farm of Shahed University in crop year of 2013-2014. Treatments included iron fertilizer at two levels of application of nano iron chelate fertilizer and iron chelate fertilizer with base of EDDHSA and the second agent was amount of fertilizer at three levels of 0,5 and 10 kg haG 1 . Number of flowers, flowers performance, yield of wet and dry stigmas, amount of chlorophyll a and b, total chlorophyll, leaf area index, yield of dry leaf, concentration of leaf iron and total iron were investigated in this study. Qualitative features of Saffron including secondary metabolites of crocin (color agent), picrocrocin (taste agent) and safranal (scent) were measured. The results showed that all traits under study except picrocrocin, safranal, crocin, chlorophyll b, total chlorophyll and leaf area were affected by type, amount or interaction of type in amount of fertilizer. With increase of iron nano fertilizer content by 10 kg of flower number and yield of wet flower increased compared to the control. But application of 5 kg nano chelate led to increased yield of dry stigma, yield of dry leaf, concentration of leaf iron and total iron, compared to control. The overall results showed that a nano-based iron fertilizer is more effective than micro and 5 kg application of this fertilizer is superior to 10 kg.

In this work, we investigate the problem of multiuser linearly separable function computation, where N servers help compute the desired functions (jobs) of K users. In this setting each desired function can be written as a linear combination of up to L (generally non-linear) sub-functions. Each server computes some of the sub-tasks, and communicates a linear combination of its computed outputs (files) in a singleshot to some of the users, then each user linearly combines its received data in order to recover its desired function. We explore the range of the optimal computation cost via establishing a novel relationship between our problem, syndrome decoding and covering codes. The work reveals that in the limit of large N , the optimal computation cost -in the form of the maximum fraction of all servers that must compute any subfunction -is lower bounded as γ ≥ H −1 q ( log q (L) N), for any fixed log q (L)/N . The result reveals the role of the computational rate log q (L)/N , which cannot exceed what one might call the computational capacity Hq(γ) of the system.

<p>In this work, we explore the problem of multi-user linearly-separable distributed computation, where N servers help compute the desired functions (jobs) of K users, and where each desired function can be written as a linear combination of up to L (generally non-linear) subtasks (or sub-functions). Each server computes some of the subtasks, communicates a function of its computed outputs to some of the users, and then each user collects its received data to recover its desired function. We explore the computation and communication relationship between how many servers compute each subtask vs. how much data each user receives.<br> For a matrix F representing the linearly-separable form of the set of requested functions, our problem becomes equivalent to the open problem of sparse matrix factorization F=DE over finite fields, where a sparse decoding matrix D and encoding matrix E imply reduced communication and computation costs respectively. This paper establishes a novel relationship between our distributed computing problem, matrix factorization, syndrome decoding and covering codes. To reduce the computation cost, the above D is drawn from covering codes or from a here-introduced class of so-called `partial covering' codes, whose study here yields computation cost results that we present. </p>

No abstract