Abstract: Determining data availability is disclosed, including: performing a data availability challenge with respect to a claimer node to determine whether the claimer node stores at least some elements included in a base layer in a digital tree corresponding to a data entity; and publishing a first set of elements associated with the base layer of the digital tree and the data availability challenge. Furthermore, encoding auditing is disclosed, including: obtaining a first set of elements associated with a base layer of a digital tree corresponding to a data entity; and generating an encoding validity determination of the digital tree based at least in part on whether the first set of elements is usable to recover a second set of elements associated with the base layer of the digital tree.

Abstract: Verifying a set of remotely stored data is disclosed, including: requesting a first element corresponding to a first leaf in a base layer of a digital tree, wherein the digital tree corresponds to a set of remotely stored data; receiving the first element corresponding to the first leaf; and receiving a membership proof comprising a set of elements from one or more layers higher in the digital tree than the base layer of the digital tree, wherein the set of elements comprises a path from the first leaf to a root of the digital tree, wherein the set of elements is dependent on a plurality of additional leaves and wherein verification of the first element corresponding to the first leaf increases a level of confidence in an availability of the plurality of additional leaves.

Abstract: Verifying a set of remotely stored data is disclosed, including: requesting a first element corresponding to a first leaf in a base layer of a digital tree, wherein the digital tree corresponds to a set of remotely stored data; receiving the first element corresponding to the first leaf; and receiving a membership proof comprising a set of elements from one or more layers higher in the digital tree than the base layer of the digital tree, wherein the set of elements comprises a path from the first leaf to a root of the digital tree, wherein the set of elements is dependent on a plurality of additional leaves and wherein verification of the first element corresponding to the first leaf increases a level of confidence in an availability of the plurality of additional leaves.

Abstract: A method for calculating a given multivariate polynomial f(Xi) is provided. The multivariate polynomial f(Xi) is to be calculated for every Xi in a large dataset X=(X1, X2, . . . ; XK) in accordance with an S-resilient, A-secure, and T-private scheme where K is an integer enumerating the number of elements in the dataset X, S is the number of stragglers to be tolerated, A is the number of adversaries to be tolerated, and T is the number of colluding workers that to be tolerated. The method is implemented by a computer system that includes a master computing device and a plurality of worker computing devices. The working computing devices can include colluding workers, malicious computing devices, and straggler computing devices.