This paper is to study the following generalized Abel's integral equationand its variant in the distributional (Schwartz) sense based on fractional calculus of distributions. We obtain a number of interesting and new results which are not achievable in the classical sense.
Two parties with private data sets can find shared elements using a Private Set Intersection (PSI) protocol without revealing any information beyond the intersection. Circuit PSI protocols privately compute an arbitrary function of the intersectionsuch as its cardinality, and are often employed in an unbalanced setting where one party has more data than the other. Existing protocols are either computationally inefficient or require extensive server-client communication on the order of the larger set. We introduce Practically Efficient PSI or PEPSI, a non-interactive solution where only the client sends its encrypted data. PEPSI can process an intersection of 1024 client items with a million server items in under a second, using less than 5 MB of communication. Our work is over 4 orders of magnitude faster than an existing non-interactive circuit PSI protocol and requires only 10% of the communication. It is also up to 20 times faster than the work of Ion et al., which computes a limited set of functions and has communication costs proportional to the larger set. Our work is the first to demonstrate that non-interactive circuit PSI can be practically applied in an unbalanced setting.
A repairable threshold scheme (which we abbreviate to RTS ) is a (τ, n)-threshold scheme in which a subset of players can "repair" another player's share in the event that their share has been lost or corrupted. This will take place without the participation of the dealer who set up the scheme. The repairing protocol should not compromise the (unconditional) security of the threshold scheme. Combinatorial repairable threshold schemes (or combinatorial RTS ) were recently introduced by Stinson and Wei [8]. In these schemes, "multiple shares" are distributed to each player, as defined by a suitable combinatorial design called the distribution design. In this paper, we study the reliability of these combinatorial repairable threshold schemes in a setting where players may not be available to take part in a repair of a given player's share. Using techniques from network reliability theory, we consider the probability of existence of an available repair set, as well as the expected number of available repair sets, for various types of distribution designs. * D.R. Stinson's Research is supported by NSERC discovery grant RGPIN-03882.Corollary 3.4. A 3-(v, k, 1)-design can be used as a distribution design to produce an RTS with threshold τ if k ≥ τ (τ − 1) + 1.Remark 3.5. In order to obtain τ = 3, we require k ≥ 7 in Corollary 3.4; to obtain τ = 4, we require k ≥ 13, etc. ReliabilityIn our analysis, to compute the reliability metrics for repair sets, we employ the use of cutsets from network reliability theory (see Colbourn [2] for basic results and terminology relating to network reliability). When using BIBDs as distribution designs, we were able to easily compute reliability formulas in Section 2 without the use of this methodology because the sets C j were disjoint. However, it is advantageous to use cutsets to analyze the reliability of the RTS constructed using distribution designs with t ≥ 3.In this section, for brevity, we will conflate the notion of players and blocks and express all our arguments in terms of blocks of the distribution design (X, B).Definition 3.6. A cutset for a block B is a minimal subset of blocks B ′ such that a repair is not possible if all the blocks in B ′ are not available. A cutset fails if every block in the cutset is not available.Lemma 3.7. Let B = {x 1 , . . . , x k } be a block in the distribution design. Then the sets C j , for 1 ≤ j ≤ k, are the cutsets. Example 3.8. Here are the blocks in an 3-(8, 4, 1)-design: A 1 = {1, 2, 3, 4} A 2 = {5, 6, 7, 8} B 1 = {1, 2, 5, 6} B 2 = {1, 2, 7, 8} B 3 = {1, 3, 5, 7} B 4 = {1, 3, 6, 8} B 5 = {1, 4, 5, 8} B 6 = {1, 4, 6, 7} B 7 = {3, 4, 7, 8} B 8 = {3, 4, 5, 6} B 9 = {2, 4, 6, 8} B 10 = {2, 4, 5, 7} B 11 = {2, 3, 6, 7} B 12 = {2, 3, 5, 8} Suppose A 1 wants to repair their share. Then, the relevant cutsets are
Secret sharing schemes are desirable across a variety of real-world settings due to the security and privacy properties they can provide, such as availability and separation of privilege. However, transitioning secret sharing schemes from theoretical research to practical use must account for gaps in achieving these properties that arise due to the realities of concrete implementations, threat models, and use cases. We present a formalization and analysis, using Ellison’s notion of ceremonies, that demonstrates how simple variations in use cases of secret sharing schemes result in the potential loss of some security properties, a result that cannot be derived from the analysis of the underlying cryptographic protocol alone. Our framework accounts for such variations in the design and analysis of secret sharing implementations by presenting a more detailed user-focused process and defining previously overlooked assumptions about user roles and actions within the scheme to support analysis when designing such ceremonies. We identify existing mechanisms that, when applied to an appropriate implementation, close the security gaps we identified. We present our implementation including these mechanisms and a corresponding security assessment using our framework.
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