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The study of probabilistic secret sharing schemes using arbitrary probability spaces and possibly infinite number of participants lets the authors' investigate abstract properties of such schemes. It highlights important properties, explains why certain definitions work better than others, connects this topic to other branches of mathematics, and might yield new design paradigms. A probabilistic secret sharing scheme is a joint probability distribution of the shares and the secret together with a collection of secret recovery functions for qualified subsets. The scheme is measurable if the recovery functions are measurable. Depending on how much information an unqualified subset might have, they define four scheme types: perfect, almost perfect, ramp, and almost ramp.
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