Often the power to use a cryptosystem has to be shared. In threshold schemes, t-out-of-l have the power to regenerate a secret key (while less than t have not). However threshold schemes cannot be used directly in many applications, such as threshold signatures in which t-out-of-l have to co-sign a message. A normal threshold scheme would require the shareholders to send their shares to a trusted person who would sign for them. But the use of such a trusted person violates the main point of threshold signatures!
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