Threshold cryptosystems

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!We first overview the research in the field and then discuss a threshold decryption/signature scheme which is as secure as RSA. We conclude by giving a list of open problems.