A conceptual example of a threshold cryptography system

Please notice that the following is not written in academic terminology. It is an unofficial note from one of my research meetings back when I was a student at UVic. I haven't had a chance to research the topic yet, but  I'd like to put it here to keep track ...


Notion: A message that is encrypted by a public key at any entity in the network can only be decrypted by cooperation of a group of entities (no single entity is able to decrypt the message on its own).



Assumptions:

I) Assume that each entity of the group has been given a private key, where the following equation is held:



Notice: The above relationship shows that by knowing the public key, the private keys will still remain unknown (an attacker cannot compute private keys by knowing the public key).

II) Encryption process of the message M:



 
III) In the decryption process, each entity should independently decode the message and send the net result of its computation to a Combiner Unit (CU). The CU will merge all the results (received from independent entities)  and compute (decrypt) the meaningful message M. The local computation process at each entity that holds a private key  is as follows:



It can be shown that no entity is able to decrypt a meaningful message.

IV) Finally, the CU computes the final message by