# Relation between Threshold Cryptosystem and Secure Multiparty Computation ?

Is there any relation between secure multiparty computation and threshold cryptosystem ?

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Yes and no.

A threshold cryptosystem means the decryption key can be split into $n$ shares such that only $t\leq n$ are required to recover it. That property in isolation is not useful for multiparty computation.

However when you combine a threshold cryptosystem with one that is at least partially homomorphic (meaning you can do some operation, like addition or multiplication, under encryption), then the two properties combined can make a useful basis for multiparty computation.

The model is that each party encrypts their inputs, you use the homomorphic property to do some computation on the ciphertexts while they are still encrypted, and the threshold property enforces that only the final value is decrypted assuming less than $t$ dishonest key share holders. Without the threshold (or at least a distributed $n$-out-of-$n$ key), the keyholder could simply decrypt the inputs and learn everyone's value.

The most famous paper on this is "Multiparty Computation from Threshold Homomorphic Encryption." This approach is used in lots of papers. A different approach in the same model is "Mix and Match: Secure Function Evaluation via Ciphertexts." It is less used (I think maybe a bit overlooked) but you can do interesting things with it.

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