What is homomorphic encryption?
Homomorphic Encryption (HE) is a class of encryption schemes that offers the ability to compute on data which is encrypted. HE therefore enables operations on the data without decrypting and exposing it, preventing the leakage of private information.  Without the proper decryption key, the original data can’t be accessed. Data can then be processed at an outsourced third party without that third party being trusted to preserve privacy or security itself.
Challenges with homomorphic encryption
This level of security has generated interest by governments and large corporations given the potential for adversarial quantum computing. However, it requires substantial computing resources, making it impractically slow even for basic computations. Additionally, a typical homomorphic encryption scheme only supports computations between data encrypted by the same key.
Therefore, when there are multiple data owners, it assumes a trusted third party who possesses a key released to each party for encryption. As a result, the key owner could decrypt and potentially infringe on each of the data owners' privacy. 
Multi-Party / Multi-Key homomorphic encryption
These specific challenges led to the development of Multi-Party Homomorphic Encryption (MPHE) and Multi-Key Homomorphic Encryption (MKHE). MPHE tends to be more efficient but requires that the computing parties be fixed throughout the operation. Meanwhile, MKHE allows you to add new parties anytime during the computation; however, it typically doesn't scale well as the number of parties increases. 
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