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Fully Homomorphic Encryption (FHE) provides a simple template for secure computation between two parties (Alice and Bob) where: Alice encrypts her input under her key, Bob homomorphically evaluates the desired function on Alice's ciphertext and his own input, and sends the encrypted output to Alice. Extending this approach to multiple parties raises the problem of which key to encrypt under; if all parties choose a key on their own, then homomorphic evaluation on ciphertexts under different keys will not be possible, and if a single party chooses the key for everyone then corrupting this party will break privacy for all.
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