Dynamic multi-key FHE in symmetric key setting from LWE without using common reference matrix

We propose a dynamic (leveled) multi-key Fully Homomorphic Encryption (FHE) scheme under the learning with errors (LWE) assumption. Multi-key FHE in a dynamic setting is the most versatile variant of FHE which requires no information about the participating parties prior to key generation. In addition, it also introduces inputs from new parties into the computation. Designing dynamic multi-key FHE without blowing up the storage and ciphertext size of homomorphic computation is a challenging task. Unlike the existing LWE based multi-key FHE, our design does not use any common reference matrix in common parameter which seems to strengthen the power of using MFHE to allow users to generate their own keys independently. The ciphertext length and public key size of our construction are comparable to the existing similar works. We use only one component in the public extension key, resulting efficient extension of the ciphertexts under additional secret keys. The ciphertext in our design is generated using the FHE scheme of Gentry, Sahai, and Waters with no additional components, allowing efficient homomorphic evaluation. Our scheme is proven to be secure under standard LWE assumption without using circular security. More interestingly, our multi-key FHE is dynamic while maintaining ciphertext compactness.