Catalano and Fiore propose a scheme to transform a linearly-homomorphic
encryption into a homomorphic encryption scheme capable of evaluating quadratic
computations on ciphertexts. Their scheme is based on the linearly-homomorphic
encryption (such as Goldwasser-Micali, Paillier and ElGamal) and need to
perform large integer operation on servers. Then, their scheme have numerous
computations on the servers. At the same time, their scheme cannot verify the
computations and cannot evaluate more than degree-4 computations. To solve
these problems, we no longer use linearly-homomorphic encryption which based on
number theory assumptions. We use label and pseudorandom function to encrypt
message, which significantly reduce the computations on the servers and enable
us to use homomorphic MACs technology to realize verifiable computations
naturally. We also extend the method to construct $d$-server schemes, which
allow the client to delegate degree-$d$ computations on outsourced data.

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