Fully homomorphic encryption (FHE) is a form of public-key encryption that allows the computation of arbitrary functions on encrypted data without decrypting the data. As a result, it is a useful tool with numerous applications. Certificateless encryption (CLE) is a type of public-key encryption that combines the advantages of PKI-based public-key encryption with those of identity-based encryption (IBE). Thus, certificateless fully homomorphic encryption (CLFHE) has aroused considerable research interest. Recently, Chen, Hu, and Lian proposed a leveled certificateless homomorphic encryption (CLHE) scheme and proved its semantic security based on the learning with errors (LWE) problem in the random oracle model. However, their scheme supports only homomorphic addition, but not homomorphic multiplication. In this work, we construct two leveled CLFHE schemes using the approximate eigenvector method presented by Gentry, Sahai, and Waters. Based on the hardness of the LWE problem, we prove that one scheme satisfies adaptive semantic security and anonymity in the random oracle model, whereas the other satisfies selective semantic security and anonymity in the standard model.
Abstract The security of most early fully homomorphic encryption schemes was based on the hardness of the Learning with Errors (LWE) problem. These schemes were inefficient in terms of per gate computations and public‐key size. More efficient schemes were later developed based on the hardness of the Ring‐LWE (RLWE) problem. While the hardness of the LWE problem is based on the hardness of the approximate shortest vector problem (GapSVP γ ) over regular lattices, the hardness of the RLWE problem is based on the hardness of the approximate shortest vector problem over ideal lattices. As of now, it has not been proved that the (GapSVP γ ) problem over ideal lattices is as difficult as the corresponding problem over regular lattices. In this work, the authors propose a multi‐bit levelled fully homomorphic encryption scheme using multivariate polynomial evaluations whose security depends on the hardness of the LWE problem. In terms of per gate computation cost, this scheme is more efficient than existing LWE‐based schemes. Further, for an appropriate choice of parameters, the per computation cost for homomorphic multiplication can be made asymptotically comparable to RLWE‐based schemes in a parallel computing environment. For homomorphic multiplication, the scheme uses a polynomial‐based technique that does not require relinearization (and key switching).
This paper proposes an efficient identity-based leveled fully homomorphic encryption (IBFHE) scheme over ideal lattice. First, we achieve an identity-based encryption (IBE) scheme in dual-LWE cryptosystem. Then, we take advantage of the eigenvector method to compile the IBE scheme into an IBFHE scheme. Compared with the previous scheme, our scheme is of shorter key size, smaller storage, and higher efficiency, owing to a new trapdoor generation function and algebraic structure of ideal lattice. The security analysis shows that our scheme is selective-identity secure against chosen-plaintext attacks in the standard model.
Fully homomorphic encryption (FHE) has important applications in cloud computing. However, almost all fully homomorphic encryption schemes share two common flaws that they all use large-scale secret keys and some operations are inefficient. In this paper, the 'special b' variant of the learning with errors problem (bLWE) is presented, and helps us construct the first circularly secure key switching process which can replace the key switching process and similar re-linearisation process used by the existing FHE schemes. Then, we present an efficient FHE. Compared with Brakerski's scheme, our scheme reduces L secret keys to one and is more efficient. Finally, we prove the chosen-plaintext attack (CPA) security of the fully homomorphic scheme and the circular security of key switching process in standard model under the learning with errors problem (LWE) assumption.
Fully homomorphic encryption (FHE) has important applications in cloud computing. However, almost all fully homomorphic encryption schemes share two common flaws that they all use large-scale secret keys and some operations are inefficient. In this paper, the 'special b' variant of the learning with errors problem (bLWE) is presented, and helps us construct the first circularly secure key switching process which can replace the key switching process and similar re-linearisation process used by the existing FHE schemes. Then, we present an efficient FHE. Compared with Brakerski's scheme, our scheme reduces L secret keys to one and is more efficient. Finally, we prove the chosen-plaintext attack (CPA) security of the fully homomorphic scheme and the circular security of key switching process in standard model under the learning with errors problem (LWE) assumption.
Identity‐based fully homomorphic encryption (IBFHE) provided a fundamental solution to the problem of huge public key size that exposed in fully homomorphic encryption (FHE) schemes, thus it is significant to make FHE become more practical. In recent years, the construction of IBFHE schemes were mainly based on lattices due to their conjectured resistance against quantum cryptanalysis, however, which makes these cryptosystems further unpractical. The main reason is the trapdoor function on which the scheme was based is rather complex for practical and the ciphertext size is too large. In this study, the authors propose an efficient batch IBFHE scheme, which can be proven secure from the standard LWE assumption in the standard model. The first contribution of this work is that the authors construct an efficient batch version of MP12 preimage sampling algorithm, which can efficiently generate identity keys for multi‐bit IBE schemes. Based on that, the authors construct an asymptotically‐faster multi‐bit IBE scheme as the second contribution. The third contribution is that the authors transform the multi‐bit IBE scheme to batch IBFHE scheme which supports to encrypt any message in . Compared with the similar schemes, the authors show their schemes are essentially improved.