Abstract
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 ( ${\mathsf {LWE}}$ ) 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 ${\mathsf {LWE}}$ 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.
Highlights
Homomorphic encryption (FHE) [1], [2] is a variant of public-key encryption
We prove that the proposed certificateless fully homomorphic encryption (CLFHE) scheme is semantically secure and anonymous for adaptive chosen-identity based on the hardness of the learning with errors (LWE) problem in the random oracle model
By exploiting the Gaussian sampling algorithm SampleRight presented in [22], we prove that the proposed CLFHE scheme is semantically secure and anonymous for selective-identity based on the hardness of the LWE problem in the standard model
Summary
Homomorphic encryption (FHE) [1], [2] is a variant of public-key encryption. It allows anyone to perform arbitrary computation on encrypted data. M. Li: Leveled Certificateless Fully Homomorphic Encryption Schemes From Learning With Errors using the re-linearization technique presented in [15] and proved that it was semantically secure based on the hardness of the LWE problem in the random oracle model. Li: Leveled Certificateless Fully Homomorphic Encryption Schemes From Learning With Errors using the re-linearization technique presented in [15] and proved that it was semantically secure based on the hardness of the LWE problem in the random oracle model Their scheme incurs a very high computational complexity. Chen, Hu, and Lian [16] proposed a leveled certificateless homomorphic encryption (CLHE) scheme using the approximate eigenvector method presented in [3] and proved its semantic security based on the hardness of the LWE problem in the random oracle model.
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