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Journal of Systems Science and Complexity

On the Security of Homomorphic Encryption Schemes with Restricted Decryption Oracles

Guangsheng Ma, Hongbo Li

Published Aug 30, 2024• 6 citations
Publisher
DOI
10.1007/s11424-024-3221-1

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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 Data can be stored and processed in an encrypted format using fully homomorphic encryption, which makes cloud computing provider can process data well without even knowing data information. However, most existing homomorphic encryption schemes based on ring learning with errors, focus on the special classes of rings, such as power-of-two cyclotomic rings, which significantly limits its application efficiency in cloud computing. Therefore, in this letter, to solve this problem above, an efficient fully homomorphic encryption scheme is proposed based on ring learning with errors in arbitrary cyclotomic rings. Particularly, a “somewhat homomorphic encryption” scheme is firstly proposed based on ring learning with errors. A novel fully homomorphic encryption scheme is further presented using key-switching. With theoretical and experimental verification, the proposed scheme has advantage of high efficiency of encryption and decryption. The security of our scheme strictly reduces to hardness of decision ring learning with errors problem in the random oracle model.
We present a two-round protocol to solve the socialist millionaire problem based on the homomorphic property of the Goldwasser-Micali (GM) cryptosystem. We require the proposed protocol to be secure against active and passive attacks. However, homomorphic encryption schemes are malleable by design [14][1]. To tackle this problem we apply an authenticated encryption scheme, called Encrypt-then-MAC, to our protocol [3]. We analyze the security of the proposed protocol, and we show that an active adversary, who has access to the ciphertext on the communication channel and the decryption oracle, cannot forge another ciphertext which leads him to guess the plaintext (IND-CCA2 security). Moreover, the active adversary cannot modify the ciphertext which leads to a desired modification of the plaintext to affect the outcome of the protocol (NM-CCA2 security). Note that our solution can be applied to other problems which are solvable with an exclusive- or homomorphic property.