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Proxy re-encryption (PRE), introduced by Blaze et al. in 1998, allows a semi-trusted proxy with the re-encryption key to translatea ciphertext under the delegator into another ciphertext, which can be decrypted by the delegatee. In this process, the proxy is required to know nothing about the plaintext. Many PRE schemes have been proposed so far, however until now almost all the unidirectional PRE schemes suffer from the transferable property. That is, if the proxy and a set of delegatees collude, they can re-delegate the delegator's decryption rights to the other ones, while the delegator has no agreement on this. Thus designing non-transferable unidirectional PRE scheme is an important open research problem in the field. In this paper, we tackle this open problem by using the composite order bilinear pairing. Concretely, we design a non-transferable unidirectional PRE scheme based on Hohenberger et al.'s unidirectional PRE scheme. Furthermore, we discuss our scheme's application to secure cloud storage, especially for sharing private multimedia content for social cloud storage users.

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