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In 2008, Libert and Vergnaud constructed the first multi-use unidirectional proxy re-signature scheme. In this scheme, the proxy can translate the signatures several times but only in one direction. Thus, two problems remain open. That is, to construct a multi-use unidirectional proxy re-signature scheme based on classical hardness assumptions, and to design a multi-use unidirectional proxy re-signature scheme with the size of signatures and the verification cost growing sub-linearly with the number of translations. This paper solves the first problem and sharply reduces the verification costs. We use the preimage sampleable algorithm to develop a multi-use unidirectional proxy re-signature scheme based on lattices, namely, the hardness of the Small Integer Solution (SIS) problem. The verification cost does not grow with the number of translations and the size of signatures grows linearly with the number of translations in this scheme. Furthermore, the proposal is secure in quantum environment.

Proxy re-encryption (PRE) allows a semi-trusted proxy to turn Alice’s ciphertexts into Bob’s ciphertexts, however, the proxy cannot get the corresponding plaintexts. Because of the special property, PRE achieves good reliability and secrecy. Nevertheless, most of the proposed proxy re-encryption schemes are based on the number theoretic problem and their security are proved in the random oracle. Therefore, this paper constructs an efficient and novel identity-based multi-bit PRE scheme based on the learning with errors (LWE) assumption and can resist to quantum attack. What’s more, it is proved to be CPA secure in the standard model and has the properties of multi-use and bidirectional.