| When a cryptosystem has been applied in practice, how can the users be sure that the system in use is secure? The approach of provable security leads us to an answer of the question. A provable secure cryptosystem can be ensured secure by security proofs under the defined model. In contrast, cryptosystems without rigorous proofs cannot be regarded as secure even in practice. Actually, there are many schemes that were originally thought as secure but eventually broken, which clearly indicates the need of formal security assurance. Nowadays, most of the design of new cryptosystems follows the modern methodology of provable security.Signcryption is a very important primitive in public key cryptography, which combines the functions of both public key encryption and digital signature in a more efficient manner. In this thesis, we deal with signcryption related public key cryptographic systems. Considering the importance of provable security mentioned above, we focus our research on the models, constructions as well as proofs of the systems.Concretely, our main contributions contain five aspects. First, we improve the Zheng signcryption to ensure its security in the single key pair setting and the send to oneself scenario. Second, we analyze the existing security model for signcryption with the property of non-interactive non-repudiation, show that it is not good enough to ensure security, then we propose improved the model, and construct two schemes and provide security proofs under the new model. Third, we generalize the model for signcryption with the property of non-interactive non-repudiation, and discuss the security for the constructions of sign-then-encrypt and the encrypt-then-sign. Fourth, we introduce key-insulated signcryption which offers security in case of key exposure. Finally, we extend the property of non-interactive opening to identity-based encryption, such that a receiver can open a ciphertext to a third party non-interactively in a secure manner.
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