Time-stamping protocols, which assure that a document was existed at a certain time, are applied to some useful and practical applications such as electronic patent applications and so on. There are two major time-stamping protocols, the simple protocol and the linking protocol. In the former, a time-stamp authority issues a time-stamp token that is the digital signature of the concatenated value of a hashed message and the present time. In the latter, the time-stamp authority issues a time-stamp token that is the hash value of the concatenated value of a hashed message and the previous hash value. Although security requirements and analysis for above time-stamping protocols has been discussed, there are no strict cryptographic security notions for them. In this paper, we reconsider the security requirements for time-stamping protocols and define security notions for them, in a universally composable security sense, which was proposed by Canetti. We also show that these notions can be achieved using combinations of a secure key exchange protocol, a secure symmetric encryption scheme, and a secure digital signature scheme.