Abstract: The goal of position-based cryptography is to use the geographical position of a party as its only credential to achieve some cryptographic tasks, such as position-based encryption. Position-based key exchange should have the property that if there is a prover at the claimed position, then at the end of the protocol, the verifiers should share a uniform key with it while for any one group of colluding adversaries should look indistinguishable from a key drawn uniformly at random. The provable security of key exchange in position-based cryptography is investigated in this paper. In the universally composable framework, the provable secure model of position-based key exchange is proposed. According to the security requirements of position-based key exchange, the ideal functionality of position-based key exchange is presented. For any one group of colluding adversaries, the shared key derived from the ideal functionality is indistinguishable from a random key. At the same time, the ideal functionality of bounded retrieval model is designed as one of the set-up assumptions in position-based cryptography. In addition, the position-based key exchange protocol in 1-dimension space, as an example, can securely realize the functionality of position-based key exchange in the bounded retrieval model.