In the IPv4/IPv6 transition process, since some legacy IPv4 networks still need to communicate with the IPv6 Internet, the stateless communication mechanism between an IPv4 network and the IPv6 Internet, which complements the current IPv4/IPv6 translation framework, was proposed. First, the communication procedures in two related scenarios were demonstrated. The two scenarios include IPv6 Internet clients accessing IPv4 servers and IPv4 clients accessing IPv6 Internet servers. The one-way IPv6-IPv4 address mapping function is the key component of the mechanism. Therefore, the requirements and three quantitative criteria of the one-way mapping function were discussed. Afterwards, multiple Hash functions as the candidates of the one-way mapping function were compared and analyzed with the real user data of large IPv6 websites and real IPv6 server addresses. The simulation results show that the FarmCity Hash function is suitable to be deployed in the above two scenarios because it has short average processing time, low collision rate and low reverse query complexity. It also verifies the validity of the stateless communication mechanism. Compared with current stateful communication mechanisms, the stateless mechanism has better scalability and traceability. Moreover, the capacity for bidirectional communication facilitates a smooth migration path towards the IPv6 Internet.