Anonymous access authentication schemes provide users with massive application services while protecting the privacy of users' identities. The identity protection schemes in 3G and 4G are not suitable for 5G anonymous access authentication due to complex computation and pseudonym asynchrony. In this paper, we consider mobile devices with limited resources in the 5G network and propose an anonymous access authentication scheme without the Public Key Infrastructure. The anonymous access authentication scheme provides users with variable shard pseudonyms to protect users' identities asynchronously. With the variable shared pseudonym, our scheme can ensure user anonymity and resist the mark attack, a novel attack aimed at the basic k-pseudonym scheme. Finally, we analyze the scheme with BAN logic analysis and verify the user anonymity.