The right to anonymous speech is can be considered (and is by many) an implication of first amendment rights to free speech. One way to look at it is that the requirement that a person be identified before being allowed to speak is itself a limitation on speech.

I'm too lazy to do my own citations, but the courts have repeatedly struck down bans on anonymous speech on these grounds.

Right there in the 1st amendment.

Backed up by the supreme court in the right for group memberships not to be revealed.

You have the right to anonymously assemble and associate with a political group. I hope you don't lose it.