Anonymous trust/reputation system
You are probably looking for what is called "anonymous credentials".

An anonymous credentials system relates three types of parties: authorities, users, and verifiers. An authority (Alice) can issue a credential to a user (Bob), that certifies that the user satisfies some property (in your case, that would be "is trusted"). Credentials are unforgeable. Then, that user (Bob) can prove to a verifier (Peter) that it possesses the credential, without revealing anything about himself other than the possession of the credential (anonymity), even if the verifier colludes with other verifiers and/or the authority.

The way anonymous credentials systems are typically implemented is that a user has a single private key sk, and multiple corresponding public keys pk, pk', etc. (pseudonyms) that can be generated on the fly as randomized commitments on sk. Then, the credential issuing protocol is a special kind of signature scheme that produces a signature on the value inside the commitment; in that way, Bob can obtain a credential from Alice via his pseudonym Bob-Work, and prove possession of it under his pseudonym Bob-Friends, in such a way that the two pseudonyms cannot be linked to each other.

A good general introduction to anonymous credentials is Anna Lysanskaya's Ph.D. thesis, especially chapter 3.

Since 2002, there has been quite a bit of research and progress on anonymous credentials, so we have efficient instantiations supporting additional features like revocation or delegation.