Public-key Encryption with Keyword Search (PEKS) allows authorized users to search by keywords on encrypted data by generating trapdoors for the desired keywords. Basically there are two requirements for the process of trapdoors generation. On one hand, it is important to allow users to privately search on encrypted data without revealing keywords to the Trapdoor Generation Entity $\mathcal{TGE}$, and solutions proposed are based on the use of blinding protocols between user and $\mathcal{TGE}$. On the other hand, in some applications it is necessary to prevent $\mathcal{TGE}$ from generating trapdoors on behalf of a legitimate user, this is done by splitting the $\mathcal{TGE}$ and generating trapdoors in distributed manner. Until now, these two requirements in PEKS were considered separately and many solutions were proposed dealing with each one. However, no solution is known to satisfy the two requirements at the same time.In this paper we present a new approach to deal with this issue. First, we extend the notion of anonymous Identity-Based Encryption with distributed Private Key Generator to handle the blind key-extraction protocols as in Blind IBE. We call such new schemes blind anonymous (n,t)-IBE, and we give concrete construction of such schemes. Second, we generically transform the blind anonymous (n,t)-IBE into a threshold PEKS with oblivious keyword search satisfying the requirements specified above.