Dynamic Searchable Symmetric Encryption allows a client to store a dynamic collection of encrypted documents with a server, and later quickly carry out keyword searches on these encrypted documents, while revealing minimal information to the server. In this paper we present a new dynamic SSE scheme that is simpler and more efficient than existing schemes while revealing less information to the server than prior schemes, achieving fully adaptive security against honest-but-curious servers. We implemented a prototype of our scheme and demonstrated its efficiency on datasets from prior work. Apart from its concrete efficiency, our scheme is also simpler: in particular, it does not require the server to support any operation other than upload and download of data. Thus the server in our scheme can be based solely on a cloud storage service, rather than a cloud computation service as well, as in prior work. In building our dynamic SSE scheme, we introduce a new primitive called Blind Storage, which allows a client to store a set of files on a remote server in such a way that the server does not learn how many files are stored, or the lengths of the individual files, as each file is retrieved, the server learns about its existence (and can notice the same file being downloaded subsequently), but the file's name and contents are not revealed. This is a primitive with several applications other than SSE, and is of independent interest.