## 1 Introduction  
The advances of Internet technologies have brought people the ability to access the plentiful resources on the cloud. The low-cost, powerful, and convenient services provided on the cloud have drawn people to move from their local computers to remote computing facilities. One of the most popular cloud services is cloud storage. The Cloud storage service allows people to access their data at anytime, from anywhere. It also allows data owners to share their data with authorized users. It has dramatically changed the way that people save their data and share their information. The convenience of cloud storages also comes with great challenges and problems. The first one is the search mechanism. Since the number of documents stored in cloud storage is typically large, users usually find the documents of their interest using several keywords. Traditional keyword search methods use a long index for each document in the dataset. The index is usually denoted by a vector to characterize the presence or absence for a bunch of selected keywords. For each document, the keywords present in the document are marked as one in the corresponding entry in its index. When a user raises a query with certain keywords, the indexes of the documents are used to match for the user's request. Since a document may contain only a few keywords to characterize its contents, most of the locations in the index are zero. The long index could be a waste, and may increase the computation and communication overhead. Obviously, a concise index scheme is desired. Another concern to prevent people from using cloud storage is the security and privacy issue, since cloud storages are usually run by a third party service provider. Data owners want to secure the risk of information leakage, and data users would like to protect their privacy when performing search. A trivial solution is to encrypt the data before outsourcing them to the cloud, but it brings difficulty in data utilization, since the encrypted data cannot be searched like in plain text. It is also impractical to download the whole dataset and decrypt it to find the documents of the users' interest. This paper proposes a Condensed Index Multi-keyword Search (CIMS) scheme for encrypted cloud data. Two aspects are considered for the secure keyword search problem. The first aims to reduce the size of document indexes for the multi-keyword rank search. Users can raise multiple keywords to search the documents of interest in the dataset. Since most of the terms in document indexes are zero, it is possible to condense the document indexes while preserving the information contained in the original index. Principal component analysis (PCA), which is one of the technologies to simplify data in multivariate statistical analysis, is adopted to reduce the size of document indexes. Essentially, PCA performs a linear transformation to transfer the document indexes to a new eigenspace such that the first coordinate in the new space preserves the most information about the document indexes. In contrast, the last coordinate in the new space contains the least information about the document indexes. Thus, if the later dimensions of the document indexes in the new space are omitted, only a commensurately small amount of information is lost. Therefore, by preserving only the principal components of the document indexes can effectively condense the document index and significantly reduce the index size. Actually, the scheme trades the search precision with the index size. The second aspect is to incorporate the condensed index to a secure multi-keyword search scheme for encrypted cloud data. Both the documents and the query are encrypted. The third party cloud server actually performs the search by a pre-designed procedure in ciphertext, and cannot discover the document contents or the query information raised by users. However, the results of the pre-designed procedure are the same as that doing in plain text.