Searchable symmetric encryption

Searchable symmetric encryption (SSE) is a is a form of encryption that allows one to efficiently search over a collection of encrypted documents or files without the ability to decrypt them. SSE can be used to outsource files to an untrusted cloud storage server without ever revealing the files in the clear but while preserving the server's ability to search over them.

Description

A searchable symmetric encryption scheme is a symmetric-key encryption scheme that encrypts a collection of documents $\mathbf {D} =(\mathrm {D_{1}} ,\dots ,\mathrm {D_{n}} )$, where each document $\mathrm {D_{i}} \subseteq \mathbb {W}$ is viewed as a set of keywords from a keyword space $\mathbb {W}$. Given the encryption key $K$ and a keyword $w\in \mathbb {W}$, one can generate a search token $tk$ with which the encrypted data collection can be searched for $w$. The result of the search is the subset of encrypted documents that contain the keyword $w$.

Static SSE

A static SSE scheme consists of three algorithms ${\mathsf {SSE=(Setup,Token,Search)}}$ that work as follows:

• ${\mathsf {Setup}}$ takes as input a security parameter $k$ and a document collection $\mathbf {D}$ and outputs a symmetric key $K$ and an encrypted document collection $\mathbf {ED}$
• ${\mathsf {Token}}$ takes as input the secret key $K$ and a keyword $w$ and outputs a search token $tk$
• ${\mathsf {Search}}$ takes as input the encrypted document collection $\mathbf {ED}$ and a search token $tk$ and outputs a set of encrypted documents $\mathbf {R} \subseteq \mathbf {ED}$

A static SSE scheme is used by a client and an untrusted server as follows. The client encrypts its data collection using the ${\mathsf {Setup}}$ algorithm which returns a secret key $K$ and an encrypted document collection $\mathbf {ED}$. The client keeps $K$ secret and sends $\mathbf {ED}$ to the untrusted server. To search for a keyword $w$, the client runs the ${\mathsf {SearchToken}}$ algorithm on $K$ and w to generate a search token $tk$ which it sends to the server. The server runs Search with $\mathbf {ED}$ and $tk$ and returns the resulting encrypted documents back to the server.

Dynamic SSE

A dynamic SSE scheme supports, in addition to search, the insertion and deletion of documents. A dynamic SSE scheme consists of seven algorithms ${\mathsf {SSE=(Setup,Token,Search,InertToken,Insert,DeleteToken,Delete)}}$ where ${\mathsf {Setup}}$, ${\mathsf {Token}}$ and ${\mathsf {Search}}$ are as in the static case and the remaining algorithms work as follows:

• ${\mathsf {InsertToken}}$ takes as input the secret key $K$ and a new document $\mathrm {D_{n+1}}$ and outputs an insert token $itk$
• ${\mathsf {Insert}}$ takes as input the encrypted document collection EDC and an insert token $itk$ and outputs an updated encrypted document collection $\mathbf {ED'}$
• ${\mathsf {DeleteToken}}$ takes as input the secret key $K$ and a document identifier $id$ and outputs a delete token $dtk$
• ${\mathsf {Delete}}$ takes as input the encrypted data collection $\mathrm {EDC}$ and a delete token $dtk$ and outputs an updated encrypted data collection $\mathbf {ED'}$

To add a new document $\mathrm {D_{n+1}}$ the client runs ${\mathsf {InsertToken}}$ on $K$ and $\mathrm {D_{n+1}}$to generate an insert token $itk$ which it sends to the server. The server runs ${\mathsf {Insert}}$ with $\mathbf {ED}$ and $itk$ and stores the updated encrypted document collection. To delete a document with identifier $id$, the client runs the ${\mathsf {DeleteToken}}$ algorithm with $K$ and $id$ to generate a delete token $dtk$ which it sends to the server. The server runs ${\mathsf {Delete}}$ with $\mathbf {ED}$ and $dtk$ and stores the updated encrypted document collection.

An SSE scheme that does not support ${\mathsf {DeleteToken}}$ and ${\mathsf {Delete}}$ is called semi-dynamic.

History of Searchable Symmetric Encryption

The problem of searching on encrypted data was considered by Song, Wagner and Perrig though previous work on Oblivious RAM by Goldreich and Ostrovsky could be used in theory to address the problem. This work proposed an SSE scheme with a search algorithm that runs in time $O(s)$, where $s=|\mathbf {D} |$. Goh and Chang and Mitzenmacher gave new SSE constructions with search algorithms that run in time $O(n)$, where $n$ is the number of documents. Curtmola, Garay, Kamara and Ostrovsky later proposed two static constructions with $O(\mathrm {opt} )$ search time, where $\mathrm {opt}$ is the number of documents that contain $w$, which is optimal. This work also proposed a semi-dynamic construction with $O(\mathrm {opt} \cdot \log(u))$ search time, where $u$ is the number of updates. An optimal dynamic SSE construction was later proposed by Kamara, Papamanthou and Roeder.

Goh and Chang and Mitzenmacher proposed security definitions for SSE but the definitions had limitations. Curtmola, Garay, Kamara and Ostrovsky proposed the notion of adaptive security for SSE which is now standard. This work also was the first to observe leakage in SSE and to formally capture it as part of the security definition.

All the previously mentioned constructions support single keyword search. Cash, Jarecki, Jutla, Krawczyk, Rosu and Steiner proposed an SSE scheme that supports conjunctive search in sub-linear time in $n$. The construction can also be extended to support disjunctive and Boolean searches that can be expressed in searchable normal form (SNF) in sub-linear time. At the same time, Pappas, Krell, Vo, Kolesnikov, Malkin, Choi, George, Keromytis and Bellovin described a construction that supports conjunctive and all disjunctive and Boolean searches in sub-linear time.

See also

• Homomorphic encryption
• Oblivious RAM
• Structured encryption
• Deterministic encryption