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Recent content in Ste on Outsourced BitsHugo -- gohugo.ioen-usThu, 16 Jun 2016 12:32:12 -0300Graph Encryption: Going Beyond Encrypted Keyword Search
http://senykam.github.io/2016/06/16/graph-encryption-going-beyond-encrypted-keyword-search
Thu, 16 Jun 2016 12:32:12 -0300http://senykam.github.io/2016/06/16/graph-encryption-going-beyond-encrypted-keyword-search<p><em>This is a guest post by <a href="http://www.xianruimeng.org/">Xianrui Meng</a> from
Boston University about a paper he presented at CCS 2015, written in
collaboration with <a href="https://www.cs.bgu.ac.il/~kobbi/">Kobbi Nissim</a>, <a href="http://www.cs.bu.edu/~gkollios/">George
Kollios</a> and myself. Note that Xianrui is on
the job market.</em></p>
<p><img src="http://senykam.github.io/img/graph21.jpg" class="alignright" width="250">
Encrypted search has attracted a lot of attention from practitioners and
researchers in academia and industry. In previous posts, Seny already described
different ways one can search on encrypted data. Here, I would like to discuss
search on encrypted <em>graph</em> databases which are gaining a lot of
popularity.</p>
<h2 id="graph-databases-and-graph-privacy">Graph Databases and Graph Privacy</h2>
<p>As today's data is getting bigger and bigger, traditional
relational database management systems (RDBMS) cannot scale to the massive
amounts of data generated by end users and organizations. In addition, RDBMSs
cannot effectively capture certain data relationships; for example in
object-oriented data structures which are used in many applications. Today,
<a href="http://nosql-database.org/">NoSQL</a> (Not Only SQL) has emerged as a good
alternative to RDBMSs. One of the many advantages of NoSQL systems is that
they are capable of storing, processing, and managing large volumes of
structured, semi-structured, and even unstructured data. NoSQL databases (e.g.,
document stores, wide-column stores, key-value (tuple) stores, object
databases, and graph databases) can provide the scale and availability needed
in cloud environments.</p>
<p>In an Internet-connected world, graph databases have become an increasingly
significant data model among NoSQL technologies. Social networks (e.g.,
Facebook, Twitter, Snapchat), protein networks, electrical grid, Web, XML
documents, networked systems can all be modeled as graphs. One nice thing
about graph databases is that they store the relations between entities
(objects) in addition to the entities themselves and their properties. This
allows the search engine to navigate both the data and their relationships
extremely efficiently. Graph databases rely on the node-link-node relationship,
where a node can be a profile or an object and the edge can be any relation
defined by the application. Usually, we are interested in the structural
characteristics of such a graph databases.</p>
<p>What do we mean by the confidentiality of a graph? And how to do we protect it?
The problem has been studied by both the security and database communities. For
example, in the database and data mining community, many solutions have been
proposed based on <em>graph anonymization</em>. The core idea here is to
anonymize the nodes and edges in the graph so that re-identification is hard.
Although this approach may be efficient, from a security point view it is hard
to tell what is achieved. Also, by leveraging auxiliary information,
researchers have studied how to attack this kind of approach. On the other
hand, cryptographers have some really compelling and provably-secure tools such
as ORAM and FHE (mentioned in Seny's previous posts) that can protect all the
information in a graph database. The problem, however, is their performance,
which is crucial for databases. In today's world, efficiency is more than
running in polynomial time; we need solutions that run and scale to massive
volumes of data. Many real world graph datasets, such as biological networks
and social networks, have millions of nodes, some even have billions of nodes
and edges. Therefore, besides security, scalability is one of main aspects we
have to consider.</p>
<h2 id="graph-encryption">Graph Encryption</h2>
<p>Previous work in encrypted search has focused on how to
search encrypted documents, e.g., doing keyword search, conjunctive queries,
etc. Graph encryption, on the other hand, focuses on performing graph queries
on encrypted graphs rather than keyword search on encrypted documents. In some
cases, this makes the problem harder since some graph queries can be extremely
complex. Another technical challenge is that the privacy of nodes and edges
needs to be protected but also the <em>structure</em> of the graph, {\bf which can
lead to many interesting research directions}.</p>
<p>Graph encryption was introduced by Melissa Chase and Seny in
[<a href="http://eprint.iacr.org/2011/010.pdf">CK10</a>]. That paper shows how
to encrypt graphs so that certain graph queries (e.g., neighborhood, adjacency
and focused subgraphs) can be performed (though the paper is more general as it
describes <em>structured encryption</em>). Seny and I, together with Kobbi Nissim
and George Kollios, followed this up with a paper last year
[<a href="http://eprint.iacr.org/2015/266.pdf">MKNK15</a>] that showed how to
handle more complex graphs queries.</p>
<h2 id="queries-on-encrypted-graph-databases">Queries on Encrypted Graph Databases</h2>
<h3 id="neighbor-queries-and-adjacency-queries">Neighbor Queries and Adjacency Queries</h3>
<p>As I mentioned earlier,
[<a href="http://eprint.iacr.org/2011/010.pdf">CK10</a>] studied some simple
graph queries, such as adjacency queries and neighbor queries. An adjacency
query is takes two nodes as input and returns whether they have an edge in
common. A neighbor query takes a node as input and returns all the nodes that
share an edge with it.</p>
<p>The construction for neighbor queries is mainly based on the searchable
symmetric encryption (SSE), where the input graph is viewed as particular kind
of document collection. Another novel technique that is proposed in the paper
is to use an efficient symmetric non-committing encryption scheme to achieve
adaptive security efficiently. The paper also proposes a nice solution for
focused subgraph queries, which are an essential part of the seminal HITS
ranking algorithm of Kleinberg but are also useful in their own right.</p>
<h3 id="approximate-shortest-distance-queries">Approximate Shortest Distance Queries</h3>
<p>Shortest distance queries are arguably one of the most fundamental and
well-studied graph queries due to their numerous applications. A shortest
distance query takes as input two nods and returns the smallest number of edges
in the shortest path between them. In social networks these queries allow you
to find the smallest number of friends (or collaborators, peers, etc) between
two people. So a graph encryption scheme that supports shortest distance
queries would potentially have many applications in graph database security,
and could be a major building block for other graph encryption schemes. In the
following, I briefly give an overview on our solution for <em>approximate</em>
shortest distance queries.</p>
<p>As I mentioned, to design a secure yet scalable graph encryption
scheme, we have to take into account many things, including the storage space on
the server side, the bandwidth for the query, the computational overhead for
both client and server, etc. Suppose we are given a graph <span class="math">\(G= (V, E)\)</span> and let
<span class="math">\(n= |V|\)</span>, <span class="math">\(m = |E|\)</span>. If we were to use traditional shortest distance algorithm such as
Dijsktra's algorithm, the query time would be <span class="math">\(O(n\log n+m)\)</span>, which can be very
slow for large graphs. The benefit of course would be that we would not need
extra storage. Another approach is to build an encrypted adjacency matrix (see
[<a href="http://eprint.iacr.org/2011/010.pdf">CK10</a>]) that somehow supports
shortest distance queries. The problem there is that we would need to pay at
least <span class="math">\(O(n^2)\)</span> storage, which is obviously expensive when the <span class="math">\(n\)</span> is, say, <span class="math">\(1\)</span>
million.</p>
<p>Fortunately, thanks to brilliant algorithmic computer scientists, there exists
a really nice and neat data structure called a <em>distance oracle</em> (DO)
[<a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.94.333&rep=rep1&type=pdf">TZ05</a>,
<a href="http://research.microsoft.com/pubs/115785/wsdm2010.pdf">SGNP10</a>,
<a href="http://research-srv.microsoft.com/pubs/201773/cosn-similarity.pdf">CDFGGW13</a>].
Using such a structure, one can use much less storage overhead (typically <span class="math">\(O(n
\log n)\)</span>) and fast query performance (typically <span class="math">\(O(\log n)\)</span>). However, most
distance oracles return the <em>approximate</em> distance rather than the exact
one. But one one can tweak the parameters in order to get the best trade-off
between performance and approximation. When I first looked at these data
structures, I felt that this was a really amazing tool; not only because of its
functionality but also due to its simplicity.</p>
<p>There are many ways of generating distance oracles. Some of them offer
better approximation while others can have better performance. Here I just
describe one kind which are <em>sketch-based</em> distance oracles. In a such an
oracle, every node <span class="math">\(v\)</span> has a sketch, <span class="math">\(Sk_v\)</span> (normally generated by some
randomized algorithm). <span class="math">\(Sk_v\)</span> is a set containing many node pairs <span class="math">\(\langle
w_i,d(v, w_i)\rangle\)</span>, where <span class="math">\(w_i\)</span> is some node id and <span class="math">\(d(v, w_i)\)</span> is the
distance between <span class="math">\(v\)</span> and <span class="math">\(w_i\)</span>. For example, the following sketch <span class="math">\(Sk_v\)</span> consists of
three pairs</p>
<p><span class="math">\[ Sk_v = \{\langle w_1, d(v, w_1)\rangle, \langle w_2, d(v, w_2)\rangle,
\langle w_3, d(v, w_3)\rangle \}.
\]</span></p>
<p>Querying the shortest distance
between <span class="math">\(u\)</span> and <span class="math">\(v\)</span> is quite simple. We only need to retrieve <span class="math">\(Sk_u\)</span> and
<span class="math">\(Sk_v\)</span>, and find the common nodes in both sketches and add up their
corresponding distances. We then return the minimum sum as the
shortest distance. Formally, let <span class="math">\(I\)</span> be the common nodes that appear in both
<span class="math">\(Sk_u\)</span> and <span class="math">\(Sk_v\)</span>. Then, the approximate shortest distance between <span class="math">\(u\)</span> and <span class="math">\(v\)</span>,
<span class="math">\(d(u,v)\)</span>, is</p>
<p><span class="math">\[d(u, v) = argmin_{s \in I}\{ d(u, s) + d(v, s)\}\]</span></p>
<p>The design of this distance oracle guarantees that the returned distance is no
greater than <span class="math">\(\alpha\times \mathsf{dist}(u,v)\)</span>, where <span class="math">\(\mathsf{dist}(u, v)\)</span> is
the true shortest distance between <span class="math">\(u\)</span> and <span class="math">\(v\)</span> and <span class="math">\(\alpha\)</span> is the
approximation ratio. Note that the approximation ration <span class="math">\(\alpha\)</span> is a function
of some parameters of the sketch so to one controls the approximation by
tweaking the sketch which in turns effects both setup and query efficiency. In
our solution, we leverage sketched-based distance oracles but we have to be
very careful not to affect their approximation ration.</p>
<p>A distance oracle encryption scheme
<span class="math">\(\mathsf{Graph} = (\mathsf{Setup}, \mathsf{DistQuery})\)</span> consists of a polynomial-time algorithm
and a polynomial-time two-party protocol that work as follows:</p>
<ul>
<li><p><span class="math">\((K, \mathsf{EO}) \leftarrow \mathsf{Setup}(1^k, \Omega, \alpha, \varepsilon)\)</span>: is a
probabilistic algorithm that takes as input a security parameter <span class="math">\(k\)</span>, an oracle<br>
<span class="math">\(\Omega\)</span>, an approximation factor <span class="math">\(\alpha\)</span>, and an error parameter <span class="math">\(\varepsilon\)</span>.
It outputs a secret key <span class="math">\(K\)</span> and an encrypted oracle <span class="math">\(\mathsf{EO}\)</span>.</p></li>
<li><p>item <span class="math">\((d, \bot) \leftarrow \mathsf{DistQuery}_{C,S}\big((K, q), \mathsf{EO}\big)\)</span>: is a
two-party protocol between a client <span class="math">\(C\)</span> that holds a key <span class="math">\(K\)</span> and a shortest
distance query <span class="math">\(q = (u, v) \in V^2\)</span> and a server <span class="math">\(S\)</span> that holds an encrypted
oracle <span class="math">\(\mathsf{EO}\)</span>. After executing the protocol, the <span class="math">\(C\)</span> receives a distance <span class="math">\(d
\geq 0\)</span> and server <span class="math">\(S\)</span> receives <span class="math">\(\bot\)</span>.</p></li>
</ul>
<p>For <span class="math">\(\alpha \geq 1\)</span> and <span class="math">\(\varepsilon \lt 1\)</span>, we say that <span class="math">\(\mathsf{Graph}\)</span> is
<span class="math">\((\alpha, \varepsilon)\)</span>-correct if for all <span class="math">\(k \in \mathbb{N}\)</span>, for all <span class="math">\(\Omega\)</span>
and for all <span class="math">\(q = (u, v) \in V^2\)</span>,</p>
<p><span class="math">\[
\mbox{Pr}\big[d \leq \alpha\cdot {\sf dist}(u, v)\big] \geq 1 - \varepsilon,
\]</span></p>
<p>where the probability is over the randomness in computing <span class="math">\((K, \mathsf{EO}) \leftarrow
\mathsf{Setup}(1^k, \Omega, \alpha, \varepsilon)\)</span> and then <span class="math">\((d, \bot) \leftarrow
\mathsf{DistQuery}\big((K, q), \mathsf{EO}\big)\)</span>. I skip the adaptive security definition
as it is similar to adaptive security for SSE and is captured by the general
notion of security for structured encryption given in
[<a href="http://eprint.iacr.org/2011/010.pdf">CK10</a>]. Next, I will go over
two solutions for the oracle encryption.</p>
<p><strong>A computationally-efficient solution.</strong>
This approach is rather straightforward, so here I briefly sketch its description. The <span class="math">\(\mathsf{Setup}\)</span> algorithm works as follows:</p>
<ol>
<li>For each node <span class="math">\(v \in V\)</span>, generate a token by applying a PRF to <span class="math">\(v\)</span>: <span class="math">\(\mathsf{tk}_v = F_K(v)\)</span>.</li>
<li>Pad the sketches to the same length and encrypt each sketch <span class="math">\(Sk_v\)</span> as <span class="math">\({\sf Enc}_K(Sk_v)\)</span> using a symmetric encryption scheme.</li>
<li>For each node <span class="math">\(v \in V\)</span>, store the pair <span class="math">\((\mathsf{tk}_v, {\sf Enc}_K(Sk_v))\)</span> in a <a href="https://en.wikipedia.org/wiki/Associative_array}">dictionary data structure</a> <span class="math">\(\mathsf{DX}\)</span> (you should do the insertions at random).</li>
</ol>
<p>The <span class="math">\(\mathsf{DistQuery}\)</span> algorithm is quite simple: given nodes <span class="math">\(u\)</span> and <span class="math">\(v\)</span>, the
client just computes <span class="math">\(F_K(u)\)</span> and <span class="math">\(F_K(v)\)</span> and sends them to the server as the
token. After receiving the token, the server just retrieves <span class="math">\(\mathsf{DX}[F_K(u)]\)</span> and
<span class="math">\(\mathsf{DX}[F_K(v)]\)</span> and sends back the encrypted sketches <span class="math">\({\sf Enc}_K(Sk_u)\)</span> and
<span class="math">\({\sf Enc}_K(Sk_v)\)</span>. Finally, the client decrypts the sketches, and computes the
approximate shortest distance as is normally done in sketch-based distance
oracles. This approach is efficient and simple since we use symmetric
encryption. We show in the paper that this scheme is adaptively secure and that
the leakage for this scheme are the size of the graph, maximum size of the
distance oracle, and the query pattern (see paper for a precise definition).</p>
<p><strong>Communication-efficient solution.</strong>
The problem with the scheme described above is that the communication
complexity is linear in the maximum sketch size. As I mentioned above, this
can be a bottleneck in practice when the graphs are large. Now, at very high
level, I briefly discuss how we can achieve a solution with optimal <span class="math">\(O(1)\)</span>
communication complexity. The scheme makes use of a PRF, a degree-<span class="math">\(2\)</span> somewhat
homomorphic encryption scheme <span class="math">\(\mathsf{SHE} = ({\sf Gen}, {\sf Enc}, {\sf Dec})\)</span>, and a hash function <span class="math">\(h:
V\to [t]\)</span>.</p>
<ul>
<li><p><span class="math">\(\mathsf{Setup}(1^k, \Omega, \alpha, \varepsilon)\)</span>: Given <span class="math">\(1^k\)</span>, <span class="math">\(\Omega\)</span>,
<span class="math">\(\alpha\)</span>, and <span class="math">\(\varepsilon\)</span> as inputs, it generates a public/secret-key pair
<span class="math">\(({\sf pk}, {\sf sk})\)</span> for <span class="math">\(\mathsf{SHE}\)</span>. Let <span class="math">\(D\)</span> be the maximum distance over
all the sketches and <span class="math">\(S\)</span> be the maximum sketch size. <span class="math">\(\mathsf{Setup}\)</span> sets <span class="math">\(N
\leftarrow 2\cdot D +1\)</span> and samples a hash function <span class="math">\(h \leftarrow \mathcal{H}\)</span>
with domain <span class="math">\(V\)</span> and co-domain <span class="math">\([t]\)</span>, where <span class="math">\(t = 2\cdot
S^2\cdot\varepsilon^{-1}\)</span>. It then creates a hash table for each node <span class="math">\(v \in
V\)</span>. More precisely, for each node <span class="math">\(v\)</span>, it processes each pair <span class="math">\((w_i, d_i) \in
sk_v\)</span> and stores <span class="math">\({\sf Enc}_{pk}(2^{N - \delta_i})\)</span> at location <span class="math">\(h(w_i)\)</span> of a
<span class="math">\(t\)</span>-size array <span class="math">\(\mathsf{T}_v\)</span>. In other words, for all <span class="math">\(v \in V\)</span>, it creates an
array <span class="math">\(\mathsf{T}_v\)</span> such that for all <span class="math">\((w_i, \delta_i) \in Sk_v\)</span>,
<span class="math">\(\mathsf{T}_v[h(w_i)] \leftarrow {\sf Enc}_{pk}(2^{N - \delta_i})\)</span>. It then fills
the empty cells of <span class="math">\(\mathsf{T}_v\)</span> with homomorphic encryptions of <span class="math">\(0\)</span> and
stores each hash table <span class="math">\(\mathsf{T}_{v_1}\)</span> through <span class="math">\(\mathsf{T}_{v_n}\)</span> in a
dictionary <span class="math">\(\mathsf{DX}\)</span> by setting, for all <span class="math">\(v \in V\)</span>, <span class="math">\(\mathsf{DX}[F_K(v)]
\leftarrow \mathsf{T}_v\)</span>. Finally, it outputs <span class="math">\(\mathsf{DX}\)</span> as the encrypted
oracle <span class="math">\(\mathsf{EO}\)</span>.</p></li>
<li><p>The <span class="math">\(\mathsf{DistQuery}\)</span> protocol works as follows. Given a query <span class="math">\(q = (u,
v)\)</span>, the client sends tokens <span class="math">\((\mathsf{tk}_1, \mathsf{tk}_2) = (F_K(u),
F_K(v))\)</span> to the server which uses them to retrieve the hash tables of nodes
<span class="math">\(u\)</span> and <span class="math">\(v\)</span> by computing <span class="math">\(\mathsf{T}_u := \mathsf{DX}[\mathsf{tk}_1]\)</span> and
<span class="math">\(\mathsf{T}_v := \mathsf{DX}[\mathsf{tk}_2]\)</span>. The server then homomorphically
evaluates an inner product over the hash tables. More precisely, it computes <span class="math">\(c
:= \sum_{i=1}^t \mathsf{T}_u[i]\cdot\mathsf{T}_v[i]\)</span>, where <span class="math">\(\sum\)</span> and <span class="math">\(\cdot\)</span>
refer to the homomorphic addition and multiplication operations of of the SHE
scheme. Finally, the server returns only <span class="math">\(c\)</span> to the client who decrypts it and
outputs <span class="math">\(2N - \log_2 \left({\sf Dec}_{\sf sk}(c)\right)\)</span>.</p></li>
</ul>
<p>See the paper for more details and an analysis of the construction. What is
important to note is that we can show that the scheme does not affect the
quality of underlying oracle's approximation too much and, in fact, in certain
cases it improves it!</p>
<p>It is also worth of mentioning that, in the paper, we also propose a third
scheme that has <span class="math">\(O(1)\)</span> communication complexity but with some additional
leakage which we call the sketch pattern leakage. This third scheme is far more
efficient than the one above. One interesting subtlety is that, unlike more
standard encrypted schemes schemes, where the leakage is over a structure that
holds all the original data (e.g., an inverted index with full indexing), the
leakage in this case is only over a data structure that holds a random subset
of the data.</p>
<p>Finally, We implemented all our constructions and verified their efficiency
experimentally.</p>
<h2 id="conclusions-and-future-work">Conclusions and Future Work</h2>
<p>I went over our graph encryption schemes with support for approximate shortest distance
queries. The solutions I described are all adaptively-secure. Of course, there
are other possible approaches based on ORAM or FHE which can provide stronger
security (even hide access pattern!) but at a higher cost. As graph databases become more and more
popular, I believe graph encryption will play an increasingly important role in
database security. We live in a data-centric world that generates network and
graph data of all kinds. There are still more challenging and exciting open
problems in graph database security: e.g., how to construct graph encryption
schemes for more complex graph queries? Can we support graph mining tasks,
e.g., can we construct graph encryption schemes that allow us to detect
communities over encrypted social networks? And of course, as is common in
encrypted search, how can we quantify the security of our graph encryption
schemes? Any briliant ideas? Talk to us! :-)</p>
Applied Crypto Highlights: Searchable Encryption with Ranked Results
http://senykam.github.io/2015/04/15/applied-crypto-highlights-searchable-encryption-with-ranked-results
Wed, 15 Apr 2015 20:57:14 -0300http://senykam.github.io/2015/04/15/applied-crypto-highlights-searchable-encryption-with-ranked-results<p><em>This is the second in a series of guest posts highlighting new research in
applied cryptography. This post is written by <a href="http://www.baldimtsi.com/">Foteini
Baldimtsi</a> who is a postdoc at Boston University and
<a href="http://research.microsoft.com/en-us/people/oohrim/">Olya Ohrimenko</a> who is a
postdoc at Microsoft Research. Note that Olya is on the job market this year.</em></p>
<p><img src="http://senykam.github.io/img/steam.jpg" class="alignright" width="250">
Modern cloud services let their users outsource data as well as request
computations on it. Due to potentially sensitive content of users' data and
distrust in cloud services, it is natural for users to outsource their data
encrypted. It is, however, important for the users to still be able to use
cloud services for performing computations on the encrypted data. In this
article we consider an important class of such computations: search over
outsourced encrypted data. Searchable Encryption has attracted a lot of
attention from research community and has been thoroughly described by Seny
in <a href="http://outsourcedbits.org/2013/10/06/how-to-search-on-encrypted-data-part-1">previous blog posts</a>.</p>
<p>Search functionality alone, however, might not be enough when one considers a
large amount of data. Ideally, users would like to not only receive the
matching results, but get them back sorted according to how relevant they are
to their query (just like a search engine does!). In this blog post we describe
our <a href="http://fc15.ifca.ai/preproceedings/paper_89.pdf">recent result</a> from
the conference on Financial Cryptography and Data Security 2015 which builds on
top of searchable encryption techniques to return <em>ranked results</em> to
users' queries. Our goal is to create a scheme that is efficient and achieves a
high level of privacy against a curious cloud server.</p>
<h2 id="ranking-search-results-on-plaintext-data">Ranking search results on plaintext data</h2>
<p>Let us start by briefly describing how ranking would be done if users did not
take into account the privacy of their data and outsourced it in an unencrypted
format. Literature on information retrieval offers an abundance of ranking
methods. For our paper, we chose the $\mbox{tf-idf}$ ranking method due to its
simplicity, popularity and the fact that it supports free text queries. This method
is effective since it is based on term/keyword frequency (tf) and inverse
document frequency (idf).</p>
<p>Let $D=D_1,\dots,D_n$ be a document collection of $n$ documents, in which there
exist $m$ unique terms/keywords $t_1,\dots,t_m$. First, for every term, $t$, we
compute its frequency ($\mbox{tf}$) in each document $D_i$ as well as its inverse
document frequency ($\mbox{idf}$) in the entire collection (it captures how common the term is
in the whole document collection). Then, for each term and document we compute</p>
<p><span class="math">\[
\mbox{tf-idf}_{t,D_i} = \mbox{tf}_{t,d} \times \mbox{idf}_{t}
\]</span></p>
<p>and store the score values in the rank table, $T$:</p>
<p><figure><img src="http://senykam.github.io/img/searchindextable.jpg" alt="" title="$\mbox{tf-idf}$ rank table, $T$, outsourced to the cloud"><figcaption>$\mbox{tf-idf}$ rank table, $T$, outsourced to the cloud</figcaption></figure></p>
<p>Note that if a term does not appear in a table, then we store $0.00$ as a rank.
This table could be either computed by the owner of the document collection and
outsourced to the cloud, or computed by the cloud itself since it
receives the actual document collection $D$ in the clear.</p>
<p>Now suppose that a user wants to query the cloud for the multi-keyword query "searchable
encryption". Then, the cloud first searches for the terms "searchable" and
"encryption" in the table, adds the corresponding rows together to get the
overall score of the query, sorts the scores, and returns the relevant
documents in a sorted order.</p>
<h2 id="ranking-search-results-on-encrypted-data">Ranking search results on encrypted data</h2>
<p>A user that wishes to protect her privacy is likely to outsource her document
collection to the cloud in an encrypted format: $E(D_1),\dots,E(D_n)$. In order
to be able to perform <em>ranked search</em>, the user has to create the rank
table $T$ and send it to the cloud (as opposed to outsourcing plaintext data
where the cloud could also compute the rank table itself). Since the rank table
contains information about the distribution of words in individual documents and
the whole collection, it has to be encrypted as well. However, in order for the
server to be able to return ranked results using the $\mbox{tf-idf}$ method
described above, encrypted$T$ should be able to support the following
operations:</p>
<ol>
<li>search for terms/keywords</li>
<li>add numerical values</li>
<li>sort a list of numerical values.</li>
</ol>
<p>For the first operation one could simply encrypt the keywords on the table
using a <a href="http://outsourcedbits.org/2013/10/06/how-to-search-on-encrypted-data-part-1/">searchable
encryption</a>
(SE) scheme. Then, whenever the user wants to search for a phrase, she sends to
the cloud an SE trapdoor for each keyword in the phrase. The server can then
use the trapdoors to locate the keywords in the table.</p>
<p>The next two operations refer to the numerical entries on the table which
should be encrypted in a way that supports addition and sorting. A natural
solution would be to encrypt these values under a <a href="http://outsourcedbits.org/2012/06/26/applying-fully-homomorphic-encryption-part-1/">fully-homomorphic
encryption</a>
scheme that can support any type of computation over encrypted data. However,
the resulting solution would be very inefficient to be applied in practice.
Another potential solution would be to encrypt the numerical values under an
<a href="http://www.cc.gatech.edu/aboldyre/papers/bclo.pdf">order-preserving
encryption</a> (OPE) scheme.
However, this would be sufficient only for single-keyword queries, since OPE
schemes cannot support homomorphic addition (and, even if they did, they would
<a href="http://luca-giuzzi.unibs.it/corsi/Support/papers-cryptography/RAD78.pdf">not be
secure</a>.
Note that for single keyword queries, OPE might not be ideal since it leaks the
rank order of the documents for each keyword (see also the discussion
<a href="http://outsourcedbits.org/2013/10/14/how-to-search-on-encrypted-data-part-2/">here</a>.</p>
<p>Given that we aim for an efficient and provably secure
solution, we propose to encrypt the numerical values of the rank table using
the <a href="http://en.wikipedia.org/wiki/Paillier_cryptosystem">Paillier encryption
scheme</a>: a semi-homomorphic
scheme that supports the addition of encrypted values. (For the rest of this
post, we use $[a]$ to denote the encryption of value $a$ using this scheme.) %a
semi-homomorphic encryption scheme. By the properties of Paillier, the server
can add the corresponding rows of $T$ when a query is received. What is still
left to discuss is, how the server can also sort these encrypted values. In
the rest of the post, we describe our private sorting mechanism over encrypted
values.</p>
<p>Our private sorting mechanism requires to equip the cloud server with a secure
co-processor (e.g., <a href="http://www-03.ibm.com/security/cryptocards/pciecc/overview.shtml">IBM
PCIe</a>, <a href="https://software.intel.com/en-us/blogs/2013/09/26/protecting-application-secrets-with-intel-sgx">Intel
SGX</a>,
<a href="https://technet.microsoft.com/en-us/library/cc749022%28v=ws.10%29.aspx">Windows
TPM</a>.
The secure co-processor is then given the decryption key of the
semi-homomorphic encryption scheme which lets him assist the cloud server in
sorting. For the protocol to proceed, we assume that the co-processor does not
collude with the cloud server and both of them are following the protocol in an
honest-but-curious way. That is, neither of them deviates from the protocol but
both are curious to learn more about user's data.</p>
<p><figure><img src="http://senykam.github.io/img/introimagesingleuserslim.jpg" alt="" title="An overview of the interactions between the user, the cloud server $S_1$ and the co-processor $S_2$."><figcaption>An overview of the interactions between the user, the cloud server $S_1$ and the co-processor $S_2$.</figcaption></figure></p>
<p>Regarding the privacy of our scheme, we design our protocol in such a way that:
(a) the co-processor learns nothing about the values being sorted and (b) the
cloud server, as in SE, learns the search pattern (i.e., whether a keyword was
queried before or not), but learns nothing about the ranking of the documents.
For example, he does not learn which document ranks higher for user's query.</p>
<h2 id="private-sort">Private Sort</h2>
<p>We now develop a sorting protocol that the cloud server and the co-processor
can use to jointly sort encrypted ranking data of the documents. From now on we
denote the cloud server by $S_1$ and the co-processor by $S_2$. Our private
sort is a two-party protocol between $S_1$ and $S_2$ where $S_1$ has an
encrypted array of $n$ elements $[A] = { [A_1], [A_2], \ldots, [A_n]}$ and
$S_2$ has the secret key that can decrypt $A$.<br>
By the end of the protocol, $S_1$ should obtain $[B] = {[B_1],
[B_2], \ldots, [B_n]}$ where $[B]$ is an encryption of$A$ sorted. Since $S_1$
and $S_2$ are both curious, we are interested in protecting the content of $A$
and$B$ from both of them and we are willing to reveal <em>only</em> the size of
$A$,$n$. Hence, $S_2$ should only assist $S_1$ in sorting without seeing the
encrypted content of $A$ or $B$, otherwise he can trivially decrypt it. On the
other side of the protocol, nothing about the decryption key nor plaintext
values of $A$ and $B$ should be leaked to $S_1$. For example, we do not want
to leak to neither $S_1$ nor $S_2$ values of elements in $A$, their comparison
result with other elements, and their new location in $B$ (in the paper we
express these properties using simulation based security definitions).</p>
<p><strong>Private Sort Construction Overview.</strong>
As can be seen from the definitions, the participation of $S_1$ and $S_2$ in
private sort should not reveal anything about the content of the data to either
of them. Hence, any method we use for comparison and sorting must appear
independent of the data. We note, however, that many sorting algorithms access
the data depending on the comparison result and data content (e.g., quicksort).
This does not fit our model where everything about the data, including
individual comparisons, should be protected from $S_1$ and $S_2$.</p>
<p>Fortunately, there are sorting algorithms where data comparisons are determined
by the size of the data to be sorted, $n$ in our case, and not the data
content. One such algorithm is a
<a href="http://dl.acm.org/citation.cfm?id=1468121">sorting network</a> by K.
Batcher which relies on Two-Element Sort circuit. This circuit takes two
elements and outputs them in a sorted order.
Then the network consists of $O(\log n)$ layers where every layer has $O(n)$ Two-Element Sort circuits,
where constants in big-O are determined by $n$.
In order to sort the data, one simply passes it through the network.
Moving the data through the network depends only on $n$ and the Two-Element Sort.
Hence, if we develop a private Two-Element Sort, the implementation
of private Batcher's network becomes trivial.</p>
<h3 id="private-twoelement-sort">Private Two-Element Sort</h3>
<p>As the name suggests, Private Two-Element Sort is a special case of Private Sort, as defined above, for the case $n=2$. That is, $S_1$ has two encrypted elements $[a]$ and $[b]$ and wishes to obtain $[c]$ and $[d]$ where $c = \min(a,b)$ and $d = \max(a,b)$. Similarly,$S_2$ has the secret key of the encryption. The security definition is also the same and informally states that neither $S_1$ nor $S_2$ learn anything about $a$ and $b$.</p>
<p>We first describe operations that are required to perform Two-Element Private Sort without encryption and then for every operation give its private version. The sorting consists of:</p>
<ol>
<li>$t := a > b$ (Set bit $t$ to the result of comparing $a$ and $b$).</li>
<li>$c := (1-t)a + tb$ (Use $t$ to select the minimum of $a$ and $b$).</li>
<li>$d := ta + (1-t)b$ (Use $t$ to select the maximum of $a$ and $b$).</li>
</ol>
<p>Note that these three operations have to be performed on encrypted data:
$a$ and $b$ are part of the encrypted input of $S_1$,
bit $t$ and values $c$ and $d$ also should be encrypted to protect their content from $S_1$.
Moreover, neither of these values should be shown to$S_2$ since he can trivially decrypt them,
violating privacy guarantees against$S_2$.</p>
<p>We show how to perform above operations over encrypted data one by one, starting first with
a <em>Private Comparison</em> protocol for computing $[t]$ and following with
a <em>Private Select</em> protocol for computing $[c]$ and $[d]$.</p>
<p><strong>Private Comparison.</strong>
This protocol is a variation of a classical Andrew Yao's <a href="http://research.cs.wisc.edu/areas/sec/yao1982-ocr.pdf">Millionaire's problem</a>:
$S_1$ has $[a]$ and $[b]$ and wishes to obtain $[t]$, where
$t = (a > b)$ and $S_2$ has the private key of the encryption scheme.
Although there is more than one way of doing so, we pick an efficient
algorithm from a recent result by <a href="http://www.internetsociety.org/sites/default/files/04_1_2.pdf">Bost et al.</a>, which is a correction of the original <a href="http://bioinformatics.tudelft.nl/sites/default/files/Comparing%20encrypted%20data.pdf">protocol</a> by T.Veugen.
This algorithm lets $S_1$ and $S_2$ compare $a$ and $b$ using
number of interactions that is logarithmic in the number of bits in each element.</p>
<p>Note that neither $S_1$ nor $S_2$ learn the values of $a$, $b$, and $t$.
In addition, $S_2$ does not learn the ciphertexts corresponding to these values.</p>
<p><strong>Private Select.</strong>
Given the comparison bit $t$, we now devise a private algorithm for using this
bit to select the minimum and the maximum value of $a$ and $b$ (that is
performing operations 2 and 3 above). Recall that $S_1$ has to obtain $[c]$
and $[d]$ with $S_2$ "blindly" assisting him in the protocol.</p>
<p>We wish to use simple cryptographic operations in order to compute $c$ and $d$.
That is, we use semi-homomorphic cryptographic techniques as opposed to
fully-homomorphic ones. To this end, we use an interesting property of layered
Paillier Encryption. We omit many details from here and only point out the
features that we need.</p>
<p>We denote messages encrypted using first and second layers of Paillier Encryption as
$[m]$ and $[![m]!]$, respectively.
We recall that Paillier Encryption supports addition of ciphertexts as well
as multiplication by a constant, i.e., $[m_1][m_2] = [m_1+m_2]$ and $[m]^{C} = [Cm]$.
The same operations hold for ciphertexts of the second layer.
However, what is more interesting is that the ciphertext of the first layer
is in the same domain as the plaintext of the second layer, which
allows the following operations:</p>
<p>This trick allows us to implement the functionality of private select for $c$,
and similarly for $d$, as follows:</p>
<p><span class="math">\[[\![[c]]\!] := [\![[a]]\!]^{[1-t]} [\![[b]]\!]^{[t]} = [\![[(1-t)a + tb]]\!]\,\]</span></p>
<p>where $c$ and $d$ are doubly encrypted.</p>
<p>Recall that the output of Two-Element Private Sort is a building block of the
general sort, where $c$ and $d$ participate in further invocations of
Two-Element Private Sort. To make the values $c$ and $d$ usable in the next
layer of Batcher's network, $S_1$ uses $S_2$ to strip off the extra layer of
encryption. $S_1$ blinds the value he needs to strip via $[![[c+r]]!]$, and
sends it to $S_2$, who decrypts the ciphertext and sends back only $[c+r]$.
Using homomorphic properties of Paillier, $S_1$ subtracts $r$ to get $[c]$.
The similar protocol is executed for $d$. %Note that this protocol requires
one interaction with $S_2$.</p>
<h3 id="private-nelement-sort">Private $n$-Element Sort</h3>
<p>Let us now show how to sort an array of $n$ elements using our Private Two-Element Sort.
%We are now ready to combine all the building blocks.
$S_1$ executes Batcher's sorting network layer by layer.
For each layer in the network and for every sorting gate in this layer,
he engages with $S_2$ in Private Two-Element Sort.
He uses the outputs of this layer as inputs to the next layer
of the network. (See Figure\ref{fig:batcher} for an illustration.)</p>
<p><figure><img src="http://senykam.github.io/img/batcher1.jpg" alt="" title="Example of privately sorting an encrypted array of four elements $5,1,2,9$ where $[m]$ denotes a Paillier encryption of message $m$ and $\mathsf{pairs}_i$ denotes a pair of elements to be sorted. Note that only $S_1$ stores values in the arrays $A_i$ while $S_2$ blindly assists $S_1$ in sorting the values."><figcaption>Example of privately sorting an encrypted array of four elements $5,1,2,9$ where $[m]$ denotes a Paillier encryption of message $m$ and $\mathsf{pairs}_i$ denotes a pair of elements to be sorted. Note that only $S_1$ stores values in the arrays $A_i$ while $S_2$ blindly assists $S_1$ in sorting the values.</figcaption></figure></p>
<p><strong>Sketch of Privacy Analysis.</strong>
We note that the number of times $S_1$ engages with $S_2$ in the protocol does
not reveal either of them anything about the data content. Each engagement is
an execution of Private Two-Element Sort which, in turn, is a call to Private
Comparison and two calls to Private Select. Private comparison guarantees
privacy against $S_1$ and $S_2$ as long as they are non-colluding honest
adversaries. Private select relies on homomorphic properties of Paillier and
requires only the re-encryption step from $S_2$. Since $S_2$ receives a
blinded value he does not learn the value of $c$ or $d$. Moreover, since the
values of $c$ and $d$ are re-randomized we can treat $O(n (\log n)^2)$ calls to
Two-Element Private Sort independently.</p>
<h2 id="conclusion">Conclusion</h2>
<p>We constructed a private sort mechanism that allows a cloud server $S_1$ to sort
a list of encrypted data without learning anything about their order (while
assisted by a non-colluding co-processor $S_2$). As discussed in the beginning
of our post, this tool lets a user store his encrypted documents in
a cloud server and receive ranked results when searching on them.</p>
<p>The method, as described in the post, assumes that the rank table has an entry
for every keyword-document pair, even if a keyword does not appear in
this document zero is stored.
In the <a href="https://eprint.iacr.org/2014/1017">full version</a> of the paper, we show that
we can relax this requirement and store only information for documents where
keyword appears, hence, significantly reducing the size of $T$ and query time for the server.
If we do so, we can add ranking to the optimal SE technique by <a href="http://research.microsoft.com/apps/pubs/?id=102088">Curtmola et al.</a> for single keyword queries or to the technique by <a href="https://eprint.iacr.org/2013/169">Cash et al.</a>
for efficiently answering Boolean queries on encrypted data (see earlier <a href="http://outsourcedbits.org/2014/08/21/how-to-search-on-encrypted-data-searchable-symmetric-encryption-part-5/#comment-2512">blog post</a> for more details on each).
Although the resulting scheme gives a significant performance
improvement and protects the ranking of the documents,
it inherits the leakage of the access pattern (i.e., identifiers of the documents where each query keyword appears)
from the corresponding SE technique.</p>
<p>Our work leaves several interesting open questions, including:
how to efficiently update the collection?
how can a user verify the ranking result it receives?
is a non-colluding co-processor provably necessary to solve multi-keyword
ranked search? Any ideas? :)</p>