Priority-Aware Private Matching Schemes for Proximity-Based Mobile Social Networks

Let Alice’s attribute set and corresponding priority vector be $X$ and $A$ respectively, and Bob’s attribute set and corresponding priority vector be $Y$ and $B$ respectively. The set of common attributes between Alice and Bob is denoted as $S=X\cap Y$, where $q=|S|$ and $S=\{s_{1},s_{2},\ldots,s_{q}\}$. Then we arrange the corresponding priorities of Alice and Bob on the common attributes into vector $V_{A}=(a_{1},a_{2},\ldots,a_{q})$ and $V_{B}=(b_{1},b_{2},\ldots,b_{q})$, respectively. The most widely applied similarity function is cosine similarity:

where $\theta$ is the angle between $V_{A}$ and $V_{B}$. It is often used to measure the angular similarity between two vectors. However, cosine similarity is orthogonal to the priorities on a common attribute. That is cosine similarity can be high when the priorities on the common attribute are quite different or exactly the same. This implies its defect. We do not accept the high similarity if the priorities differs far from each other. Thus, it is not the best choice in our scenario.

Different from cosine similarity, Jaccard similarity coefficient better fits our scenario, which considers the quotient of the size of the intersection over the size of the union of $X$ and $Y$. To simplify the computation we employ its variant form, Tanimoto similarity coefficient [22]:

where $V_{A}$ and $V_{B}$ are the same as in (3). The inner product appears in the numerator and the denominator of (4) displays the difference between $V_{A}$ and $V_{B}$, and the norm term adjusts the size of the unit. After these refinement, Tanimoto similarity coefficient can embody the difference between two priority set on common attributes.

Based on the commutative encryption function [17] and Tanimoto similarity coefficient, we design our basic privacy-aware private matching scheme, P-match, which considers both the number of common attributes and the corresponding priorities on them. As an initialization, users encrypt their attributes and priorities under their secret keys. Fig. 2 shows the details and the procedure is as follows:

1. (i)

   When two users are within the communication range of each other, the initiator Alice begins the matching process by broadcasting a message $\langle h_{p}(x_{i})^{K_{A}},a_{i}^{k_{A}}\rangle$;

2. (ii)

   as one of the responders, Bob replies Alice with $h_{p}(y_{i})^{K_{B}}$;

3. (iii)

   based the received message, Alice computes $(h_{p}(y_{i})^{K_{B}})^{K_{A}}$ and sends the two-tuple $\langle h_{p}(y_{i})^{K_{B}},(h_{p}(y_{i})^{K_{B}})^{K_{A}}\rangle$ together to Bob;

4. (iv)

   Bob finds matches and creates a list $L_{1}=\langle(h_{p}(y_{i})^{K_{B}})^{K_{A}},y_{i}\rangle$, and computes a set $F_{1}=\langle(h_{p}(x_{i})^{K_{A}})^{K_{B}}\rangle$. Then, he compares the first element in $L_{1}$ with $F_{1}$ to compute the common attributes with Alice, the common ones form as set $S$, i.e., $S=X\cap Y$. Bob creates another list $L_{2}=\langle(h_{p}(s_{i})^{K_{A}})^{K_{B}},s_{i},b_{i}\rangle,\forall s_{i}\in S$ unless $S=\emptyset$, and sends back the $\langle h_{p}(x_{i})^{K_{A}},(a_{i}^{k_{A}})^{k_{B}}\rangle$;

5. (v)

   Alice computes her $k_{A}^{\prime}$ by Extended Euclidean algorithm and decrypts the second part of the received message, then sends out the results $\langle h_{p}(x_{i})^{K_{A}},a_{i}^{k_{B}}\rangle$;

6. (vi)

   upon the received messages, Bob creates another list $L_{3}=\langle(h_{p}(x_{i})^{K_{A}})^{K_{B}},a_{i}^{k_{B}}\rangle$, then compares the first element in $L_{3}$ with $(h_{p}(s_{i})^{K_{A}})^{K_{B}}$ in $L_{2}$. He obtains the corresponding $a_{i}$ by comparing if the received $a_{i}^{k_{B}}=b_{i}^{k_{B}}$ when he can find $(h_{p}(x_{i})^{K_{A}})^{K_{B}}\in L_{2}$, otherwise, discards them. When he gets these information, another list is created as $L_{4}=\langle(h_{p}(s_{i})^{K_{A}})^{K_{B}},a_{i},b_{i}\rangle$. Then for $q=|S|$, Bob uses priorities on each common attribute of Alice and himself to build two vectors $V_{A}=(a_{1},a_{2},\cdots,a_{q})$ and $V_{B}=(b_{1},b_{2},\cdots,b_{q})$, respectively. To measure the similarity, he computes the Tanimoto Coefficient $T(V_{A},V_{B})=\frac{V_{A}\cdot V_{B}}{||V_{A}||^{2}+||V_{B}||^{2}-V_{A}\cdot V_{B}}$, and compares with the threshold $t$ which is predefined by Bob. If $T(V_{A},V_{B})<t$, the process terminates, otherwise, Bob replies Alice with $T(V_{A},V_{B})$.

As the initiator may receive several replies from others, a ranking of Tanimoto Coefficient is provided for the initiator to choose the best match.

By utilizing P-match, we achieve Privacy Level I. Bob can learn the common attributes with Alice, as well as the corresponding priorities, while Alice learns nothing except the similarity value. However, there may be a problem when the scenario changes, i.e., Alice wants to know some knowledge since Bob may cheat on her. Thus, we propose an enhanced version P-match⁺.

When we only consider the common attributes and their priorities, Tanimoto similarity coefficient is energetic, but not effective enough if all the attributes and priorities are taken into account, because it is impossible to invent an efficient way to put Alice’s and Bob’s all attributes in the same order if $X\neq Y$. Hence we need another function, Ochiai similarity coefficient [23]

This coefficient was firstly applied in biology and is useful in comparing the faunistic feature between two different localities. It is also a ratio, where the numerator is the size of the intersection of two sets $A$, $B$, and the denominator is the geometric average between the size of $A$ and $B$. However, since all the priorities, on common and uncommon attributes, are considered here, and we cannot do the intersection between two ordered priority sets, we need some more effective similarity function. For that purpose, we notice that our priority is a kind of weight function. For a finite set $Z=\{z_{1},z_{2},\ldots,z_{\lambda}\}$ where every $z_{i}$ has the weight $w(z_{i})$, $i=1,2,\ldots,\lambda$, let the weight function be $w:Z\rightarrow\mathbb{I}$, then the un-weighted sum on $Z$ is defined by $\sum^{\lambda}_{i=1}z_{i}$ while the weighted sum on $Z$ is defined by $\sum^{\lambda}_{i=1}{z_{i}w(z_{i})}$. If we let $Z=X$, $\mathbb{I}=\{1,\ldots,9\}$ and $f(x_{i})=a_{i}$, then the weighted model becomes our scenario. This implies that we can regard the priority as a counting rule of an attribute, which is, we count an attribute $x_{i}$ $a_{i}$ times.

Therefore, we construct a new priority-aware coefficient $P(A,\,B)$ based on Ochiai similarity coefficient and the weighted sample. For two attribute sets $X=\{x_{1},x_{2},\ldots,x_{n_{1}}\}$ and $Y=\{y_{1},y_{2},\ldots,y_{n_{2}}\}$, the corresponding priority vectors are $A=(a_{1},a_{2},\cdots,a_{n_{1}})$ and $B=(b_{1},b_{2},\cdots,b_{n_{2}})$, let $S=X\cap Y=\{s_{1},\ldots,s_{q}\}$. First we generate two counting sets:

Then

Next we find the intersection of $X^{\prime}$ and $Y^{\prime}$,

where every $c_{i}=\min\{a_{i},b_{i}\}$ for $i=1,\ldots,q$. Then

Provided (8) and (10), now we apply the Ochiai similarity coefficient to the two counting sets $X^{\prime}$ and $Y^{\prime}$ given by (4.2.1) and (4.2.1), and define it to be our new similarity function:

The range of the priority-aware similarity coefficient (11) is from $0$ to $1$, which are corresponding to the cases ”no common attribute at all” for 0 and ”same attributes, same priorities” for 1.

To achieve Privacy Level II, we change some procedures from step vi in our basic version.

Algorithm 1 shows the changes on responder side in our P-match⁺. When Bob receives $\langle h_{p}(x_{i})^{K_{A}},a_{i}^{k_{B}}\rangle$ from Alice, he computes $(h_{p}(x_{i})^{K_{A}})^{K_{B}}$ and sends them back to Alice together with $h_{p}(x_{i})^{K_{A}}$. He can also decrypt $a_{i}$ by computing $k_{B}^{\prime}$. Then he creates a list $L_{3}=\langle(h_{p}(x_{i})^{K_{A}})^{K_{B}},a_{i}\rangle$ and compares the first element in $L_{3}$ with $(h_{p}(s_{i})^{K_{A}})^{K_{B}}$ in $L_{2}$. Followed, he computes $c_{i}=\min\{a_{i},b_{i}\}$ for those $(h_{p}(x_{i})^{K_{A}})^{K_{B}}$ in $L_{2}$, the priority-aware Ochiai Coefficient can be computed as $P(A,B)=\frac{\sum_{i=1}^{q}c_{i}}{\sqrt{\sum_{i=1}^{|X|}a_{i}}\cdot\sqrt{\sum_{i=1}^{|Y|}b_{i}}}$, where $q=|S|$. If $P(A,B)<t$, the algorithm terminates, otherwise, Bob replies $P(A,B)$ to Alice.

While on the initiator side, Alice needs to do some computation work to obtain the number of the common attributes, the details are shown in Algorithm 2. Based on the received message $\langle h_{p}(x_{i})^{K_{A}},(h_{p}(x_{i})^{K_{A}})^{K_{B}}\rangle$ from Bob, the algorithm creates a list of $\langle x_{i},(h_{p}(x_{i})^{K_{A}})^{K_{B}}\rangle$, and obtains the set of common attributes by comparing $(h_{p}(x_{i})^{K_{A}})^{K_{B}}$ in this list with $(h_{p}(y_{i})^{K_{B}})^{K_{A}}$, which is received before. It terminates if there is no match found by computing $(h_{p}(x_{i})^{K_{A}})^{K_{B}}=(h_{p}(y_{i})^{K_{B}})^{K_{A}}$, otherwise, outputs $|S|$ to Alice.

Suppose that the public database is $R=\{r_{i}\}^{n}_{i=1}$ consisting of attributes of all users. We let $\{h_{i}(\cdot)\}_{i=1}^{l}$ be a family of hash functions with $h_{j}(r_{i})\in[1,\lambda]$ for an attribute $r_{i}\in R$, and $\mathcal{H}$ be a large public pool of hash functions of such $h_{i}$ with each indexed by a unique identifier. An empty $\lambda$-bit Bloom filter is an array of $\lambda$ bits, all setting to $0$ bits. To add an element $r_{i}$ to the $\lambda$-bit Bloom filter, we set all the bits in $h_{j}(r_{i})$ positions $1$, $1\leq j\leq l$. To check whether an $r_{i}$ is some user’s attribute, we verify whether all the bits in positions $h_{j}(r_{i})$ are $1$. If not, $r_{i}$ is not this user’s attribute; otherwise, $r_{i}$ is his/her attribute with a probability determined by $n,\lambda$ and $l$.

We label each $r_{i}\in R$ by the 2-tuple $\{j,r_{i}(j)\}^{\kappa}_{j=1}$, where $r_{i}(j)=r_{i}+j-1$ is the counting function of $r_{i}$. Then the database $R$ is extended to the set $\{i,\{j,r_{i}(j)\}^{\kappa}_{j=1}\}^{n}_{i=1}$. Moreover, this extended set can be identified with the indexed set $R^{\prime}=\{\{i,j,r_{i}(j)\}^{\kappa}_{j=1}\}^{n}_{i=1}$. If Alice has an attribute set $X=\{x_{i}\}_{i=1}^{n_{1}}$ with the priority set $\{a_{i}\}^{n_{1}}_{i=1}$, we can assign her a personal set $S_{A}=\{\{i,j,x_{i}(j)\}^{a_{i}}_{j=1}\}^{n_{1}}_{i=1}$, where $x_{i}=r_{i^{\prime}}$ for an attribute $r_{i^{\prime}}\in R$, and $a_{i}\in\{1,2,\cdots,\kappa\}$ is the priority of $x_{i}$. The same technique can be applied to Bob, we get $S_{B}=\{\{i,j,y_{i}(j)\}^{b_{i}}_{j=1}\}^{n_{2}}_{i=1}$, where $y_{i}$ is in Bob’s attribute set $Y$, $|Y|=n_{2}$, $b_{i}$ is the priority of $y_{i}$. Denote $q_{1}:=|S_{A}|$, $q_{2}:=|S_{B}|$, then we have $q_{1}=\Sigma_{i=1}^{n_{1}}a_{i}=|X^{\prime}|$, $q_{2}=\Sigma_{i=1}^{n_{2}}b_{i}=|Y^{\prime}|$ with $X^{\prime}$ given by Equation 4.2.1, and $Y^{\prime}$ given by Equation 4.2.1.

We pair every $l$ a random number $l^{\prime}$, $1<l^{\prime}<l$, and publish the $2$-tuple $(l,l^{\prime})$ to all users. Now we can use a $\lambda$-bit Bloom filter to check how many attributes are in Alice’ set $X$ with her priorities, and how many are in Bob’s set $Y$ with his priorities, so that one of them can learn the similarity level.

We then present our proposed E-match shown in Fig. 3 in details.

1. (i)

   Alice sends a request to Bob.

2. (ii)

   Bob agrees.

3. (iii)

   Alice randomly chooses $\{h_{i}\}_{i=1}^{l}\subset\mathcal{H}$ with indexes denoted by $\mathcal{H}_{A}$. Alice adds each $\{i,j,x_{i}(j)\}$ in $S_{A}$ into a $\lambda$-bit Bloom filter array, denoted by $BF_{A}$, with $l^{\prime}$ different random hash functions in $\mathcal{H}_{A}$ and $(l-l^{\prime})$ random hash functions out of $\mathcal{H}$ (Computation offline above). Alice sends $\mathcal{H}_{A}$ and $BF_{A}$ to Bob.

4. (iv)

   Bob counts the number of $0$-bits in $BF_{A}$, denoted by $d_{1}$, then he adds every $\{i,j,y_{i}(j)\}$ in $S_{B}$ to $BF_{A}$ using $\mathcal{H}_{A}$ to get a $\lambda$-bit Bloom filter array, denoted by $BF_{B}$. He counts the number of $0$ bits $BF_{B}$, say, $d_{0}$, and computes

   $\displaystyle P^{*}(A,B)=\frac{\sqrt{l}[q_{2}-\lambda(\ln d_{1}-\ln d_{0})]}{l^{\prime}\sqrt{\lambda q_{2}(\ln\lambda-\ln d_{1})}},$

   (12)

   Bob compares $P^{*}(A,B)$ with his pre-defined threshold $t$, so that he decides whether to match Alice or not.