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The secure link prediction problem

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  • Link Prediction is an important and well-studied problem for social networks. Given a snapshot of a graph, the link prediction problem predicts which new interactions between members are most likely to occur in the near future. As networks grow in size, data owners are forced to store the data in remote cloud servers which reveals sensitive information about the network. The graphs are therefore stored in encrypted form.

    We study the link prediction problem on encrypted graphs. To the best of our knowledge, this secure link prediction problem has not been studied before. We use the number of common neighbors for prediction. We present three algorithms for the secure link prediction problem. We design prototypes of the schemes and formally prove their security. We execute our algorithms in real-life datasets.

    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

    Citation:

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  • Figure 1.  System model

    Figure 2.  Example of a Maximum circuit with $ N = 7 $

    Figure 3.  Different max blocks used in $ \mathtt{MAXIMUM} $ circuit

    Figure 4.  Few circuit blocks

    Figure 5.  Number of vertices and edges of the subgraphs

    Figure 6.  comparison between $ \mathtt{SLP} $-$ \mathtt{I} $ and $ \mathtt{SLP} $-$ \mathtt{II} $ w.r.t. computation time when the primes are of 128 bits each

    Figure 7.  Time taken by the proxy in $ \mathtt{SLP} $-$ \mathtt{II} $ for different datasets considering 128-bit primes

    Figure 8.  Computational time in $ \mathtt{SLP} $-$ \mathtt{I} $ with 128,256 and 512-bit primes

    Figure 9.  Computational time in $ \mathtt{SLP} $-$ \mathtt{II} $ with 128,256 and 512-bit primes

    Table 1.  Complexity Comparison Table

    Param Entity $ \mathtt{SLP} $-$ \mathtt{I} $ $ \mathtt{SLP} $-$ \mathtt{II} $ $ \mathtt{SLP} $-$ \mathtt{III} $
    Leakage CS $ |V| $, $ \tau_{v_1},\tau_{v_2},\ldots $ $ |V| $, $ \tau_{v_1},\tau_{v_2},\ldots $ $ |V| $, $ \tau_{v_1},\tau_{v_2},\ldots $
    PS $ S_{v},i_{res} $ $ S'_{v},i_{res} $ $ i_{res} $
    client $ \lambda $ bits $ \lambda $ bits $ \lambda $ bits
    Storage CS $ |V|^2\rho $ bits $ 2|V|^2\rho $ bits $ |V|^2\rho $ bits
    PS $ \rho $ bits $ \rho $ bits $ \rho $ bits
    client $ |V|^2(\mathsf{M}+\mathsf{A}) $ $ |V|^2(\mathsf{M}+\mathsf{A}+\mathsf{M_1}+\mathsf{A_1}) $ $ |V|^2(\mathsf{M}+\mathsf{A}) $
    Compu- CS $ |V|^2 $ $ \mathsf{P} $ + $ |V| $ $ \mathsf{E} $ $ |V|^2 $ $ \mathsf{P} $ + $ |V|^2 $ $ \mathsf{P} $ + $ 4|V| $ $ \mathsf{E} $
    tation + ($ |V|^2+ |V| $) $ \mathsf{M} $ ($ |V|^2+ 2|V| $) $ \mathsf{M} $ + ($ |V|^2+ 3|V| $) $ \mathsf{M} $ +
    $ MGC_{const}{(\log |V|,|V|)} $
    PS $ |V|log|V| (\mathsf{M+C}+\mathsf{M_1+C_1}) $ $ |V| (\mathsf{M_1+C_1}) $ + $ |V| (\mathsf{M+C}+\mathsf{M_1+C_1}) $+
    +$ |V|log|V| \mathsf{C} $ +$ |V| log|V| \mathsf{C} $ $ MGC_{eval}{(\log |V|,|V|)} $
    client$ \rightarrow $CS $ |V|^2 \rho $ bits $ 2 |V|^2 \rho $ bits $ |V|^2\rho $ bits
    Commu- CS$ \rightarrow $PS $ 2|V|\rho $ bits $ |V|\rho $ bits $ 2|V|\rho $ bits + $ |V|OT ^{(\log |V| +1)}_{snd} $+
    nication $ MGC_{size}{(\log |V|,|V|)} $ bits
    PS$ \rightarrow $CS - - $ |V|OT ^{(\log |V| +1)}_{rcv} $
    PS$ \rightarrow $client $ \log |V| $ bits $ 2|V| \log |V| $ bits $ \log |V| $ bits
    $S_{v}$ - Set of scores of $v$ with all other vertices, $S'_{v} $- a subset of $ S_{v}$, $\rho $- length of elements in $\mathbb{G}$ or $\mathbb{G}_1$, $\mathsf{C}$- comparison in $\mathbb{G}$, $\mathsf{C_1}$- comparison in $\mathbb{G}_1$, $\mathsf{M}$- multiplication in $\mathbb{G}$, $\mathsf{M_1}$- multiplication in $\mathbb{G}_1$, $\mathsf{E}$- exponentiation in $\mathbb{G}$, $\mathsf{E_1}$- exponentiation in $\mathbb{G}_1$, $\mathsf{P}$- pairing/ bilinear map computation, $MGC_{size}{(\log |V|,|V|)}$- size of $MGC$ with $|V|$ $\log |V|$-bit inputs, $MGC_{const}{(\log |V|,|V|)}$- $MGC$ contraction with $|V|$ $\log |V|$-bit inputs, $MGC_{eval}{(\log |V|,|V|)}$- $MGC$ evaluation with $|V|$ $\log |V|$-bit inputs, $OT ^{(\log |V| +1)}_{snd}$- information to send for $(\log |V|+1) $-bit $OT$, $OT ^{(\log |V| +1)}_{rcv}$- information to receive for $\log |V| $-bit $OT$.
     | Show Table
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    Table 2.  Detail of the graph datasets

    Dataset Name #Nodes #Edges
    bitcoin-alpha 3,783 24,186
    ego-facebook 4,039 88,234
    email-Enron 36,692 183,831
    email-Eu-core 1,005 25,571
    Wiki-Vote 7,115 103,689
     | Show Table
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