A Secure and Lightweight MultiParty Private IntersectionSum Scheme over a Symmetric Cryptosystem
Abstract
:1. Introduction
 We propose a secure and lightweight multiparty private intersectionsum scheme, called SLMPPIS, which avoids the data privacy leakage problem of only repeatedly conducting existing twoparty PIS schemes.
 SLMPPIS is clientagnostic. The requester can ask the cloud server to obtain the computation result without the help of data owners, and data owners can maintain their offline status as long as their data have been outsourced securely to the cloud.
 SLMPPIS is based on symmetric cryptosystem only. Therefore, the larger the number of participants, the more efficient SLMPPIS is. Specifically, when the number of participants is five, the efficiency can be increased by 22.98%.
2. Related Work
3. Preliminaries
3.1. Oblivious Transfer
 (1)
 Receiver’s indistinguishability security. For any $\sigma ,\tau \in \left\{0,1\right\}$ and for any probabilistic polynomial time (PPT) adversary $\mathcal{A}$ executing the sender’s part, the views that $\mathcal{A}$ sees in case the receiver tries to obtain ${m}_{\sigma}$ and in case the receiver tries to obtain ${m}_{\tau}$ are computationally indistinguishable given ${m}_{0}$ and ${m}_{1}$.
 (2)
 Sender’s indistinguishability security.For any adversary $\mathcal{A}$ substituting the receiver and a simulator ${\mathcal{A}}^{{}^{\prime}}$ playing the receiver’s role in the ideal model, the outputs of $\mathcal{A}$ and ${\mathcal{A}}^{{}^{\prime}}$ are statistically indistinguishable given ${m}_{0}$ and ${m}_{1}$.
3.2. Oblivious Pseudorandom Function
3.3. Arithmetic Sharing
 Shared Values: on input secret $x(x\in {Z}_{N})$, output sharing values ${\u2329x\u232a}_{0}^{A},{\u2329x\u232a}_{1}^{A}$ satisfying ${\u2329x\u232a}_{0}^{A}+{\u2329x\u232a}_{1}^{A}=x\phantom{\rule{3.33333pt}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}N$.
 Sharing: ${P}_{i}(i\in \left\{0,1\right\})$ chooses $r(r\in {Z}_{N})$ and sets ${\u2329x\u232a}_{i}^{A}=(xr)\phantom{\rule{3.33333pt}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}N$. Then, ${P}_{i}$ sends r to ${P}_{1i}$, that is, ${\u2329x\u232a}_{1i}^{A}=r\phantom{\rule{3.33333pt}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}N$.
 Reconstruction: ${P}_{1i}$ sends ${\u2329x\u232a}_{1i}^{A}$ to ${P}_{i}$, which computes $x=({\u2329x\u232a}_{0}^{A}+{\u2329x\u232a}_{1}^{A})\phantom{\rule{3.33333pt}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}N$.
4. Problem Formulations
4.1. System Model
 (1)
 Requester. The Re is responsible for submitting the processed data set to CS. In addition, the Re obtains the sharing value of the intersectionsum from the CS and computes the private intersectionsum result.
 (2)
 Cloud Server. The CS acts as a connection between the Re and DOs. After receiving the processed data set from the Re, the CS assists the DOs in performing data processing and obtaining the processed data sets from the DOs. In addition, the CS computes the sharing value of the intersectionsum and sends it to the Re.
 (3)
 Data Owners. Each DO holds a private set of data, and additionally holds a private integer value associated with each element. Each DO is responsible for submitting the processed data set to the CS.
4.2. Adversary Model
 (1)
 ${\mathcal{A}}_{1}$,${\mathcal{A}}_{2}$, and ${\mathcal{A}}_{3}$ may eavesdrop on all communication links to obtain data owned by participants.
 (2)
 ${\mathcal{A}}_{1}$ may compromise the CS to learn the data of the Re.
 (3)
 ${\mathcal{A}}_{2}$ may compromise the CS to learn the DOs’ data and associated values, or compromise one DO to learn the data and associated values of the other DOs.
 (4)
 ${\mathcal{A}}_{3}$ may compromise the CS and DOs to obtain the intersectionsum result.
5. The Protocol Framework
5.1. Initialization
5.2. Outsourcing Request
 (1)
 Generate the pseudorandom function value. The Re computes the pseudorandom function value $rand{f}_{{x}_{i}}$ of the data ${x}_{i}$ based on the group key K; namely, $rand{f}_{{x}_{i}}=F(K,{x}_{i})$.
 (2)
 Generate the intersection sharing value. For each datum ${x}_{i}$, the Re first computes the pseudorandom function value ${F}_{j}$ based on the symmetric key ${k}_{0j}(j\in \left[n\right])$; namely, ${F}_{j}=F({k}_{0j},{x}_{i})$. Then, the Re XORs multiple ${F}_{j}$ according to the formula (1) to generate the intersection sharing value $S{h}_{{x}_{i}}$:$$S{h}_{{x}_{i}}={F}_{1}\oplus \cdots \oplus {F}_{n}$$
 (3)
 Generate ciphertext and verification information for pseudorandom function values and intersection sharing values. The Re calculates the ciphertext $\left\{ciphe{r}_{R{e}_{1}},\dots ,ciphe{r}_{R{e}_{m}}\right\}$ and the root $roo{t}_{Re}$ of the Merkle tree according to the formula (2) based on pseudorandom function values and intersection sharing values $\left\{rand{f}_{{x}_{1}}\left\rightS{h}_{{x}_{1}},\dots ,rand{f}_{{x}_{m}}\left\rightS{h}_{{x}_{m}}\right\}$:$$\left\{\begin{array}{c}ciphe{r}_{R{e}_{i}}=Enc(rand{f}_{{x}_{i}}\left\rightS{h}_{{x}_{i}},{k}_{ac}),i\in \left[m\right]\hfill \\ roo{t}_{Re}=MTgen(rand{f}_{{x}_{1}}\left\rightS{h}_{{x}_{1}},\dots ,rand{f}_{{x}_{m}}\left\rightS{h}_{{x}_{m}})\hfill \end{array}\right.$$
Algorithm 1 Re’s data processing 

5.3. Data Submission
Algorithm 2 DO’s Data Processing 

 (1)
 Generate calculation value. For each datum ${y}_{i}$, the DO first computes the pseudorandom function value $rand{f}_{{y}_{i}}$ based on the group key K; namely, $rand{f}_{{y}_{i}}=F(K,{y}_{i})$. Then, the DO chooses a random number ${r}_{i}$ for associated datum ${t}_{i}$ and encrypts the difference between ${t}_{i}$ and ${r}_{i}$ based on $rand{f}_{{y}_{i}}$ to generate the calculation value $E{S}_{{y}_{i}}$; namely, $E{S}_{{y}_{i}}=Enc(RP{F}_{{y}_{i}},{t}_{i}{r}_{i})$.
 (2)
 Generate intersection sharing value. For each datum ${y}_{i}$, the DO first computes the pseudorandom function value ${F}_{j}$ based on the symmetric key negotiated with the Re and other DOs ${k}_{uj}(j\in \left[n\right],j\ne u)$; namely, ${F}_{j}=F({k}_{uj},{y}_{i})$. Then, the DO XORs multiple ${F}_{j}$ according to the formula (5) to generate the intersection sharing value $S{h}_{{y}_{i}}$:$$S{h}_{{y}_{i}}={F}_{0}\oplus \cdots \oplus {F}_{u1}\oplus {F}_{u+1}\cdots \oplus {F}_{n}$$
 (3)
 Generate calculation sharing value. For each datum ${y}_{i}$, the DO chooses a random number ${r}_{i}$ and generates calculation sharing value $E{S}_{{y}_{i}}$ based on the symmetric key ${k}_{0u}$ negotiated with the Re; namely, $E{R}_{{y}_{i}}=Enc({k}_{0u},{r}_{i})$.
Algorithm 3 Obtaining Shared Values 

 (1)
 The CS and DO perform an oblivious pseudorandom function algorithm ${F}_{OPRF}$. First, the CS takes the pseudorandom function value $rand{f}_{{x}_{i}}$.Then, as the output of ${F}_{OPRF}$, the DO receives the key ${k}_{OPRF}$, and the CS receives the value $O{F}_{{x}_{i}}$ corresponding to $rand{f}_{{x}_{i}}$; namely, $O{F}_{{x}_{i}}=F({k}_{OPRF},S{E}_{{x}_{i}})$. In particular, the CS can only obtain $O{F}_{{x}_{i}}$ corresponding to $rand{f}_{{x}_{i}}$, and cannot obtain the key ${k}_{OPRF}$.
 (2)
 The DO generates an oblivious pseudorandom function value. Based on the ${k}_{OPRF}$ output by ${F}_{OPRF}$ and the pseudorandom function value $rand{f}_{{y}_{i}}$, the DO generates the corresponding oblivious pseudorandom function value $O{F}_{{y}_{i}}$; namely, $O{F}_{{y}_{i}}=F({k}_{OPRF},S{E}_{{y}_{i}})$.
 (3)
 The DO generates the hash table T. Firstly, the DO generates hash value ${H}_{D{O}_{i}}$ based on the oblivious pseudorandom function value $O{F}_{{y}_{i}}$; namely, ${H}_{D{O}_{i}}=H\left(O{F}_{{y}_{i}}\right)$. Then, $T{h}_{D{O}_{i}}$ is generated based on $O{F}_{{y}_{i}}$ and intersection sharing value $S{h}_{{y}_{i}}$; namely, $T{h}_{D{O}_{i}}=O{F}_{{y}_{i}}\oplus S{h}_{{y}_{i}}$. In addition, for the binary bit string ${h}_{d}$, but not in $\left\{{H}_{D{O}_{1}},\dots ,{H}_{D{O}_{m}}\right\}$, the DO selects the random number r as ${h}_{d}$ corresponding to $T{h}_{d}$. Finally, the DO encrypts H and generates $ciphe{r}_{H}$ based on the session key ${k}_{ab}$, where $H=\left\{{H}_{D{O}_{1}},\dots ,{H}_{D{O}_{m}},{h}_{d}\right\}$. The DO takes $ciphe{r}_{H},Th$ as two columns to generate a hash table T and sends it to the CS.
 (4)
 The CS obtains the intersection sharing value. Firstly, the CS obtains the hash table T based on $ciphe{r}_{H}$ and ${k}_{ab}$. Then, based on the oblivious pseudorandom function value $O{F}_{{x}_{i}}$ and T, the CS obtains the intersection sharing value $S{h}_{C{S}_{i}}$; namely, $S{h}_{C{S}_{i}}=T{h}_{C{S}_{i}}\oplus O{F}_{{x}_{i}}$. According to the steps of the DO generation hash table T, it can be proven that $S{h}_{C{S}_{i}}$ satisfies the formula (6):$$S{h}_{C{S}_{i}}=\left\{\begin{array}{cc}T{h}_{CS}\oplus O{F}_{x}=O{F}_{y}\oplus S{h}_{y}\oplus O{F}_{x}=S{h}_{y},\hfill & x=y\hfill \\ T{h}_{CS}\oplus O{F}_{x}=r\oplus O{F}_{x},\hfill & x\ne y.\hfill \end{array}\right.$$
5.4. Outsourcing Response
 (1)
 Compute private set intersection. The CS, according to formula (7), executes XOR based on the intersection sharing value $Sh{x}_{i}$ generated by the Re and $S{h}_{C{S}_{i}}^{\left(u\right)}$, which is obtained by the interaction with each $D{O}^{\left(u\right)}$, and then generates $re{s}_{{x}_{i}}$. If $re{s}_{{x}_{i}}=0$, then the datum ${x}_{i}$ belong to the set intersection; namely, ${x}_{i}\in X\bigcap \left({\bigcap}_{u=1}^{n}{Y}^{\left(u\right)}\right)$:$$re{s}_{{x}_{i}}=Sh{x}_{i}\oplus S{h}_{C{S}_{i}}^{\left(1\right)}\oplus \cdots \oplus S{h}_{C{S}_{i}}^{\left(n\right)}$$
 (2)
 Compute private intersectionsum. The CS decrypts the calculation value $E{S}_{{x}_{i}}$ based on pseudorandom function value $rand{f}_{{x}_{i}}$; namely, $D{S}_{{x}_{i}}=Dec(rand{f}_{{x}_{i}},E{S}_{{x}_{i}})$. Then, the CS sums multiple calculation values to obtain $sum$.
 (3)
 Generate the ciphertext of the private sum result. Based on the private intersectionsum result $sum$ and the session key ${k}_{ac}$ negotiated with the Re, the CS generates the ciphertext of the private sum result; namely, $ciphe{r}_{res}=Enc(sum,{k}_{ac})$. Then, the CS generates $E{R}_{Re}$ based on calculated sharing values $E{R}_{{x}_{i}}$ corresponding to the intersection data ${x}_{i}$ and sends $\left\{ciphe{r}_{res},E{R}_{Re}\right\}$ to the Re.
Algorithm 4 Private Sum Computation 

6. Security Analysis
6.1. Security of Re’s Data
6.2. Security of DOs’ Data
6.3. Security of IntersectionSum Result
7. Performance Analysis
7.1. Experimental Settings
7.2. Experimental Results
7.2.1. Computational Costs in Different Phases
7.2.2. The Effect of $AES$ and $Hash$
7.2.3. Online and Offline
7.2.4. Comparison with Related Work
8. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Notation  Description 

$\leftX\right$  The size of the data set X 
${x}_{i}$  The ith data of X 
$\left[m\right]$  The set $\left\{1,2,\dots ,m\right\}$ 
$D{O}^{\left(u\right)}$  The uth DO of the DOs 
${Y}^{\left(u\right)}$  The data set of the $D{O}^{\left(u\right)}$ 
${y}_{i}^{\left(u\right)}$  The ith data of ${Y}^{\left(u\right)}$ 
$Enc$  The asymmetric encryption algorithm 
$Dec$  The asymmetric decryption algorithm 
$MTgen$  Merkle tree generation algorithm 
$F(\xb7)$  Pseudorandom function 
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Zhang, J.; Kang, X.; Liu, Y.; Ma, H.; Li, T.; Ma, Z.; Gataullin, S. A Secure and Lightweight MultiParty Private IntersectionSum Scheme over a Symmetric Cryptosystem. Symmetry 2023, 15, 319. https://doi.org/10.3390/sym15020319
Zhang J, Kang X, Liu Y, Ma H, Li T, Ma Z, Gataullin S. A Secure and Lightweight MultiParty Private IntersectionSum Scheme over a Symmetric Cryptosystem. Symmetry. 2023; 15(2):319. https://doi.org/10.3390/sym15020319
Chicago/Turabian StyleZhang, Junwei, Xin Kang, Yang Liu, Huawei Ma, Teng Li, Zhuo Ma, and Sergey Gataullin. 2023. "A Secure and Lightweight MultiParty Private IntersectionSum Scheme over a Symmetric Cryptosystem" Symmetry 15, no. 2: 319. https://doi.org/10.3390/sym15020319