# Privacy-Preserving Ride-Sharing System with Minimum-Detouring Route
## 4 Proposed Approach
### Protocol Details
**
```
1: Construct the rider set \(\mathcal{R}\)
2: Compute \(\mathcal{R}.EZpath\) with \(\mathcal{P}\)
3:for every \(r\in\mathcal{R}_{near}\)do
4:if\(r.z(s),r.z(d)\in\mathcal{R}.EZpath\)and\(Dir(r,\mathcal{R})<0\)then
5: Add \(r\) to \(\mathcal{R}\);
6:endif
7:endfor
8:return\(\mathcal{R}\) ```
**Algorithm 7** Filtering Algorithm with Existing Schedule (_FAES_). **Note**: **The algorithm is based on the following algorithm. **
```
1: Construct the rider set \(\mathcal{R}\)
2: Compute \(\mathcal{R}.EZpath\) with \(\mathcal{P}\)
3:for every \(r\in\mathcal{R}_{near}\)do
4:if\(r.z(s),r.z(d)\in\mathcal{R}.EZpath\)and\(Dir(r,\mathcal{R})<0\)then
5: Add \(r\) to \(\mathcal{R}\);
6:endif
7:endfor
8:return\(\mathcal{R}\) ```
**Algorithm 8** Filtering Algorithm with Existing Schedule (_FAES_). **Note**: **The algorithm is based on the following algorithm. **
```
1: Construct the rider set \(\mathcal{R}\)
2: Compute \(\mathcal{R}.EZpath\) with \(\mathcal{P}\)
3:for every \(r\in\mathcal{R}_{near}\)do
5:if\(r.z(s),r.z(d)\in\mathcal{R}.EZpath\)and\(Dir(r,\mathcal{R})<0\)then
6: Add \(r\) to \(\mathcal{R}\);
7:endif
8:endif
9:endif
10:endif
11:endif
12:endfor
13:endfor
14:endfor
15:endfor
16:endfor
17:endfor
18:return\(\mathcal{R}\) ```
**Algorithm 9** Filtering Algorithm with Existing Schedule (_FAES_). **Note**: **The algorithm is based on the following algorithm. **
```
1: Construct the rider set \(\mathcal{R}\)
2: Compute \(\mathcal{R}.EZpath\) with \(\mathcal{P}\)
3:for every \(r\in\mathcal{R}_{near}\)do
5:if\(r.z(s),r.z(d)\in\mathcal{R}.EZpath\)and\(Dir(r,\mathcal{R})<0\)then
6: Add \(r\) to \(\mathcal{R}\);



For each path, the RS compares each point with its deadline time. Suppose \(T_{1}\), \(T_{2}\), \(\cdots T_{2v}\) are \(p\)-bit integers. Then the RS randomly chooses \(2v\)\((k-1)\)-bit length integers \(\mathcal{A}=\{a_{1},a_{2},\cdots a_{2v}\}\)\((k-1>p)\) and applies the following operations:

\[S([\mathcal{T}(\mathcal{P})])=\prod_{i=1}^{2v}[\mathbb{T}_{i}]^{2^{k(2v-i)}} \tag{15}\]

\[S(\mathcal{A})=S(a_{1},a_{2},\cdots a_{2v})=\sum_{i=1}^{2v}a_{i}2^{k(2v-i)} \tag{16}\]

\[S\big{(}[\widetilde{\mathcal{T}}(\mathcal{P})]\big{)}=E(S(\mathcal{A}))S([ \mathcal{T}(\mathcal{P})]) \tag{17}\]

The RS first packs \([\mathbb{T}_{1}]\), \([\mathbb{T}_{2}]\cdots[\mathbb{T}_{2v}]\) and \(a_{1},a_{2},\cdots a_{2v}\) in (15), (16) and then adds the encrypted integers to each item of \(\{[T_{i}]\}_{i=1}^{2v}\) in (17) and obtains \([\widetilde{\mathcal{T}}(\mathcal{P})]\), which is the packing ciphertexts after blinding operation. Similarly, for the deadline time of each point \([\mathbb{T}_{V}(\mathcal{P})]=\ \big{[}[\mathbb{t}_{V}]\big{]}_{i=1}^{2V}\), the RS generates random integers \(\mathcal{A}^{\prime}=\left\{a_{1}^{\prime},a_{2}^{\prime},\cdots a_{2V}^{ \prime}\right\}\) and executes the same blinding operations in (15)-(17). Then the RS sends \([\widetilde{\mathcal{T}}(\mathcal{P})]\) and \([\widetilde{\mathcal{T}}_{V}(\mathcal{P})]\) to the CS, and they jointly finish the judging for each path. On receiving \([\widetilde{\mathcal{T}}(\mathcal{P})]\) and \([\widetilde{\mathcal{T}}_{V}(\mathcal{P})]\), the CS decrypts the ciphertext and unpacks the data. For each path \(\mathcal{P}_{j}\), the RS holds \(\mathcal{A}\) and \(\mathcal{A}^{\prime}\); the CS holds \(\widetilde{\mathcal{T}}(\mathcal{P}_{j})\) and \(\widetilde{\mathcal{T}}_{V}(\mathcal{P}_{j})\); and they execute the garbled circuit \(TC\) as shown in Figure 4, in which the _SCMP_ circuit is shown in Figure 5. CMP, MUX, SUB, and MIN denote a comparator circuit, a multiplexer circuit, a subtractor circuit, and a minimum circuit, respectively. We present the details of the circuit \(TC\) in Algorithm 3. The circuit first restores the real values from blind values (1) and then compares the arriving time and the deadline time for each point (2-8). Only if all deadlines are satisfied will the circuit output the total time consumption of the ride-sharing process (9-10), or the circuit outputs a big enough integer \(\{1\}^{p}\) (12).