f(x_1, x_2, x_3, \cdots) = (y_1, y_2, y_3, \cdots)

f(x_1, x_2, x_3, \cdots) = (y_1, y_2, y_3, \cdots)

\begin{aligned} & Alice: v_0, v_1, s, r_0, r_1 \\ & Bob: i \in \{0,1\}, k \\ & Bob \text{ want to get } v_i \\ & \text{Generate } g \in Z_p, g \text{ is large integer}, p \text{ is large prime number}\\\\ 1. & Alice \to Bob : g_s \\ 2. & Bob \text{ calculate } L_i = \begin{cases} g^k & \text{if }i=0 \\ g^{s-k} & \text{if } i = 1 \end{cases} \\ 3. & Bob \to Alice :L_i \\ 4. & Alice \text{ calculate } C_0 = (g^{r_0},(L_i)^{r_0}\oplus v_0), C_1=(g^{r_1},(g^s/L_i)^{r_1}\oplus v_1) \\ 5. & Alice \to Bob C_0, C_1 \\ 6. & Bob \text{ get } v_i = \begin{cases} C_0[0]^k \oplus C_0[1] = (g^{r_0})^k \oplus (L_i)^{r_0} \oplus v_0 = (g^{r_0})^k \oplus (g^k)^{r_0} \oplus v0 & \text{if }i=0 \\ C_1[0]^k \oplus C_1[1] = (g^{r1})^k \oplus (g^s/L_i)^{r_1} \oplus v_1 = (g^{r_1})^k \oplus (g^k)^{r_1} \oplus v_1 & \text{if } i = 1 \end{cases} \end{aligned}