Secure Multiparty Delegated Classical Computation

It provides a method for computing nonlinear multivariable functions using only linear classical computing and limited manipulation of quantum information To demonstrate this protocol, the pairwise AND function is computed and can be used as a building block for other functions.

Assumptions

• The clients have limited computational capabilities, namely access to linear XOR functionalities.

Outline

Main Routine

The server sends an ancilla bit to the first client. The first client performs the $\pi /2$ rotation along Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} -axis according to his input bit and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi} rotation accordingto his random bit for security. He then sends the qubit to the next client who performs the same rotation according to his bits. This process if followed until all clients have performed their operations. Now, one of the client performs the conjugate transpose of the $\pi /2$ rotation on the qubit based on the global XOR of all the inputs which he gets by the XOR routine. The state now prepared is the value of the function XORed with the XOR of the random bits of all clients. The clients now announce the random bits with the help of which the final result is calculated.

XOR Routine

The clients choose random bits whose XOR is their input bit and send each such random bit to each client. The clients now perform the XOR of the received bits. To calculate the global XOR, the send their results to the designated client who then performs the XOR of all the received bits to get the global XOR.

Notations

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_i} : Client with index Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} .
• $x_{i}$: Input bit of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i^{th}} client.
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_i} : Random bit of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i^{th}} client.
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_y(\theta)} : Rotation around $y$-axis in Bloch sphere by angle $\theta$.
• $U$: Operator for performing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi/2} rotation around Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} -axis in Bloch Sphere.
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} : Operator for performing $\pi$ rotation around $y$-axis in Bloch Sphere.

Hardware Requirements

• Basic state preparation and measurement devices.
• Access to secure classical channels.

Properties

• The input of each client remains hidden from the other clients and from the server.
• The server performs the computation without learning anything about the result.
• As long as at least two clients are honest, it is enough to guarantee the secrecy of the independent inputs.

Pseudo Code

To compute Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x_1, x_2,... , x_n) = \Sigma_{i,j = 1}^{n} x_i x_j \forall i \neq j} ,

Main Routine

1. The server generates an ancilla bit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |0\rangle} and sends it to client $C_{1}$.
2. For $i=1$ to $n-1$:
   1. $C_{i}$ applies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^{r_i}U^{x_i}} on the received qubit and sends it to client Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_{i+1}} .
3. $C_{n}$ applies $V^{r_{n}}U^{x_{n}}$ on the received qubit.
4. Any client then applies $(U^{\dagger })^{\oplus _{i}x_{i}}$.
   $(U^{\dagger })^{\oplus _{i}x_{i}}\underbrace {V^{r_{n}}U^{x_{n}}} _{{\mathcal {C}}_{n}}...\underbrace {V^{r_{2}}U^{x_{2}}} _{{\mathcal {C}}_{2}}\underbrace {V^{r_{1}}U^{x_{1}}} _{{\mathcal {C}}_{1}}|0\rangle =|r\oplus f\rangle$
5. The resulting state is now sent to the server who measures the outcome Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r \oplus f} and announces it.
6. The clients locally compute XOR of the random bits of other clients.
7. They then perform the operation $f=r\oplus (r\oplus f)$ to get the result.

XOR Routine

1. For $i,j=1,\dots ,n$
   1. Each client $C_{j}$ chooses random bits $x_{j}^{i},~r_{j}^{i}\in \{0,1\}$, such that $x_{j}=\bigoplus _{i=1}^{n}x_{j}^{i}$ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_j=\bigoplus_{i=1}^nr_j^i} and sends $x_{j}^{i}$ and $r_{j}^{i}$ to client $C_{i}$.
   2. Each client $C_{i}$ then computes Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde{x}_i=\bigoplus_{j=1}^n x_j^i} and ${\tilde {r}}_{i}=\bigoplus _{j=1}^{n}r_{j}^{i}$.
2. To perform the operation $U^{\dagger }$, the clients send ${\tilde {x}}_{i}$ to the designated client, who computes the global XOR.
   Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bigoplus_{i=1}^n x_i=\bigoplus_{i=1}^n \tilde{x}_i }
3. When server announces </math>r \oplus f</math>, all clients broadcast </math>\tilde{r}_i</math> to calculate </math>r</math> and know the value of </math>f</math>.

Further Information

*contributed by Natansh Mathur