Device-Independent Oblivious Transfer

This example protocol achieves the task of device-independent oblivious transfer in the bounded quantum storage model using a computational assumption.

Assumptions

• The quantum storage of the receiver is bounded during the execution of the protocol
• The device used is computationally bounded - it cannot solve the Learning with Errors (LWE) problem during the execution of the protocol
• The device behaves in an IID manner - it behaves independently and identically during each round of the protocol

Protocol Description

Protocol 1: Rand 1-2 OT$^{l}$

1. A device prepares $n$ uniformly random Bell pairs Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle |\phi ^{(v_{i}^{\alpha },v_{i}^{\beta })}\rangle ,i=1,...,n} , where the first qubit of each pair goes to $S$ along with the string Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle v^{\alpha }} , and the second qubit of each pair goes to $R$ along with the string $v^{\beta }$.
2. R measures all qubits in the basis $y=[$Computational,Hadamard$]_{c}$ where $c$ is $R$'s choice bit. Let $b\in \{0,1\}^{n}$ be the outcome. $R$ then computes Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle b\oplus w^{\beta }} , where the $i$-th entry of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle w^{\beta }} is defined by

   Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle w_{i}^{\beta }:={\begin{cases}0,{\mbox{if }}y={\mbox{ Hadamard}}\\v_{i}^{\beta },{\mbox{if }}y={\mbox{ Computational}}\end{cases}}}

3. $S$ picks uniformly random Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x\in \{} Computational, HadamardFailed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \}^{n}} , and measures the $i$-th qubit in basis $x_{i}$. Let Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle a\in \{0,1\}^{n}} be the outcome. $S$ then computes Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle a\oplus w^{\alpha }} , where the $i$-th entry of $w^{\alpha }$ is defined by

   $w_{i}^{\alpha }:={\begin{cases}v_{i}^{\alpha },{\mbox{if }}x_{i}={\mbox{ Hadamard}}\\0,{\mbox{if }}x_{i}={\mbox{ Computational}}\end{cases}}$

4. $S$ picks two uniformly random hash functions $f_{0},f_{1}\in F$, announces $x$ and $f_{0},f_{1}$ to $R$ and outputs Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle s_{0}:=f_{0}(a\oplus w^{\alpha }|_{I_{0}})} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle s_{1}:=f_{1}(a\oplus w^{\alpha }|_{I_{1}})} where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle I_{r}:=\{i\in I:x_{i}=[} Computational,Hadamard$]_{r}\}$
5. $R$ outputs Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle s_{c}=f_{c}(b\oplus w^{\beta }|_{I_{c}})}

Protocol 2: Self-testing with a single verifier

1. Alice chooses the state bases Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \theta ^{A},\theta ^{B}\in } {Computational,Hadamard} uniformly at random and generates key-trapdoor pairs Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (k^{A},t^{A}),(k^{B},t^{B})} , where the generation procedure for Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle k^{A}} and $t^{A}$ depends on Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \theta ^{A}} and a security parameter $\eta$, and likewise for Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle k^{B}} and $t^{B}$. Alice supplies Bob with Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle k^{B}} . Alice and Bob then respectively send Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle k^{A},k^{B}} to the device.
2. Alice and Bob receive strings Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle c^{A}} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle c^{B}} , respectively, from the device.
3. Alice chooses a challenge type Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle CT\in \{a,b\}} , uniformly at random and sends it to Bob. Alice and Bob then send $CT$ to each component of their device.
4. If Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle CT=a} :
   1. Alice and Bob receive strings $z^{A}$ and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle z^{B}} , respectively, from the device.
5. If Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle CT=b} :
   1. Alice and Bob receive strings Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle d^{A}} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle d^{B}} , respectively, from the device.
   2. Alice chooses uniformly random measurement bases (questions) Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x,y\in } {Computational,Hadamard} and sends $y$ to Bob. Alice and Bob then, respectively, send $x$ and $y$ to the device.
   3. Alice and Bob receive answer bits $a$ and $b$, respectively, from the device. Alice and Bob also receive bits Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle h^{A}} and $h^{B}$, respectively, from the device.

Protocol 3: DI Rand 1-2 OT$^{l}$

Data generation:

1. The sender and receiver execute $n$ rounds of Protocol 2 (Self-testing) with the sender as Alice and receiver as Bob, and with the following modification:

   If $CT_{i}=b$, then with probability $p$, the receiver does not use the measurement basis question supplied by the sender and instead inputs Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y_{i}=[} Computational, Hadamard$]_{c}$ where $c$ is the receiver's choice bit. Let $I$ be the set of indices marking the rounds where this has been done.

   For each round Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle i\in \{1,...,n\}} , the receiver stores:

   • Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle c_{i}^{B}}
   • Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle z_{i}^{B}} if $CT_{i}=a$
   • or Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (d_{i}^{B},y_{i},b_{i},h_{i}^{B})} if $CT_{i}=b$

   The sender stores Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \theta _{i}^{A},\theta _{i}^{B},(k_{i}^{A},t_{i}^{A}),(k_{i}^{B},t_{i}^{B}),c_{i}^{A},CT_{i};} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle z_{i}^{A}} if $CT_{i}=a$ or $(d_{i}^{A},x_{i},a_{i},h_{i}^{A})$ and $y_{i}$ if $CT_{i}=b$

2. For every Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle i\in \{1,...,n\},} the sender stores the variable Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle RT_{i}} (round type), defined as follows:
   • if $CT_{i}=b$ and $\theta _{i}^{A}=\theta _{i}^{B}=$Hadamard, then $RT_{i}=$ Bell
   • else, set $RT_{i}=$ Product
3. For every Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle i\in \{1,...,n\},} the sender chooses Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle T_{i}} , indicating a test round or generation round, as follows:
   • if $RT_{i}=$ Bell, choose Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle T_{i}\in } {Test, Generate} uniformly at random
   • else, set Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle T_{i}=} Test

   The sender sends (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 T_1,...,T_n} ) to the receiver

   Testing:

4. The receiver sends the set of indices 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} to the sender. The receiver publishes their output for all 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 T_i = } Test rounds where 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 \notin I} . Using this published data, the sender determines the bits which an honest device would have returned.
5. The sender computes the fraction of test rounds (for which the receiver has published data for) that failed. If this exceeds some 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 \epsilon} , the protocol aborts

   Preparing data:

6. Let 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{I} := \{i : i \in 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 T_i = } Generate} 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 n^{\prime} = |\tilde{I}|} . The sender checks if there exists a 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 k > 0 } such that 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 \gamma n^{\prime} \leq n^{\prime}/4 - 2l -kn^{\prime}} . If such a 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 k} exists, the sender publishes 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{I}} and, for each 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 \in \tilde{I}} , the trapdoor 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 t_i^B} corresponding to the key 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 k_i^B} (given by the sender in the execution of Protocol 2,Step 1); otherwise the protocol aborts.
7. For each 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 \in \tilde{I},} the sender calculates 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_i^{\alpha} = d^A_i.(x_{i,0}^A \oplus x_{i,1}^A)} and defines 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 w^{\alpha}} by

   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 w_i^{\alpha} = \begin{cases} v_i^{\alpha}, \mbox{if } x_i = \mbox{Hadamard}\\ 0, \mbox{if } x_i = \mbox{Computational}\end{cases}}

   and the receiver calculates 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_i^{\beta} = = d^B_i.(x_{i,0}^B \oplus x_{i,1}^B)} and defines 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 w^{\beta}} by

   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 w_i^{\beta} = \begin{cases} 0, \mbox{if } y_i = \mbox{Hadamard}\\ v_i^{\beta}, \mbox{if } y_i = \mbox{Computational}\end{cases}}

   Obtaining output:

8. The sender randomly picks two hash functions 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_0,f_1 \in F} , announces 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_0,f_1} 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 x_i} for each 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 \in \tilde{I}} , and outputs 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 s_0 = f_0(a \oplus w^{\alpha}|_{\tilde{I}_0})} 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 s_1 = f_1(a \oplus w^{\alpha}|_{\tilde{I}_1})} , where 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{I}_r := \{i \in \tilde{I}: x_i = [} Computational,HadamardFailed 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\}}
9. Receiver outputs 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 s_c = f_c(a \oplus w^{\beta}|_{\tilde{I}_c})}

References

*contributed by Chirag Wadhwa