Editing Device-Independent Oblivious Transfer

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This [https://arxiv.org/abs/2111.08595 example protocol] achieves the task of device-independent oblivious transfer in the bounded quantum storage model using a computational assumption.
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==Assumptions==
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* 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

==Outline==
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* The protocol consists of multiple rounds, which are randomly chosen for testing or string generation
* The testing rounds are carried out to ensure that the devices used are following the expected behaviour. The self-testing protocol used is a modification of the one used in [[Device-Independent Quantum Key Distribution | DIQKD]]. This modification is necessary as, unlike the DIQKD scenario, the parties involved in OT may not trust each other to cooperate. The self-testing protocol uses the computational assumptions associated with ''Extended noisy trapdoor claw-free'' (ENTCF) function families to certify that the device has created the desired quantum states. If the fraction of failed testing rounds exceeds a certain limit, the protocol is aborted.
* At the end of the protocol, the honest sender outputs two randomly generated strings of equal length, and the honest receiver outputs their chosen string out of the two.

==Notation==
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* <math>S</math>: The sender
* <math>R</math>: The receiver
* <math>l</math>: Length of the output strings
* <math>s_0, s_1</math>: The strings output by the sender
* <math>c</math>: A bit denoting the receiver's choice 
* For any bit <math>r</math>, ['''Computational, Hadamard''']<math>_r = \begin{cases}\mbox{Computational, if } r = 0\\ \mbox{Hadamard,         if } r = 1\end{cases}</math>
* <math>\sigma_X = \begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix} </math>
* <math>\sigma_Z = \begin{pmatrix}1 & 0 \\ 0 & -1 \end{pmatrix} </math>
* For bits <math>v^{\alpha},v^{\beta}: |\phi^{(v^{\alpha},v^{\beta})}\rangle = (\sigma_Z^{v^{\alpha}}\sigma_X^{v^{\beta}} \otimes I) \frac{|00\rangle+|11\rangle}{\sqrt{2}}</math>
* An ENTCF family consists of two families of function pairs: <math>F</math> and <math>G</math>. A function pair <math>(f_{k,0},f_{k,1})</math>is indexed by a public key <math>k</math>. If <math>(f_{k,0},f_{k,1}) \in F</math>, then it is a ''claw-free pair''; and if <math>(f_{k,0},f_{k,1}) \in G</math>, then it is called an ''injective pair''. ENTCF families satisfy the following properties:
*# For a fixed <math>k \in K_F, f_{k,0}</math> and <math>f_{k,1}</math> are bijections with the same image; for every image <math>y</math>, there exists a unique pair <math>(x_0,x_1)</math>, called a ''claw'', such that <math>f_{k,0}(x_0) = f_{k,1}(x_1) = y</math>
*# Given a ''key'' <math>k \in K_F</math>, for a claw-free pair, it is quantum-computationally intractable (without access to ''trapdoor'' information) to compute both a <math>x_i</math> and a single generalized bit of <math>x_0 \oplus x_1</math>, where <math>(x_0,x_1)</math> forms a valid claw. This is known as the ''adaptive hardcore bit'' property.
*# For a fixed <math>k \in K_G, f_{k,0}</math> and <math>f_{k_1}</math> are injunctive functions with disjoint images.
*# Given a key <math>k \in K_F \cup K_G</math>, it is quantum-computationally hard (without access to ''trapdoor'' information) to determine whether <math>k</math> is a key for a claw-free or an injective pair. This property is known as ''injective invariance''.
*# For every <math>k \in K_F \cup K_G</math>, there exists a trapdoor <math>t_k</math> which can be sampled together with <math>k</math> and with which 2 and 4 are computationally easy.
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==Protocol Description==
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===Protocol 1: Rand 1-2 OT<math>^l</math>===
'''Requirements:''' Entanglement distribution, classical communication

'''Input:''' Receiver - a bit <math>c</math>

'''Output:''' Sender outputs randomly generated  <math>s_0,s_1 \in \{0,1\}^l</math>, Receiver outputs <math>s_c</math>
# A device prepares <math>n</math> uniformly random Bell pairs <math>|\phi^{(v_i^{\alpha},v_i^{\beta})}\rangle, i = 1,...,n</math>, where the first qubit of each pair goes to <math>S</math> along with the string <math>v^{\alpha}</math>, and the second qubit of each pair goes to <math>R</math> along with the string <math>v^{\beta}</math>.
# R measures all qubits in the basis <math>y = [</math>'''Computational,Hadamard'''<math>]_c</math> where <math>c</math> is <math>R</math>'s choice bit. Let <math>b \in \{0,1\}^n</math> be the outcome. <math>R</math> then computes <math>b \oplus w^{\beta}</math>, where the <math>i</math>-th entry of <math>w^{\beta}</math> is defined by 
#: <math>w_i^{\beta} := \begin{cases} 0, \mbox{if } y = \mbox{ Hadamard}\\ v_i^{\beta}, \mbox{if } y = \mbox{ Computational}\end{cases}</math>
# <math>S</math> picks uniformly random <math>x \in \{</math> '''Computational, Hadamard'''<math>\}^n</math>, and measures the <math>i</math>-th qubit in basis <math>x_i</math>. Let <math>a \in \{0,1\}^n</math> be the outcome. <math>S</math> then computes <math>a \oplus w^{\alpha}</math>, where the <math>i</math>-th entry of <math>w^{\alpha}</math> is defined by
#: <math>w_i^{\alpha} := \begin{cases} v_i^{\alpha}, \mbox{if } x_i = \mbox{ Hadamard}\\ 0, \mbox{if } x_i = \mbox{ Computational}\end{cases}</math>
# <math>S</math> picks two uniformly random hash functions <math>f_0,f_1 \in F</math>, announces <math>x</math> and <math>f_0,f_1</math> to <math>R</math> and outputs <math>s_0 := f_0(a \oplus w^{\alpha} |_{I_0})</math> and <math>s_1 := f_1(a \oplus w^{\alpha} |_{I_1})</math> where <math>I_r := \{i \in I: x_i = [</math>'''Computational,Hadamard'''<math>]_r\}</math>
# <math>R</math> outputs <math>s_c = f_c(b \oplus w^{\beta} |_{I_c})</math>


===Protocol 2: Self-testing with a single verifier===
'''Requirements:''' ENTCF function family, classical communication

# Alice chooses the state bases <math>\theta^A,\theta^B \in </math> {'''Computational,Hadamard'''} uniformly at random and generates key-trapdoor pairs <math>(k^A,t^A),(k^B,t^B)</math>, where the generation procedure for <math>k^A</math> and <math>t^A</math> depends on <math>\theta^A</math> and a security parameter <math>\eta</math>, and likewise for <math>k^B</math> and <math>t^B</math>. Alice supplies Bob with <math>k^B</math>. Alice and Bob then respectively send <math>k^A, k^B</math> to the device.
# Alice and Bob receive strings <math>c^A</math> and <math>c^B</math>, respectively, from the device.
# Alice chooses a ''challenge type'' <math>CT \in \{a,b\}</math>, uniformly at random and sends it to Bob. Alice and Bob then send <math>CT</math> to each component of their device.
# If <math>CT = a</math>:
## Alice and Bob receive strings <math>z^A</math> and <math>z^B</math>, respectively, from the device.
# If  <math>CT = b</math>:
## Alice and Bob receive strings <math>d^A</math> and <math>d^B</math>, respectively, from the device.
## Alice chooses uniformly random ''measurement bases (questions)'' <math>x,y \in</math> {'''Computational,Hadamard'''} and sends <math>y</math> to Bob. Alice and Bob then, respectively, send <math>x</math> and <math>y</math> to the device.
## Alice and Bob receive answer bits <math>a</math> and <math>b</math>, respectively, from the device. Alice and Bob also receive bits <math>h^A</math> and <math>h^B</math>, respectively, from the device.

===Protocol 3: DI Rand 1-2 OT<math>^l</math>===
'''Requirements:''' Entanglement distribution, ENTCF function family, classical communication

'''Input:''' Receiver - a bit <math>c</math>

'''Output:''' Sender outputs randomly generated  <math>s_0,s_1 \in \{0,1\}^l</math>, Receiver outputs <math>s_c</math>
::'''Data generation:'''
# The sender and receiver execute <math>n</math> rounds of '''Protocol 2''' (Self-testing) with the sender as Alice and receiver as Bob, and with the following modification:
#: If <math>CT_i = b</math>, then with probability <math>p</math>, the receiver does not use the measurement basis question supplied by the sender and instead inputs <math>y_i=[</math>'''Computational, Hadamard'''<math>]_c</math> where <math>c</math> is the receiver's choice bit. Let <math>I</math> be the set of indices marking the rounds where this has been done. 
#: For each round <math> i \in \{1,...,n\} </math>, the receiver stores:
#:* <math>c_i^B</math>
#:* <math>z_i^B</math> if <math>CT_i = a</math>
#:* or <math>(d_i^B,y_i,b_i,h_i^B)</math> if <math>CT_i = b</math>
#: The sender stores <math>\theta_i^A,\theta_i^B,(k_i^A,t_i^A),(k_i^B,t_i^B),c_i^A,CT_i;</math> and <math>z_i^A</math> if <math>CT_i = a</math> or <math>(d_i^A,x_i,a_i,h_i^A)</math> and <math>y_i</math> if <math>CT_i = b</math>
# For every <math>i \in \{1,...,n\},</math> the sender stores the variable <math>RT_i</math> (round type), defined as follows:
#* if <math>CT_i = b</math> and <math>\theta_i^A = \theta_i^B = </math>'''Hadamard''', then <math>RT_i =</math> '''Bell'''
#* else, set <math>RT_i = </math> '''Product'''
# For every <math>i \in \{1,...,n\},</math> the sender chooses <math>T_i</math>, indicating a test round or generation round, as follows:
#* if <math>RT_i = </math> '''Bell''', choose <math>T_i \in</math> {'''Test, Generate'''} uniformly at random
#* else, set <math>T_i = </math> '''Test'''
#: The sender sends (<math>T_1,...,T_n</math>) to the receiver
#: 
#: '''Testing:'''
# The receiver sends the set of indices <math>I</math> to the sender. The receiver publishes their output for all <math>T_i = </math> '''Test''' rounds where <math>i \notin I</math>. Using this published data, the sender determines the bits which an honest device would have returned.
# The sender computes the fraction of test rounds (for which the receiver has published data for) that failed. If this exceeds some <math>\epsilon</math>, the protocol aborts
#: 
#: '''Preparing data:'''
# Let <math>\tilde{I} := \{i : i \in I</math> and <math>T_i = </math> '''Generate'''} and <math>n^{\prime} = |\tilde{I}|</math>. The sender checks if there exists a <math> k > 0 </math> such that <math>\gamma n^{\prime} \leq n^{\prime}/4 - 2l -kn^{\prime}</math>. If such a <math>k</math> exists, the sender publishes <math>\tilde{I}</math> and, for each <math>i \in \tilde{I}</math>, the trapdoor <math>t_i^B</math> corresponding to the key <math>k_i^B</math> (given by the sender in the execution of '''Protocol 2,Step 1'''); otherwise the protocol aborts.
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# For each <math>i \in \tilde{I},</math> the sender calculates <math>v_i^{\alpha} = d^A_i.(x_{i,0}^A \oplus x_{i,1}^A)</math> and defines <math>w^{\alpha}</math> by
#:<math>w_i^{\alpha} = \begin{cases} v_i^{\alpha}, \mbox{if } x_i = \mbox{Hadamard}\\ 0, \mbox{if } x_i = \mbox{Computational}\end{cases}</math>
#: and the receiver calculates <math>v_i^{\beta} =  = d^B_i.(x_{i,0}^B \oplus x_{i,1}^B)</math> and defines <math>w^{\beta}</math> by
#:<math>w_i^{\beta} = \begin{cases} 0, \mbox{if } y_i = \mbox{Hadamard}\\ v_i^{\beta}, \mbox{if } y_i = \mbox{Computational}\end{cases}</math>
#: '''Obtaining output:'''
# The sender randomly picks two hash functions <math>f_0,f_1 \in F</math>, announces <math>f_0,f_1</math> and <math>x_i</math> for each <math>i \in \tilde{I}</math>, and outputs <math>s_0 = f_0(a \oplus w^{\alpha}|_{\tilde{I}_0})</math> and <math>s_1 = f_1(a \oplus w^{\alpha}|_{\tilde{I}_1})</math>, where <math>\tilde{I}_r := \{i \in \tilde{I}: x_i = [</math>'''Computational,Hadamard'''<math>]_r\}</math>
# Receiver outputs <math>s_c = f_c(a \oplus w^{\beta}|_{\tilde{I}_c})</math> 


==Properties==
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* <math>\epsilon</math>-'''Receiver security:''' If <math>R</math> is honest, then for any <math>\tilde{S}</math>, there exist random variables <math>S_0^{\prime}, S_1^{\prime}</math> such that Pr[<math>Y = S_c^{\prime}] \geq 1 - \epsilon</math> and <math>D(\rho_{c,S_0^{\prime}, S_1^{\prime},\tilde{S}}, \rho_c \otimes \rho_{S_0^{\prime}, S_1^{\prime},\tilde{S}}) \leq \epsilon</math>
*: Protocol 3 is perfectly receiver secure, i.e. <math>\epsilon</math> = 0
* <math>\epsilon</math>-'''Sender security:''' If S is honest, then for any <math>\tilde{R}</math>, there exist a random variable <math>c^{\prime}</math> such that <math>D(\rho_{S_{1-c^{\prime}},S_{c^{\prime}},c^{\prime},\tilde{R}}, \frac{1}{2^l}I \otimes \rho_{S_{c^{\prime}},c^{\prime},\tilde{R}}) \leq \epsilon</math>
*: Protocol 3 is <math>\epsilon^{\prime}</math>-sender secure, where <math>\epsilon^{\prime}</math> can be made negligible in certain conditions.

==References==
* The protocol and its security proofs can be found in [https://arxiv.org/abs/2111.08595 Broadbent and Yuen(2021)] 
<div style='text-align: right;'>''*contributed by Chirag Wadhwa''</div>