Editing Device-Independent Oblivious Transfer (section)

===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.
<!-- INCLUDE V_i^ALPHA CALCULATION -->
# 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>