Device-Independent Oblivious Transfer: Difference between revisions

Device-Independent Oblivious Transfer (edit)

Revision as of 16:35, 22 January 2022

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 ==Outline==
 <!-- A non-mathematical detailed outline which provides a rough idea of the concerned protocol -->
+* 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 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.
+* Following the generation rounds, the honest sender outputs two randomly generated strings of equal length, and the honest receiver outputs their chosen string out of the two.
 ==Notation==
 <!--  Connects the non-mathematical outline with further sections. -->
+* <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_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.
 <!-- ==Knowledge Graph== -->
 <!-- Add this part if the protocol is already in the graph -->