Method and system for spacetime-constrained oblivious transfer

1. A method for performing spacetime-constrained oblivious transfer between a party A and a party B, the method comprising:

synchronizing a plurality of agents A, B, A i , B i to a common reference frame F in a spacetime that is at least approximately Minkowski,

wherein i∈{0, 1, . . . , m−1},

wherein m corresponds to a cardinality of a plurality of messages x i from which spacetime-constrained oblivious transfer is to be performed,

wherein the agents A and A i of the plurality of agents are associated with the party A, and

wherein the agents B and B i of the plurality of agents are associated with the party B;

specifying output spacetime regions R 0 , R 1 , . . . , R m−1 , wherein no connection by a causal curve in spacetime exists for any pair of output spacetime regions selected from the output spacetime regions R 0 , R 1 , . . . , R m−1 ;

generating random numbers r∈Ω outcome and s∈Ω basis by the agent A,

wherein different elements s of a set Ω basis label different orthogonal bases of a Hilbert space, and

wherein, for any fixed s∈Ω basis , different elements r of a set Ω outcome label different orthogonal quantum states |ψ r s from an orthogonal basis s;

generating a quantum state |ψ r s of a quantum system, by the agent A;

transmitting the quantum state to the agent B, by the agent A;

transmitting the random numbers r and the random numbers s to the agent A i , by the agent A, wherein i∈{0, 1, . . . , m−1};

obtaining an associated message x i ∈Ω outcome by the agent A i , wherein i∈{0, 1, . . . , m−1};

encoding, by the agent A i , the plurality of messages x i using the random numbers r to obtain an encoded t i , wherein i∈{0, 1, . . . , m−1};

transmitting, by the agent A i , the encoded t i and the random numbers s to the agent B i at a spacetime point that is in the causal past of at least one spacetime point of an output spacetime region R i and that is not causally connected to any spacetime point of output spacetime region R k for any k∈{0, 1, . . . , m−1}/{i}, wherein i∈{0, 1, . . . , m−1};

obtaining, by the agent B, a selector input b∈{0, 1, . . . , m−1};

forwarding, by the agent B, the quantum state to an agent B b ;

measuring, by the agent B b , the quantum state in the orthogonal basis s to obtain outcome r′∈Ω outcome ;

and computing a received message y b ∈Ω outcome by decoding an encoded x b using r′ and t b , in output spacetime region R b .

2. The method of claim 1 , wherein obtaining the selector input b is performed in a spacetime region G, wherein the spacetime region G is the intersection of causal pasts of all spacetime points of all of the output spacetime regions R 0 , R 1 , . . . , R m−1 .

3. The method of claim 1 , wherein obtaining, by the agent A i , the associated message x i is performed in an associated spacetime region E i , wherein the associated spacetime region E i is the intersection of causal pasts of all spacetime points of the output spacetime region R i , and wherein i∈{0, 1, . . . , m−1}.

4. The method of claim 1 , wherein the received message y b differs no more than a specified threshold from x b .

5. The method of claim 1 , wherein the common reference frame F in the spacetime that is at least approximately the Minkowski applies in at least one selected from a group consisting of Earth's surface, regions near the Earth's surface, and Earth orbits of satellites.

6. The method of claim 1 ,

wherein Ω outcome ={0, 1} n ;

wherein then is a specified natural number;

wherein encoding the associated message x i using the random numbers r comprises an operation t i =x i ⊗r;

wherein decoding the encoded x b using r′ comprises an operation y b =t b └r′;

and wherein the ‘⊗’ denotes bit-wise sum modulo 2.

7. The method of claim 6 , wherein the quantum state |ψ r 2 is a tensor product of n qubit quantum states |ψ r j s j ;

wherein j∈{0, 1, . . . , n−1};

wherein Ω basis ={0, 1} n ;

wherein r j is the jth bit entry of the random numbers r;

wherein s j is the jth bit entry of the random numbers s;

wherein the orthogonal basis s corresponds to n-qubit basis in which a basis of the jth qubit is a qubit orthogonal basis D 0 if s j =0, or a qubit orthogonal basis D 1 if s j =1;

wherein D 0 and D 1 are two qubit orthogonal bases that are not mutually orthogonal.

8. The method of claim 1 , further comprising storing the quantum state by at least one of the plurality of agents A, B, A i , B i .

9. The method of claim 8 , wherein storing the quantum state by at least one of the plurality of agents A, B, A i , B i applies in at least one selected from a group consisting of storing the quantum state by the agent A before the quantum state is transmitted, storing the quantum state by the agent B before the quantum state is forwarded, and storing the quantum state by at least one of the plurality of agents A, B, A i , B i before the quantum state is measured.

10. The method of claim 1 , further comprising verifying whether a message y i obtained by the party B correctly represents the associated message x i provided by the party A, for i∈{0, 1, . . . , m−1}, by:

providing, by the party A, the associated message x i to a verification device C i , at the intersection of the causal pasts of spacetime points in the output spacetime region R i ;

providing, by the party B, the message y i to the verification device C i in a spacetime region R′ i ; and

determining, by the verification device, whether the message y i correctly represents the associated message x i .

11. The method of claim 10 , wherein determining whether the message y i correctly represents the associated message x i comprises:

making a first determination that the spacetime region R′ i is contained in the output spacetime region R i ;

making a second determination that the message y i is sufficiently close to the associated message x i ; and

based on the first and the second determinations, concluding that the message y i correctly represents the associated message x i .

12. The method of claim 10 , wherein determining whether the message y i correctly represents the associated message x i comprises concluding that the message y i does not correctly represent the associated message x i based on one selected from a group consisting of:

making a first determination that R′ i is contained in the output spacetime region R i and making a second determination that the message y i is not sufficiently close to the associated message x i ,

making a first determination that R′ i is not contained in the output spacetime region R i and making a second determination that the message y i is sufficiently close to the associated message x i , and

making a first determination that R′ i is not contained in the output spacetime region R i and making a second determination that the message y i is not sufficiently close to the associated message x i .

13. The method of claim 1 where the different elements s from the set Ω basis label different bases that are mutually unbiased.

14. The method of claim 1 , wherein at least one of the plurality of agents A, B, A i , B i is a satellite orbiting the Earth.

15. A system for performing spacetime-constrained oblivious transfer between a party A and a party B, the system comprising:

a laboratory L A , associated with the party A, the laboratory L A comprising:

a random number generator configured to generate random numbers r∈Ω outcome and random numbers s∈Ω basis ;

a quantum state source configured to generate quantum state |ψ r 2 ,

wherein different elements s of a set Ω basis label different orthogonal bases of a Hilbert space, and

wherein, for any fixed s∈Ω basis , different elements r of a set Ω outcome label different orthogonal quantum states |ψ r 2 from an orthogonal basis s;

a preparation device that is communicatively connected to a laboratory L B using a quantum channel, and configured to transmit the quantum state to the laboratory L B ; and

a first communication module that is communicatively connected to a plurality of laboratories L 0 A , L 1 A , . . . L m−1 A using a plurality of secure classical channels, and configured to transmit the random numbers r and the random numbers s to the plurality of laboratories L 0 A , L 1 A , . . . L m−1 A ;

the plurality of laboratories L 0 A , L 1 A , . . . , L m−1 A , associated with the party A,

wherein each laboratory L i A , i∈{0, 1, . . . , m−1}, comprises:

a second communication module that is communicatively connected to laboratory L A using one of the plurality of secure classical channels, and that is communicatively connected to a laboratory L i B of a plurality of laboratories L 0 B , L 1 B , . . . , L m−1 B using a classical channel, and wherein the second communication module is configured to receive r and s;

a computing device configured to:

obtain a message x i ∈Ω outcome ; and

encode the message x i using r to obtain an encoded t i ; and

wherein the second communication module is further configured to transmit the encoded t i and the random numbers s to laboratory L i B using the classical channel;

the laboratory L B , associated with the party B, comprising:

a redirection unit that is communicatively connected to laboratory L A using the quantum channel, and configured to receive the quantum state from laboratory L A ;

a computing device configured to obtain a selector input b∈{0, 1, . . . , m−1};

a quantum communication module that is communicatively connected to the plurality of laboratories L 0 B , L 1 B , . . . L m−1 B using a plurality of secure quantum channels, wherein the quantum communication module is configured to forward the quantum state to laboratory L b B ;

a laboratory L b B of the plurality of laboratories a L 0 B , L 1 B , . . . , L m−1 B , associated with the party B, comprising:

a third communication module that is communicatively connected to laboratory L b A using one of a plurality of classical channels, and configured to receive t b and s;

a quantum measurement device communicatively connected to laboratory L B using one of the plurality of secure quantum channels, and configured to measure the quantum state in basis s to obtain outcome r′∈Ω outcome ; and

a computing device configured to compute a received message y b ∈Ω outcome by decoding an encoded x b using r′ and t b , in output spacetime region R b ;

wherein a plurality of laboratories L A , L B , L i A and L i B is synchronized to a common reference frame F in a spacetime that is at least approximately Minkowski,

wherein i∈{0, 1, . . . , m−1},

wherein m corresponds to a cardinality of a plurality of messages x i from which spacetime-constrained oblivious transfer is to be performed, and

wherein output spacetime regions R 0 , R 1 , . . . , R m−1 are specified, wherein no connection by a causal curve in spacetime exists for any pair of output spacetime regions selected from the output spacetime regions R 0 , R 1 , . . . , R m−1 , and

wherein the transmission of the encoded t i and the random numbers s by laboratory L i A occurs at a spacetime point that is in the causal past of at least one spacetime point of an output spacetime region R i and that is not causally connected to any spacetime point of output spacetime region R k for any k∈{0, 1, . . . , m−1}/{i}, wherein i∈{0, 1, . . . , m−1}.

16. The system of claim 15 , wherein the plurality of classical channels and the plurality of secure classical channels comprise at least one selected from a group of electrical and optical network links.

17. The system of claim 15 , wherein the quantum state is encoded in a quantum system represented by one selected from a group consisting of polarization degrees of freedom of photons, time-bin degrees of freedom of photons, spin degrees of freedom of electrons, spin degrees of freedom of atoms, spin degree of freedom of nucleus, spin degrees of freedom of molecules.

18. The system of claim 15 , wherein each of the plurality of laboratories L A , L B , L i A and L i B comprises and atomic clock, used for synchronization to the common reference frame F.

19. The system of claim 15 , further comprising a plurality of verification devices C 0 , C 1 , . . . , C m−1 , wherein each of the plurality of verification devices C i , with i∈{0, 1, . . . , m−1}, is configured to verify whether a message y i obtained by the party B correctly represents the message x i provided by the party A, for i∈{0, 1, . . . , m−1}, by:

obtaining, from laboratory L i A , the message x i , at the intersection of the causal pasts of spacetime points in the output spacetime region R i ;

obtaining, from laboratory L i B , the message y i in a spacetime region R′ i ; and

determining, by the verification device, whether the message y i correctly represents the message x i .

20. The system of claim 19 , wherein determining whether the message y i correctly represents the message x i comprises:

making a first determination that the spacetime region R′ i is contained in the output spacetime region R i ;

making a second determination that the message y i is sufficiently close to x i ; and

based on the first and the second determinations, concluding that the message y i correctly represents the message x i .