Single input single output (SISO) physical layer key exchange

1. A system, comprising:

a first communication device;

a second communication device;

at least one processor operatively coupled to the first communication device, the at least one processor for the first communication device configured to:

produce a first encoded vector,

transmit a signal representing a first symbol of the first encoded vector to the second communication device, through a communication channel that applies a channel transformation to the first symbol during transmission to produce a first transformed symbol, and

transmit a signal representing a second symbol of the first encoded vector to the second communication device, through a communication channel that applies a channel transformation to the second symbol during transmission to produce a second transformed symbol; and

at least one processor operatively coupled to the second communication device, the at least one processor for the second communication device configured to:

receive a first transformed signal including the first transformed symbol,

receive a second transformed signal including the second transformed symbol,

construct a matrix based on the first transformed signal and the second transformed signal,

detect a representation of an effective channel based on the matrix, the effective channel associated with the communication channel,

perform a singular value decomposition of the representation of the effective channel to identify a singular vector of the representation of the effective channel,

select a precoding matrix based on a message for transmission, the precoding matrix associated with an index for the message for transmission,

produce a second encoded vector based on the precoding matrix and the singular vector, and

transmit (1) a signal representing a first symbol of the second encoded vector, and (2) a signal representing a second symbol of the second encoded vector, through the communication channel, to the first communication device for identification of the message.

2. The system of claim 1 , wherein the at least one processor operatively coupled to the second communication device is configured to produce the second encoded vector by:

multiplying the complex conjugate of the singular vector by the precoding matrix to produce an intermediate matrix; and

multiplying the intermediate matrix by training values to produce the second encoded vector.

3. The system of claim 1 , wherein the at least one processor for the second communication device is configured to select the precoding matrix from a codebook of unitary matrices.

4. The system of claim 1 , wherein the precoding matrix is a first precoding matrix, the message is a first message, and the index is a first index, the at least one processor operatively coupled to the second communication device is further configured to:

select a second precoding matrix associated with a second index for a second message for transmission,

produce a third encoded vector, and

transmit (1) a signal representing a first symbol of the third encoded vector, and (2) a signal representing a second symbol of the third encoded vector, through the communication channel, to the first communication device for identification of the second message.

5. The system of claim 1 , wherein the at least one processor operatively coupled to the second communication device is further configured to transmit signals representing a plurality of additional encoded vectors through the communication channel to the first communication device until a predetermined number of messages have been sent.

6. A system, comprising:

a first communication device;

a second communication device;

at least one processor operatively coupled to the first communication device, the at least one processor for the first communication device configured to:

produce a first encoded vector, transmit a signal representing a first symbol of the first encoded vector to the second communication device, through a communication channel, the communication channel applying a channel transformation to the first symbol during transmission, and

transmit a signal representing a second symbol of the first encoded vector to the second communication device, through a communication channel, the communication channel applying a channel transformation to the second symbol during transmission; and

at least one processor operatively coupled to the second communication device, the at least one processor for the second communication device configured to:

receive a first transformed signal including a version of the first symbol that has been transformed by the channel transformation,

receive a second transformed signal including a version of the second symbol that has been transformed by the channel transformation,

construct a matrix based on the first transformed signal and the second transformed signal,

detect a representation of an effective channel based on the matrix, the effective channel associated with the communication channel,

perform a singular value decomposition of the representation of the effective channel to identify a left singular vector of the representation of the effective channel and a right singular vector of the representation of the effective channel,

select a precoding matrix based on a message for transmission, the precoding matrix associated with an index for the message for transmission,

produce a second encoded vector based on a complex conjugate of the left singular vector, and the right singular vector of the representation of the effective channel, and

transmit (1) a signal representing a first symbol of the second encoded vector, and (2) a signal representing a second symbol of the second encoded vector, through the communication channel, to the first communication device for identification of the message.

7. The system of claim 6 , wherein the at least one processor operatively coupled to the second communication device is configured to produce the second encoded vector by:

multiplying the complex conjugate of the left singular vector by the precoding matrix to produce an intermediate matrix; and

multiplying the intermediate matrix by the right singular vector of the representation of the effective channel to produce the second encoded vector.

8. The system of claim 6 , wherein the at least one processor for the second communication device is configured to select the precoding matrix from a codebook of unitary matrices.

9. The system of claim 6 , wherein the precoding matrix is a first precoding matrix, the message is a first message, and the index is a first index, the at least one processor operatively coupled to the second communication device is further configured to:

select a second precoding matrix associated with a second index for a second message for transmission,

produce a third encoded vector, and

transmit (1) a signal representing a first symbol of the third encoded vector, and (2) a signal representing a second symbol of the third encoded vector, through the communication channel, to the first communication device for identification of the second message.

10. The system of claim 6 , wherein the at least one processor operatively coupled to the second communication device is further configured to transmit signals representing a plurality of additional encoded vectors through the communication channel to the first communication device until a predetermined number of messages have been sent.

11. The system of claim 6 , wherein:

the at least one processor operatively coupled to the first communication device is further configured to:

receive a third transformed signal including a version of the first symbol of the second encoded vector that has been transformed by the channel transformation; and

receive a fourth transformed signal including a version of the second symbol of the second encoded vector that has been transformed by the channel transformation,

identification of the message includes removing a representation of the right singular vector of the representation of the effective channel from each of the third transformed signal and the fourth transformed signal.

12. A method, comprising:

receiving, via a first communication device and at a first processor, a first signal representing a first symbol of a first encoded vector and a channel transformation;

receiving, via the first communication device and at the first processor, a second signal representing a second symbol of the first encoded vector and a channel transformation;

detecting, via the first processor, a representation of an effective channel based on the first signal and the second signal;

performing, via the first processor, a singular value decomposition of the representation of the effective channel to identify a left singular vector of the representation of the effective channel and a right singular vector of the representation of the effective channel;

selecting, via the first processor, a precoding matrix associated with an index for a message for transmission;

producing, via the first processor, a second encoded vector based on a complex conjugate of the left singular vector, and the right singular vector of the representation of the effective channel; and

transmitting (1) a signal representing a first symbol of the second encoded vector and (2) a signal representing a second symbol of the second encoded vector, through a communication channel, to a second communication device, for identification of the message at a second processor associated with the second communication device.

13. The method of claim 12 , wherein producing the second encoded vector includes:

multiplying the complex conjugate of the left singular vector by the precoding matrix to produce an intermediate matrix; and

multiplying the intermediate matrix by the right singular vector of the representation of the effective channel to produce the second encoded vector.

14. The method of claim 12 , wherein the precoding matrix is selected from a codebook of unitary matrices.

15. The method of claim 12 , wherein the precoding matrix is a first precoding matrix, the message is a first message, and the index is a first index, the method further comprising:

selecting a second precoding matrix associated with a second index for a second message for transmission,

producing a third encoded vector, and

transmitting a signal representing the third encoded vector, through the communication channel, to the second communication device for identification of the second message.

16. The method of claim 12 , further comprising transmitting signals representing a plurality of additional encoded vectors through the communication channel to the second communication device until a predetermined number of messages have been sent.

17. A method, comprising:

generating, at a processor of a first communication device, a first encoded vector;

transmitting, to a second communication device and through a communication channel, a first signal representing a first symbol of the first encoded vector, the communication channel applying a channel transformation to the first signal during transmission;

transmitting, to the second communication device and through the communication channel, a second signal representing a second symbol of the first encoded vector, the communication channel applying a channel transformation to the second signal during transmission;

receiving, from the second communication device and at the processor, a third signal representing a first symbol of a second encoded vector and the channel transformation;

receiving, from the second communication device and at the processor, a fourth signal representing a second symbol of the second encoded vector and the channel transformation;

detecting, via the processor, a representation of an effective channel based on the third signal and the fourth signal;

performing, via the processor, a singular value decomposition of the representation of the effective channel to identify a singular vector of the representation of the effective channel; and

identifying a message associated with the third signal and the fourth signal based on the singular vector of the representation of the effective channel.

18. The method of claim 17 , wherein the message is identified based on a codebook of unitary matrices.

19. The method of claim 17 , further comprising receiving, from the second communication device and at the processor, a plurality of additional signals representing a plurality of additional encoded vectors via the communication channel from the second communication device until a predetermined number of messages have been received.

20. The method of claim 17 , further comprising:

detecting, via the processor, a precoding matrix associated with an index for the message,

the identifying the message being based on the precoding matrix.