Imported: 17 Feb '17 | Published: 23 Sep '14

USPTO - Utility Patents

A method and apparatus are provided for performing information-theoretically secure cryptography using joint randomness not shared by others. Two valid communicating entities independently generate samples of a shared source that is not available to an illegitimate entity. The shared source may be a satellite signal, and each legitimate entity may generate uniformly distributed samples from a binary phase-shift keying signal received on an independent channel. Alternatively, the shared source may be a channel between the two legitimate entities, such that each legitimate entity generates samples of unknown distribution based on the channel impulse response of the channel. One legitimate entity generates an encryption key, a quantization error, and a syndrome from its samples. The quantization error and the syndrome are reported to the other legitimate entity. The other legitimate entity generates a matching encryption key using its samples, the quantization error, and the syndrome.

This application claims the benefit of U.S. provisional application No. 61/052,333 filed May 12, 2008, which is incorporated by reference as if fully set forth.

This application is related to wireless communications.

In a typical wireless communication scenario two wireless transmit/receive units (WTRUs), Alice and Bob, communicate with each other on a channel. To exclude an eavesdropper, Eve, Alice and Bob cryptographically protect their communications. However, traditional cryptographic techniques, which rely on computational difficulty, are increasingly ineffective as the availability of computing power increases. Therefore, it would be advantageous to provide a cryptographic technique that is information-theoretically secure, rather than one based on computational difficulty.

A method and apparatus are provided for performing information-theoretically secure cryptography using joint randomness not shared by others. Two valid communicating entities independently generate samples of a shared source that is not available to an illegitimate entity. The shared source may be a satellite signal, and each legitimate entity may generate uniformly distributed samples from a binary phase-shift keying signal received on an independent channel. Alternatively, the shared source may be a channel between the two legitimate entities, such that each legitimate entity generates samples of unknown distribution based on the channel impulse response of the channel. One legitimate entity generates an encryption key, a quantization error, and a syndrome from its samples. The quantization error and the syndrome are reported to the other legitimate entity. The other legitimate entity generates a matching encryption key using its samples, the quantization error, and the syndrome.

When referred to hereafter, the term “wireless transmit/receive unit (WTRU)” includes but is not limited to a user equipment (UE), a mobile station, a fixed or mobile subscriber unit, a pager, a cellular telephone, a personal digital assistant (PDA), a computer, or any other type of user device capable of operating in a wireless environment. When referred to hereafter, the terminology “base station” includes but is not limited to a Node-B, a site controller, an access point (AP), or any other type of interfacing device capable of operating in a wireless environment. When referred to hereafter, the term “Alice” includes a WTRU or a base station that is a legitimate communicating entity. When referred to hereafter, the term “Bob” includes a WTRU or a base station that is a legitimate communicating entity. When referred to hereafter, the term “information-theoretically secure” includes but is not limited to perfectly secure, unconditionally secure, and nearly information-theoretically secure.

FIG. 1 is an example block diagram of a wireless transmit/receive unit (WTRU) **100** for performing wireless communications using joint randomness not shared by others (JRNSO). The WTRU **100** includes a processor **110**, an antenna **120**, a user interface **125**, a display **130**, and a transceiver **135**. Optionally, the transceiver may include a transmitter, a receiver, or a satellite receiver. The WTRU **100** also includes a channel estimator **140**, a post processing unit **145**, a data conversion unit **150**, a quantization unit **155**, a source coding unit **160**, a channel coding unit **165**, a decoding unit **170**, and a privacy amplification (PA) processor **175**. The configuration shown is for illustrative purposes. A WTRU may include only a subset of the elements described herein. For example, a particular embodiment of a WTRU **100** may only include a channel estimator **140**, a post processing unit **145**, a decoding unit **165**, and a PA processor **170**. The processor **110** is shown as including the channel estimator **140**, the post processing unit **145**, the data conversion unit **150**, the quantization unit **155**, the source coding unit **160**, the channel coding unit **165**, the decoding unit **170**, and the privacy amplification (PA) processor **175** for illustration; however, one or more of these elements may be a separate unit.

FIG. 2 shows an example block diagram of a network **200** for performing wireless communications using joint randomness not shared by others (JRNSO) with a source of uniformly distributed correlated random values. The network **200** includes two legitimate WTRUs, Alice **201** and Bob **202**, and an illegitimate eavesdropper, Eve **203**. The network **200** also includes a source of uniformly distributed correlated random values **205**. For example, the source may be a binary phase-shift keying (BPSK) satellite signal that is received by Alice **201** and Bob **202** through independent channels. Although Alice **201**, Bob **202**, and Eve **203** are shown individually, the communications may involve multiple legitimate and illegitimate entities.

Alice **201** and Bob **202** each generate input samples at **210**, **212**. The input samples are based on contemporaneous estimates of the source of uniformly distributed correlated random values **205** and is not available to Eve **203**. Although Alice's channel is independent from Bob's, their estimates are highly correlated. The input samples taken by Alice **201** and Bob **202** may be expressed as X^{n }and Y^{n }respectively, where X^{n }and Y^{n }include n independent and identically distributed repetitions of the correlated random values X and Y, such that X^{n}=(X_{1}, . . . , X_{n}) and Y^{n}=(Y_{1}, . . . , Y_{n}). Where the distribution of the discrete random variable U is such that the probability that U=1 is equal to the probability that U=−1, and the distribution of the random values Z_{A }and Z_{B }is Gaussian, such that N(0, N_{A}) and N(0, N_{B}), the correlation between the random values X and Y may be expressed as X=U+Z_{A }and Y=U+Z_{B}. The random values U, Z_{A}, and Z_{B }are mutually independent. Although equal values of N_{A }and N_{B}, and a signal to noise ratio (SNR) of 1/N, are used for simplicity herein, other values are applicable.

Alice **201** and Bob **202** communicate with each other over a public channel and each generates a shared secret key K, at **220**, **222**. The public keys are based on their respective input samples X^{n}, Y^{n }and their public communications V. Alice **201** generates the bit string K_{A }based on (X^{n}, V), and Bob **202** generates the bit string K_{B }based on (Y^{n}, V). Where ε>0 and κ is the finite range of the shared secret key K, the shared secret key K is uniformly distributed, such that

nearly statistically independent of V, such that

and is almost the same at Alice **201** and Bob **202**, such that Pr(K=K_{A}=K_{B})>=1−ε. Alice **201** and Bob **202** use the shared secret key K to perform information-theoretically secure communications, at **230**, **232**.

Although Eve **203** may observe the public communications V, Eve **203** does not observe the signal **205** and cannot decrypt the information-theoretically secure communications between Alice **201** and Bob **202**. For example, where Eve **203** does not have a receiver capable of receiving the signal, is out of range of the signal, does not know which signal to sample, or does not know the sampling time period, Eve **203** cannot receive the signal.

FIG. 3 shows an example of a method for shared secret key generation **220** by Alice **201** using uniformly distributed input samples. The input samples X^{n }are quantized, at **310**. The quantization produces quantization values X_{b}^{n}, and quantization errors E^{n}. The quantization values X_{b}^{n }are channel coded to generate syndrome bits S, in terms of a given block error correction code, at **320**. The syndrome bits S and the quantization errors E^{n }are transmitted through an error-free public channel to Bob **202**, at **330**. The publicly transmitted syndrome bits S include |S| bits of information, and the publicly transmitted quantization errors E^{n }are independent of X_{b}^{n}.

Privacy amplification is performed on the quantization values X_{b}^{n}, at **340**. The publicly revealed information is hashed out of the quantization values X_{b}^{n}, leaving perfectly secret bits for the shared secret key K_{A}. The probability density function (PDF) of X is an even function and the quantized values X_{b}^{n }are a full entropy sequence, such that H(X_{b}^{n})=|X_{b}^{n}|. Therefore, X_{b}^{n }includes n bits of information and at least n″|S| perfectly secret bits are generated for the shared secret key K_{A}.

Quantization **310** includes identifying a partition, corresponding quanta, and quantization errors. Quantization produces quantization values X_{b}^{n}, and quantization errors E^{n}. Each quantized value X_{b,i }includes the v bits corresponding to the v most significant values of the input, and the corresponding quantization error E_{i }includes A−v bits corresponding to the remaining A−v least significant values of the input. Although equiprobable quantization is shown, any suitable quantization method may be applied.

The partition includes a set of disjoint intervals Q_{1 }. . . Q_{v}, which cover the sample range. The quantization boundaries of each interval Q_{i }is determined based on the samples. The corresponding quanta, which represent the quantized values, include a set of numbers q_{1 }. . . q_{v}, such that q_{i}∈Q_{i}. Where a quantization level v which indicates the number of bits per quantization value and 0<=i<=2^{v}, the determination of quantization boundaries may be expressed as

The quantization error E^{n }may be expressed as (E_{i}, . . . , E_{n})=(|X_{i}|, . . . , |X_{n}|), and the generation of a quantized value uses a binary (single bit) quantizer and may be expressed as:

FIG. 4 shows an example of a method for shared secret key generation **222** by Bob **202** using uniformly distributed input samples. The syndrome bits S and the quantization errors E^{n }are received from Alice **201**, at **410**. The input samples Y^{n }are decoded using the syndrome bits S and the quantization errors E^{n }to produce the decoded samples X_{b}^{n}, at **420**. Privacy amplification is performed on the decoded samples X_{b}^{n }at **430**.

Decoding **420** includes performing a modified belief-propagation algorithm to produce the decoded samples X_{b}^{n}. The modified belief-propagation algorithm includes calculating a per-bit log-likelihood ratio (LLR). The LLR is a logarithm of a ratio between the probability that X_{b,i }is 0 and the probability that X_{b,i }is 1. More generally, the LLR is related to the distances from V to the possible U values that cause X_{b,i }to be 0 and that cause X_{b,i }to be 1.

Where the inputs are based on a uniformly distributed binary signal, such as a BPSK modulated satellite signal, N is the noise power, 1<=i<=n, E_{i}=e, and Y_{i}=y, the LLR (the uniform distribution LLR algorithm) may be expressed as:

During privacy amplification **430** publicly revealed information is hashed out of the decoded samples X_{b}^{n}, leaving the perfectly secret bits for the shared secret key K_{B}. The PDF of X is an even function and the decoded samples X_{b}^{n }are a full entropy sequence, such that H(X_{b}^{n})=|X_{b}^{n}|. Therefore, the decoded samples X_{b}^{n }include n bits of information and at least n−|S| perfectly secret bits are generated for the shared secret key K_{B}.

FIG. 5 shows an example block diagram of a network **500** for performing wireless communications using JRNSO performed on a reciprocal wireless channel. The network **500** includes two legitimate WTRUs, Alice **501** and Bob **502**, and an illegitimate eavesdropper, Eve **503**. Although Alice **501**, Bob **502**, and Eve **503** are shown individually, the communications may involve multiple legitimate and illegitimate entities.

Alice **501** and Bob **502** each generate input samples, at **510**, **512**. The input samples are based on contemporaneous channel impulse response (CIR) measurements of their reciprocal wireless channel **505**, which is not available to Eve **503**. Due to channel reciprocity, the CIR estimates are composed of highly correlated samples of unknown distribution. Any two consecutive observations, at either Alice **501** or Bob **502**, are independent if the time between observations is larger than the channel coherence time. The input samples taken by Alice **501** and Bob **502** may be expressed as X^{n }and Y^{n }respectively, where X^{n }and Y^{n }include n independent and identically distributed repetitions of the correlated random values X and Y, such that X^{n}=(X_{1}, . . . , X_{n}) and Y^{n}=(Y_{1}, . . . , Y_{n}).

Alice **501** and Bob **502** communicate with each other over a public channel and each generates a shared secret key K, at **520**, **522**. The shared secret keys K are based on their respective input samples X^{n}, Y^{n }and their public communications V. Alice **501** generates the bit string K_{A }based on (X^{n}, V), and Bob **502** generates the bit string K_{B }based on (Y^{n}, V). Where ε>0 and κ is the finite range of the shared secret key K, the shared secret key K is uniformly distributed, such that

nearly statistically independent of V, such that

and is almost the same at Alice **201** and Bob **202**, such that Pr(K=K_{A}=K_{B})>=1−ε. Alice **501** and Bob **502** use the shared secret key K to perform information-theoretically secure communications, at **530**, **532**.

Although Eve **503** may observe the public communications V, Eve **503** does not observe the shared reciprocal wireless channel **505** between Alice **501** and Bob **502**, and cannot decrypt the information-theoretically secure communications between Alice **501** and Bob **502**. For example, where Eve **503** is at least a few wavelengths away from Alice **501** and Bob **502**, Eve's CIR measurements are not correlated with Alice's or Bob's.

FIG. 6 shows an example of a method for universal shared secret key generation **520** by Alice **501**. The input samples X^{n }are converted to uniformly distributed samples U^{n}, such that each uniformly distributed sample U_{i}∈[0,1), at **605**. Each uniformly distributed sample U_{i }has a sample size of A, indicating A bits per sample. The uniformly distributed samples U^{n }are quantized, at **610**. The quantization produces quantization values X_{q}^{n}, and quantization errors E^{n}.

Quantization **410** includes identifying a partition, corresponding quanta, and quantization errors. Quantization produces quantization values X_{q}^{n}, and quantization errors E^{n}. Each quantized value X_{q,i }includes the v bits corresponding to the v most significant values of the input, and the corresponding quantization error E_{i }includes A−v bits corresponding to the remaining A−v least significant values of the input. Although equiprobable quantization is shown, any suitable quantization method may be applied.

The partition includes a set of disjoint intervals Q_{1 }. . . Q_{v}, which cover the sample range. The quantization boundaries of each interval Q_{i }is determined based on the samples. The corresponding quanta, which represent the quantized values, include a set of numbers q_{1 }. . . q_{v}, such that q_{i}∈Q_{i}. Where a quantization level v which indicates the number of bits per quantization value and 0<i<=i2^{v}, the determination of quantization boundaries may be expressed as

Where the inputs are fixed point inputs, such as the uniformly distributed samples U^{n}, and v is less than A, the quantized value X_{q,i }is the v most significant bits in U_{i}, and the quantization error E_{i }is the remaining A−v least significant bits in U_{i}.

The quantization values X_{q}^{n }are source coded into bit strings X_{b}^{n}, at **615**. For example, gray coding may be used to convert X_{q}^{n }to X_{b}^{n}. The source coded bit strings X_{b}^{n }are channel coded to generate syndrome bits S. Block Error Correction Coding is applied X_{b}^{n }to generate syndrome bits S, in terms of a given low density parity code (LDPC), at **620**. The syndrome bits S and the quantization errors E^{n }are transmitted through an error-free public channel to Bob **502**, at **630**. The publicly transmitted syndrome bits S include |S| bits of information, and the publicly transmitted quantization errors E^{n }are independent of X_{b}^{n}. The publicly revealed information is hashed out of the common bit string X_{b}^{n}, leaving perfectly secret bits for the shared secret key K_{A}, at **640**.

FIG. 7 shows an example of a method for universal shared secret key generation **522** by Bob **502**. The input samples Y^{n }are converted to uniformly distributed samples V^{n}, such that each uniformly distributed sample V_{i}∈[0,1), at **705**. The syndrome bits S and the quantization errors E^{n }are received from Alice **501**, at **710**. The uniformly distributed samples V^{n }are decoded using the syndrome bits S and the quantization errors E^{n }to produce the decoded samples X_{b}^{n}, at **720**. The publicly revealed information is hashed out of the decoded samples X_{b}^{n}, leaving the perfectly secret bits for the shared secret key K_{B}, at **730**.

Decoding **720** includes performing a modified belief-propagation algorithm to produce the decoded samples X_{b}^{n}. The modified belief-propagation algorithm includes calculating a per-bit log-likelihood ratio (LLR). The LLR is a logarithm of a ratio between the probability that X_{b,i }is 0 and the probability that X_{b,i }is 1. More generally, the LLR is related to the distances from V to the possible U values that cause X_{b,i }to be 0 and that cause X_{b,i }to be 1.

Where 1<=i<=v, the LLR L_{i }for X_{b,i }may computed using an unknown distribution LLR algorithm as shown in FIG. 8. For example, where the quantization error E_{i }is 0.2 and the number of bits per quantized value v is 1, either U is 0.2 and X_{b,1 }is 0 or U is 0.7 and X_{b,1 }is 1. The value of U that is closer to V represents the more likely value of X_{b,1}. Where V is 0.3, X_{b,1 }is 0 and the LLR for X_{b,1 }is positive. Similarly, where V is 0.5, X_{b,1 }is 1 and the LLR for X_{b,1 }is negative. Therefore, the LLR L_{1 }is −0.1. Optionally, L_{i }may be scaled to the operational range of the modified belief-propagation algorithm by multiplying L_{i }with a constant.

FIG. 9 shows an example of a method for converting input samples of unknown distribution to uniformly distributed samples. For simplicity, the conversion is described in reference to Alice's input samples X^{n}; however, one skilled in the art would recognize that the method may be applied to Bob's input samples Y^{n}.

The input samples X^{n }are sorted according to the observed value of each input sample X_{i}, at **910**. For example, the values X_{i }may be sorted in ascending order such that {tilde over (X)}_{l}<= . . . <={tilde over (X)}_{n}. Each sorted sample {tilde over (X)}_{i }retains an association with the pre-sort index of the corresponding input sample X_{i}. The sorted samples {tilde over (X)}^{n }are converted (transformed) into sorted uniformly distributed samples Ũ^{n}, at **920**. The sorted uniformly distributed samples Ũ^{n }are sorted into uniformly distributed samples U^{n }by sorting each sorted uniformly distributed sample Ũ_{i }according to the index of the corresponding input samples X_{i}, at **930**. The resultant uniformly distributed samples U^{n }are inherently fixed-point values.

The conversion, which is based on an empirical distribution of the input samples, associates the sorted sample {tilde over (X)}_{i }with the sorted uniformly distributed sample Ũ_{i}. For example, the conversion is performed using a rate-matching method as shown in FIG. 10. Where X_{i }indicates the i^{th }sample of the input samples X^{n}, F(X_{i}) indicates the number of samples in X^{n }that are less than the input sample X_{i }added to the number of samples in the input samples X^{n }that are equal to X_{i }and have an index less than i. The corresponding uniformly distributed sample U_{i }may be expressed as:

Where 0<=j<=2^{A}−1, the value of a uniformly distributed sample U_{i }may be expressed as

Where 0<=j<=2^{A}, the C(j) uniformly distributed samples have a value of

Where └x┘ is the largest integer less than x, the uniformly distributed samples U_{i }in C(j) may be generated as shown in FIG. 10. Thus, the uniformly distributed sample U_{i }corresponding to an input sample X_{i }may be expressed as

where the input sample X_{i }may be expressed as:

Although conversion has been shown in terms of 2^{A }for simplicity, any integer number of values may be converted such that the unit interval [0,1) is partitioned into equal sub-intervals with the data distributed among them as uniformly as possible.

Although features and elements are described above in particular combinations, each feature or element can be used alone without the other features and elements or in various combinations with or without other features and elements. The methods or flow charts provided herein may be implemented in a computer program, software, or firmware incorporated in a computer-readable storage medium for execution by a general purpose computer or a processor. Examples of computer-readable storage mediums include a read only memory (ROM), a random access memory (RAM), a register, cache memory, semiconductor memory devices, magnetic media such as internal hard disks and removable disks, magneto-optical media, and optical media such as CD-ROM disks, and digital versatile disks (DVDs).

Suitable processors include, by way of example, a general purpose processor, a special purpose processor, a conventional processor, a digital signal processor (DSP), a plurality of microprocessors, one or more microprocessors in association with a DSP core, a controller, a microcontroller, Application Specific Integrated Circuits (ASICs), Field Programmable Gate Arrays (FPGAs) circuits, any other type of integrated circuit (IC), and/or a state machine.

A processor in association with software may be used to implement a radio frequency transceiver for use in a wireless transmit receive unit (WTRU), user equipment (UE), terminal, base station, radio network controller (RNC), or any host computer. The WTRU may be used in conjunction with modules, implemented in hardware and/or software, such as a camera, a video camera module, a videophone, a speakerphone, a vibration device, a speaker, a microphone, a television transceiver, a hands free headset, a keyboard, a Bluetooth® module, a frequency modulated (FM) radio unit, a liquid crystal display (LCD) display unit, an organic light-emitting diode (OLED) display unit, a digital music player, a media player, a video game player module, an Internet browser, and/or any wireless local area network (WLAN) or Ultra Wide Band (UWB) module.

1. A method of performing information-theoretically secure wireless communication, the method comprising:

receiving, at a first wireless transmit/receive unit (WTRU), a signal comprising uniformly distributed correlated random values, wherein the signal is also available to a second WTRU, and wherein the signal is received from a device that is not the first WTRU or the second WTRU;

generating, at the first WTRU, a plurality of input samples from the signal;

deriving, at the first WTRU, an information-theoretically secure cryptographic key using the plurality of input samples and communications exchanged between the first WTRU and the second WTRU over a public communications channel, wherein the deriving comprises conversion comprising:

sorting the plurality of input samples to produce a plurality of ordered samples and assigning each ordered sample an index corresponding to a corresponding input sample of the plurality of input samples,

transforming the plurality of ordered samples to generate a plurality of uniformly distributed ordered samples, and

resorting the plurality of uniformly distributed ordered samples based on the assigned index of each of the uniformly distributed ordered samples to produce a plurality of uniformly distributed samples; and

the first WTRU establishing a communications session with the second WTRU using the cryptographic key.

receiving, at a first wireless transmit/receive unit (WTRU), a signal comprising uniformly distributed correlated random values, wherein the signal is also available to a second WTRU, and wherein the signal is received from a device that is not the first WTRU or the second WTRU;

generating, at the first WTRU, a plurality of input samples from the signal;

deriving, at the first WTRU, an information-theoretically secure cryptographic key using the plurality of input samples and communications exchanged between the first WTRU and the second WTRU over a public communications channel, wherein the deriving comprises conversion comprising:

sorting the plurality of input samples to produce a plurality of ordered samples and assigning each ordered sample an index corresponding to a corresponding input sample of the plurality of input samples,

transforming the plurality of ordered samples to generate a plurality of uniformly distributed ordered samples, and

resorting the plurality of uniformly distributed ordered samples based on the assigned index of each of the uniformly distributed ordered samples to produce a plurality of uniformly distributed samples; and

the first WTRU establishing a communications session with the second WTRU using the cryptographic key.

sorting the plurality of input samples to produce a plurality of ordered samples and assigning each ordered sample an index corresponding to a corresponding input sample of the plurality of input samples,

transforming the plurality of ordered samples to generate a plurality of uniformly distributed ordered samples, and

resorting the plurality of uniformly distributed ordered samples based on the assigned index of each of the uniformly distributed ordered samples to produce a plurality of uniformly distributed samples; and

2. The method of claim 1, wherein the deriving includes privacy amplification.

3. The method of claim 1, wherein the transforming includes rate-matching.

4. The method of claim 1, wherein the deriving includes decoding.

5. The method of claim 4, wherein the decoding includes:

receiving a public component of the plurality of input samples from the second WTRU on the public communications channel; and

applying a modified belief-propagation algorithm on the plurality of input samples.

receiving a public component of the plurality of input samples from the second WTRU on the public communications channel; and

applying a modified belief-propagation algorithm on the plurality of input samples.

6. The method of claim 5, wherein applying the modified belief-propagation algorithm includes calculating a log-likelihood ratio (LLR) of the plurality of input samples.

7. The method of claim 6, wherein calculating the LLR includes performing a uniform-distribution LLR algorithm.

8. The method of claim 6, wherein calculating the LLR includes performing an unknown-distribution LLR algorithm.

9. The method of claim 1, wherein the deriving includes encoding.

10. The method of claim 9, wherein encoding includes reporting a public component of the plurality of input samples to the second WTRU on the public communications channel.

11. The method of claim 10, wherein the encoding includes quantization and channel coding.

12. The method of claim 11, wherein the quantization includes producing a quantization value and producing a quantization error.

13. The method of claim 12, wherein the producing a quantization value includes using a binary quantizer.

14. The method of claim 12, wherein the producing a quantization error includes calculating an absolute value of a corresponding random value.

15. The method of claim 12, wherein the producing a quantization value includes selecting a predetermined number of most significant bits of a corresponding random value.

16. The method of claim 12, wherein the producing a quantization error includes selecting a predetermined number of least significant bits of a corresponding random value.

17. The method of claim 12, wherein the reporting a public component includes transmitting the quantization error.

18. The method of claim 12, wherein the channel coding includes generating a syndrome.

19. The method of claim 18, wherein the reporting a public component includes transmitting the syndrome.

20. The method of claim 19, wherein the encoding includes source coding.

21. The method of claim 20, wherein the source coding includes grey coding.

22. A wireless transmit/receive unit (WTRU), the WTRU comprising:

a processor configured to:

generate a plurality of input samples from a signal comprising uniformly distributed correlated random values; and

derive an information-theoretically secure cryptographic key using the plurality of input samples and communications exchanged between the WTRU and a second WTRU over a public communications channel, wherein the deriving comprises conversion comprising:
sorting the plurality of input samples to produce a plurality of ordered samples and assigning each ordered sample an index corresponding to a corresponding input sample of the plurality of input samples,
transforming the plurality of ordered samples to generate a plurality of uniformly distributed ordered samples, and
resorting the plurality of uniformly distributed ordered samples based on the assigned index of each of the uniformly distributed ordered samples to produce a plurality of uniformly distributed samples; and

a transceiver configured to:

exchange the communications between the WTRU and a second WTRU over the public communications channel;

receive the signal comprising the uniformly distributed correlated random values, wherein the signal is also available to the second WTRU, and wherein the signal is received from a device that is not the first WTRU or the second WTRU; and

communicate with the second WTRU using the information-theoretically secure cryptographic key.

a processor configured to:

generate a plurality of input samples from a signal comprising uniformly distributed correlated random values; and

derive an information-theoretically secure cryptographic key using the plurality of input samples and communications exchanged between the WTRU and a second WTRU over a public communications channel, wherein the deriving comprises conversion comprising:
sorting the plurality of input samples to produce a plurality of ordered samples and assigning each ordered sample an index corresponding to a corresponding input sample of the plurality of input samples,
transforming the plurality of ordered samples to generate a plurality of uniformly distributed ordered samples, and
resorting the plurality of uniformly distributed ordered samples based on the assigned index of each of the uniformly distributed ordered samples to produce a plurality of uniformly distributed samples; and

a transceiver configured to:

exchange the communications between the WTRU and a second WTRU over the public communications channel;

receive the signal comprising the uniformly distributed correlated random values, wherein the signal is also available to the second WTRU, and wherein the signal is received from a device that is not the first WTRU or the second WTRU; and

communicate with the second WTRU using the information-theoretically secure cryptographic key.

generate a plurality of input samples from a signal comprising uniformly distributed correlated random values; and

derive an information-theoretically secure cryptographic key using the plurality of input samples and communications exchanged between the WTRU and a second WTRU over a public communications channel, wherein the deriving comprises conversion comprising:
sorting the plurality of input samples to produce a plurality of ordered samples and assigning each ordered sample an index corresponding to a corresponding input sample of the plurality of input samples,
transforming the plurality of ordered samples to generate a plurality of uniformly distributed ordered samples, and
resorting the plurality of uniformly distributed ordered samples based on the assigned index of each of the uniformly distributed ordered samples to produce a plurality of uniformly distributed samples; and

exchange the communications between the WTRU and a second WTRU over the public communications channel;

receive the signal comprising the uniformly distributed correlated random values, wherein the signal is also available to the second WTRU, and wherein the signal is received from a device that is not the first WTRU or the second WTRU; and

communicate with the second WTRU using the information-theoretically secure cryptographic key.

23. The WTRU of claim 22, wherein the processor includes a privacy amplification processor configured to remove a public component from the plurality of input samples.

24. The WTRU of claim 22, wherein the data conversion unit is configured to transform using rate-matching.

25. The WTRU of claim 22, wherein the processor includes a decoding unit configured to:

receive a public component of the plurality of input samples from the WTRU on the public communications channel; and

apply a modified belief-propagation algorithm to the plurality of input samples.

receive a public component of the plurality of input samples from the WTRU on the public communications channel; and

apply a modified belief-propagation algorithm to the plurality of input samples.

26. The WTRU of claim 25, wherein the decoding unit is configured to calculate a log-likelihood ratio (LLR) of the plurality of input samples for use in the modified belief-propagation algorithm.

27. The WTRU of claim 26, wherein the decoding unit is configured to calculate the LLR using a uniform-distribution LLR.

28. The WTRU of claim 26, wherein the decoding unit is configured to calculate the LLR using an unknown-distribution LLR.

29. The WTRU of claim 22, wherein the processor includes:

a quantization unit configured to produce a quantization value and a quantization error; and

a channel coding unit configured to generate a syndrome.

a quantization unit configured to produce a quantization value and a quantization error; and

a channel coding unit configured to generate a syndrome.

30. The WTRU of claim 29, wherein the transceiver is configured to report the quantization error and the syndrome to the second WTRU on the public communications channel.

31. The WTRU of claim 29, wherein the quantization unit is configured to produce a quantization value using a binary quantizer.

32. The WTRU of claim 29, wherein the quantization unit is configured to produce a quantization error by calculating an absolute value of a corresponding random value.

33. The WTRU of claim 29, wherein the quantization unit is configured to produce a quantization value by selecting a predetermined number of most significant bits of a corresponding random value.

34. The WTRU of claim 29, wherein the quantization unit is configured to produce a quantization error by selecting a predetermined number of least significant bits of a corresponding random value.

35. The WTRU of claim 29, wherein the processor includes a source coding unit configured to convert the quantization value to a bit string.

36. The WTRU of claim 35, wherein the source coding unit is configured to convert using grey coding.