Patent Publication Number: US-10331411-B2

Title: Systems and methods for generating random numbers using physical variations present in material samples

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application is a national stage filing under 35 U.S.C. 371 of PCT/US2015/063184, filed Dec. 1, 2015, which claims the benefit of U.S. Application No. 62/086,961, filed Dec. 3, 2014, the disclosure of which is incorporated by reference in its/their entirety herein. 
    
    
     TECHNICAL FIELD 
     The present disclosure relates to generating random numbers using physical variations present in material samples. 
     BACKGROUND 
     Random number generators (RNGs) are important for applications which require the creation and use of a sequence of numbers or symbols lacking any order or pattern. RNGs can be implemented in applications such as cryptography, gambling, computer modeling or simulation, and statistical sampling. In many cryptographic applications, for example, the underlying encryption protocols are only as good (e.g., secure and/or robust) as the random numbers they use. As such, randomness is a desirable feature and having access to many cheap high quality random bits is often essential. RNGs may typically fall into two categories: pseudo random number generators (PRNGs) and physical random number generators. A PRNG inputs a small random seed and generates a longer string which appears random. However, a PRNG uses an entirely deterministic process and so the knowledge of the algorithm used together with a seed is sufficient to guess the exact longer sequence. Cryptographic algorithms using poorly designed PRNGs may be subject to cryptoanalytic attack. In contrast, physical RNGs use physical phenomena such as entropic or acoustical variation or radioactive decay, for example, to produce random numbers. Theoretically the bit stream they produce is completely unpredictable. 
     SUMMARY 
     There is a desire to generate high-quality random bits or random numbers based on their prevalence in industrial or consumer applications and cryptographic importance. Many materials have inherent physical variability in their properties which can be used to generate high quality random bits. Briefly, in one aspect, the present disclosure describes systems and methods for generating random bits by using physical variations present in material samples. The use of inherent physical variability of materials provides a distinct, cost-effective way to obtain high-quality random numbers. Some embodiments in this disclosure also leverage the randomness inherent in the manufacturing process to generate random numbers. 
     In one aspect, a method of generating random numbers includes providing one or more material samples. One or more material properties for each of the material samples are measured where the measured material properties have respective variability. One or more initial random bit streams are derived from the measured material properties for each of the material samples. 
     In another aspect, a method of extracting random numbers from a batch of material samples, is provided. The batch of material samples can have substantially the same composition and can be produced by substantially the same process. One or more material properties for each of the material samples are measured. The measured material properties have respective variability. One or more initial random bit streams are derived from the measured material properties for each of the material samples. The derived initial random bit streams for the batch of material samples are combined to arrive at a combined random bit stream. One or more random numbers are generated by applying a randomness extraction algorithm to the combined random bit stream. 
     Various unexpected results and advantages are obtained in exemplary embodiments of the disclosure. One such advantage of exemplary embodiments of the present disclosure is that the utilized randomness described herein is originated from inherent physical variability of material properties, which provides distinct, cost-effective, high-quality random bit streams that are useful in a variety of applications. 
     LISTING OF EXEMPLARY EMBODIMENTS 
     Exemplary embodiments are listed below. It is to be understood that any of embodiments A to K, L to P, and Q to X can be combined. 
     Embodiment A. A method of generating random numbers, comprising:
         providing one or more material samples;   measuring one or more material properties for each of the material samples, the measured material properties having variability; and   deriving one or more initial random bit streams from the measured material properties for each of the material samples.       

     Embodiment B. The method of embodiment A, further comprising generating one or more secondary random bit streams by applying a randomness extraction algorithm to the derived initial random bit streams. 
     Embodiment C. The method of embodiment A or B, wherein measuring the material properties comprises capturing an image for a surface of the material samples. 
     Embodiment D. The method of embodiment C, wherein deriving the initial random bit streams further comprises determining surface variations of a characteristic feature from the image of the material samples, the characteristic feature related to a substructure or a texture of the surface of the material samples, and converting the surface variations into the initial random bit streams. 
     Embodiment E. The method of embodiment D, wherein determining the surface variations further comprises converting the image to a binary representation based on intensity values of pixels of the image. 
     Embodiment F. The method of any one of preceding embodiments, wherein the material properties include an optical feature, an acoustical feature, an elastic feature, a structural feature, an electronic feature, a magnetic feature, an electrets related feature, or a mechanical feature, and the variability of material properties is naturally formed or related to a specific manufacturing process for making the material samples. 
     Embodiment G. The method of any one of preceding embodiments, wherein the randomness extraction algorithm includes a block parity extractor. 
     Embodiment H. The method of any one of preceding embodiments, wherein the secondary random bit streams are capable of passing a test of independent, identically distributed (IID) random bits. 
     Embodiment I. The method of any one of preceding embodiments, wherein the one or more material samples are a batch of material samples, wherein the compositions of the batch of material samples are substantially the same, and are produced by substantially the same process. 
     Embodiment J. The method of embodiment I, further comprising combining the initial random bit streams from the batch of material samples to arrive at a combined random bit stream, and generating the secondary random bit streams by applying the randomness extraction algorithm to the combined random bit stream. 
     Embodiment K. The method of embodiment I or J, wherein the batch of material samples is selected from the group consisting of abrasives, optical films, and nonwovens. 
     Embodiment L. A method of extracting random numbers from a batch of sample materials, the batch of materials consisting of substantially the same composition and being produced by substantially the same process, the method comprising:
         measuring one or more material properties for each of the material samples, the measured material properties having variability;   deriving one or more initial random bit streams from the measured material properties for each of the material samples;   combining the derived initial random bit streams for the batch of material samples to arrive at a combined random bit stream; and   generating one or more random numbers by applying a randomness extraction algorithm to the combined random bit stream.       

     Embodiment M. The method of embodiment L, wherein measuring the material properties comprises capturing surface images for the batch of material samples. 
     Embodiment N. The method of embodiment M, wherein deriving the initial random bit streams further comprises determining surface variations of a characteristic feature from the surface images. 
     Embodiment O. The method of embodiment N, wherein determining the surface variations further comprises converting the images to binary representations based on intensity values of pixels of the images. 
     Embodiment P. The method of any one of embodiments L to O, wherein the material properties include an optical feature, an acoustical feature, an elastic feature, a structural feature, an electronic feature, a magnetic feature, an electrets related feature, or a mechanical feature. 
     Embodiment Q. A random number generator, comprising:
         a measurement component configured to measure one or more material properties for one or more material samples, the measured material properties having variability; and   a computation component functionally connected to the measurement component, the computation component including a processor configured to deriving one or more initial random bit streams from the measured material properties for each of the material samples.       

     Embodiment R. The random number generator of embodiment Q, wherein the processor is configured to generate one or more secondary random bit streams by applying a randomness extraction algorithm to the derived initial random bit streams. 
     Embodiment S. The random number generator of embodiment Q or R, further comprises a memory configured to store material properties data. 
     Embodiment T. The random number generator of any one of embodiments Q to S, wherein the measurement component includes a camera configured to capture one or more images of the material samples. 
     Embodiment U. The random number generator of embodiment T, wherein the processor is configured to determine surface variations of a characteristic feature from the one or more images of the material samples, the characteristic feature related to a substructure or a texture of the surface of the material samples, and the processor is configured to convert the surface variations into the initial random bit streams. 
     Embodiment V. The random number generator of embodiment U, wherein the processor is configured to convert the one or more images to a binary representation based on intensity values of pixels of the respective images. 
     Embodiment W. The random number generator of any one of embodiments Q to V, wherein the material properties include an optical feature, an acoustical feature, an elastic feature, a structural feature, an electronic feature, a magnetic feature, an electrets related feature, or a mechanical feature. 
     Embodiment X. The random number generator of any one of embodiments R to V, wherein the randomness extraction algorithm includes a block parity extractor. 
     Various aspects and advantages of exemplary embodiments of the disclosure have been summarized. The above Summary is not intended to describe each illustrated embodiment or every implementation of the present certain exemplary embodiments of the present disclosure. The Drawings and the Detailed Description that follow more particularly exemplify certain preferred embodiments using the principles disclosed herein. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The disclosure may be more completely understood in consideration of the following detailed description of various embodiments of the disclosure in connection with the accompanying figures, in which: 
         FIG. 1  is a flow diagram of a method for generating random numbers, according to one embodiment. 
         FIG. 2  illustrates a block diagram for a random number generator, according to one embodiment. 
         FIG. 3A  illustrates an optical image of a material sample and intensity values of image pixels of the optical image, according to one embodiment. 
         FIG. 3B  illustrates a converted optical image and binary representation of intensity values thereof, according to one embodiment. 
         FIG. 3C  illustrates obtaining initial bits based on the binary representation of  FIG. 3B , according to one embodiment. 
         FIG. 3D  illustrates generating secondary bits based on the initial bits of  FIG. 3C , according to one embodiment. 
         FIG. 3E  illustrates processing an exemplary initial random bit stream by applying a block parity extractor to generate a secondary bit stream, according to one embodiment. 
         FIG. 4A  illustrates optical images of a batch of material samples and intensity values of image pixels of the respective optical images, according to one embodiment. 
         FIG. 4B  illustrates converted optical images and binary representations of intensity values for the batch of material samples of  FIG. 4A , according to one embodiment. 
         FIG. 4C  illustrates deriving initial random bit streams based on the binary representations of  FIG. 4B , according to one embodiment. 
         FIG. 4D  illustrates concatenating the initial random bit streams of  FIG. 4C , according to one embodiment. 
         FIG. 4E  illustrates generating a secondary random bit stream based on the concatenated random bit stream of  FIG. 4D , according to one embodiment. 
         FIG. 5  illustrates an optical image of a flame-embossed film sample. 
         FIG. 6  illustrates an optical image of a nonwoven material sample. 
     
    
    
     In the drawings, like reference numerals indicate like elements. While the above-identified drawing, which may not be drawn to scale, sets forth various embodiments of the present disclosure, other embodiments are also contemplated, as noted in the Detailed Description. In all cases, this disclosure describes the presently disclosed disclosure by way of representation of exemplary embodiments and not by express limitations. It should be understood that numerous other modifications and embodiments can be devised by those skilled in the art, which fall within the scope and spirit of this disclosure. 
     DETAILED DESCRIPTION 
     Many materials have inherent physical variability in their properties which can be used to generate high quality random bits. Briefly, in one aspect, the present disclosure describes systems and methods for generating random bits by using physical variations present in material samples. The use of inherent physical variability of materials provides a cost-effective way to obtain high quality random numbers. In some embodiments, the physical variability of material properties may naturally formed and present in the materials, while in other embodiments, specific manufacturing processes may generate or modify the physical variability of material properties. Some embodiments in this disclosure also leverage the randomness inherent in the manufacturing process to generate random numbers. 
       FIG. 1  illustrates a method  100  of generating random numbers or random bit streams. At  110 , one or more material samples are provided. The material samples can be various material samples that are commercially available from, for example, 3M Company, St. Paul, Minn. The material samples can include, for example, abrasives, optical films, nonwovens, etc. The material samples exhibit at least one material property having variability which can be naturally formed, or originated from specific manufacturing processes. The material property can include, for example, an optical feature, an acoustical feature, an elastic feature, a structural feature, an electronic feature, a magnetic feature, an electrets related feature, a mechanical feature, etc. The method  100  then proceeds to  120 . 
     At  120 , one or more material properties are measured for each of the material samples. In some embodiments, a surface image of the material samples can be captured by, for example, a digital camera. The method  100  then proceeds to  130 . 
     At  130 , one or more initial random bit streams are derived from the measured material properties for each of the material samples. In some embodiments, surface variations of a characteristic feature can be determined from the surface image of the material samples. The characteristic feature can be related to, for example, a substructure or texture of the surface of material samples. In some embodiments, the surface image of the material sample can be converted into a binary representation based on, for example, intensity values of the pixels of the image. The method  100  then proceeds to  140 . 
     Optionally, at  140 , one or more secondary random bit streams are generated by applying a randomness extraction algorithm to the initial random bit streams. The randomness extraction algorithm can include one or more of, for example, a block parity extractor, a von Neumann extractor, a random walk extractor, etc. In some embodiments, two or more randomness extraction algorithms can be applied to the initial random bit streams simultaneously or sequentially to generate the secondary random bit streams. 
       FIG. 2  illustrates a random number generator  200  for generating random numbers from material samples by implementing, for example, the method  100 , according to one embodiment. The random number generator  200  includes a measurement component  224 , a computation component  226 , and one or more input/output devices  216 . 
     The measurement component  224  is configured to measure one or more material properties of the material samples. The measurement component  224  can be various measurement tools to measure material properties having inherent variability including, for example, one or more of an optical feature, an acoustical feature, an elastic feature, a structural feature, an electronic feature, a magnetic feature, electrets, or a mechanical feature. In some embodiments, the measurement component  224  can include, for example, a camera for capturing one or more images of the material samples. 
     In the embodiment of  FIG. 2 , the computation component  226  includes a processor  212  and a memory  214 . The computation component  226  is functionally connected to the measurement component  224 , receives signals related to the measured material properties from the measurement component  224 , and derives one or more initial random bit streams from the received signals. The processor  212  may apply one or more randomness extraction algorithms to the derived initial random bit streams to generate one or more secondary random bit streams. The randomness extraction algorithm may include one or more of, for example, a block parity extractor, a von Neumann extractor, a random walk extractor, etc. In some embodiments, the randomness extraction algorithm can be predetermined and stored in the memory  214 . In some embodiments, the randomness extraction algorithm can be selected from a list of randomness extraction algorithms stored in the memory  214  according to the derived initial random bit streams. The generated random numbers may be stored in the memory  214 . The processor  212  may also perform additional processing on a random bit stream such as, for example, binarization and vectorization to be discussed further below. 
     In some embodiments, the measurement component  224  may be a portable device that can work in field. The measurement component  224  can wirelessly communicate with a remote computing device such as, for example, the computation component  226  by sending and receiving signals. The computation component  226  may be integrated with, for example, a computer, a server, a mobile phone, etc. The computation component  226  can process the received material property signals and send the generated random numbers to the input/output device  216  to display thereon. 
     The memory  214  stores information. In some embodiments, the memory  214  can store instructions for performing the methods or processes described herein. In some embodiments, material properties data can be pre-stored in the memory  214 . One or more properties from the material samples, for example, an optical feature, an acoustical feature, an elastic feature, a structural feature, an electronic feature, a magnetic feature, an electrets related feature, or a mechanical feature, may be stored as the material properties data. 
     The memory  214  may include any volatile or non-volatile storage elements. Examples may include random access memory (RAM) such as synchronous dynamic random access memory (SDRAM), read-only memory (ROM), non-volatile random access memory (NVRAM), electrically erasable programmable read-only memory (EEPROM), and FLASH memory. Examples may also include hard-disk, magnetic tape, a magnetic or optical data storage media, a compact disk (CD), a digital versatile disk (DVD), a Blu-ray disk, and a holographic data storage media. 
     The processor  212  may include, for example, one or more general-purpose microprocessors, specially designed processors, application specific integrated circuits (ASIC), field programmable gate arrays (FPGA), a collection of discrete logic, and/or any type of processing device capable of executing the techniques described herein. In some embodiments, the processor  212  (or any other processors described herein) may be described as a computing device. In some embodiments, the memory  214  may be configured to store program instructions (e.g., software instructions) that are executed by the processor  212  to carry out the processes or methods described herein. In other embodiments, the processes or methods described herein may be executed by specifically programmed circuitry of the processor  212 . In some embodiments, the processor  212  may thus be configured to execute the techniques for generating random numbers described herein. The processor  212  (or any other processors described herein) may include one or more processors. 
     Input/output device  216  may include one or more devices configured to input or output information from or to a user or other device. In some embodiments, the input/output device  216  may present a user interface  218  where a user may control the assessment of the generation of a random number. For example, user interface  218  may include a display screen for presenting visual information to a user. In some embodiments, the display screen includes a touch sensitive display. In some embodiments, a user interface  218  may include one or more different types of devices for presenting information to a user. The user interface  218  may include, for example, any number of visual (e.g., display devices, lights, etc.), audible (e.g., one or more speakers), and/or tactile (e.g., keyboards, touch screens, or mice) feedback devices. In some embodiments, the input/output devices  216  may represent one or more of a display screen (e.g., a liquid crystal display or light emitting diode display) and/or a printer (e.g., a printing device or component for outputting instructions to a printing device). In some embodiments, the input/output device  116  may be configured to accept or receive program instructions (e.g., software instructions) that are executed by the processor  112  to carry out the embodiments described herein. 
     The random number generator  200  may also include other components and the functions of any of the illustrated components including the processor  212 , the memory  214 , and the input/output devices  216  may be distributed across multiple components and separate devices such as, for example, computers. The random number generator  200  may be configured as a workstation, desktop computing device, notebook computer, tablet computer, mobile computing device, or any other suitable computing device or collection of computing devices. The random number generator  200  may operate on a local network or be hosted in a Cloud computing environment. The illustrated components of  FIG. 2  are shown merely to explain various aspects of the present disclosure and the addition or removal of components would be apparent to one of skill in the art. 
     In some embodiments, the random number generator  200  allows a user to take a material sample, e.g., a 3M manufactured material sample, and insert the material sample into the generator  200 . The inherent randomness present in the material sample can be measured and a randomness extraction algorithm can be applied which, in turn, can produce a sequence of random bits. The random bits can then be used by any computing device for any application that needs, for example, encryption. 
     In some embodiments, a material-based random number generator such as the random number generator  200  can be used in object identification. Random numbers uniquely derived from material features can allow for digital material authentication. For example, the method  100  of  FIG. 1  and the random number generator  200  of  FIG. 2  can be used to add a cryptographic or steganographic features to, for example, personal documents such as passports or identification cards. 
       FIGS. 3A to 3E  illustrates generating random numbers from optical images of material samples, according to one embodiment.  FIG. 3A  shows an optical image  32  measured for a material sample. The optical image  32  includes pixels having various intensities which may relate to, for example, a substructure or a texture on the surface of the material sample. The optical image  32  can be captured by, for example, the measurement component  226  of  FIG. 2 . The various intensities of the pixels reflect, for example, the inherent randomness of the substructure or texture of the material sample surface. The intensity values of image pixels in an exemplary portion  322  of the optical image  32  are shown in Table  34 . The intensity values can be normalized, via for example the computation component  226  of  FIG. 2 , to be between 0 and 1 and are listed in an array m×n. It is to be understood that any portion of the optical image  32  or the whole optical image  32  can be used to obtain the corresponding intensity values. 
     The optical image  32  can be converted to a binary representation  32 ′ shown in  FIG. 3B . Accordingly, the intensity values of image pixels in Table  34  are converted into binary values 0 or 1 in Table  34 ′. The conversion can be conducted by, for example, the computation component  226  of  FIG. 2 . The array of binary values of Table  34 ′ is then vectorized by the computation component  226  to obtain an initial bit stream  36 . The initial bit stream  26  includes a series of bits  0  or  1  as shown in  FIG. 3C . In some embodiments, the initial random bit stream  36  may include the (m×n) number of binary bits which correspond to the binary representation  32 ′ of the optical image  32  as shown in  FIG. 3B . It is to be understood that the initial bit stream  36  can be obtained from at least a portion of the array of binary values in Table  34 ′ by any appropriate ways. Instructions for obtaining the initial bit stream  36  from the array of binary values in Table  34 ′ can be stored in the memory  214  and executed by the processor  212  of the computation component  226 . 
     The initial random bit stream  36  is then processed by applying a randomness extraction algorithm to generate a secondary random bit stream  38  as shown in  FIG. 3D . The randomness extraction algorithm can include one or more of, for example, a block parity extractor, a von Neumann extractor, a random walk extractor, etc. It is to be understood that a randomness extraction algorithm can be iteratively applied to the initial bit stream and/or the secondary random bit stream to generate a final random bit stream. It is also to be understood that two or more randomness extraction algorithms can be combined to apply to the initial random bit stream sequentially or at the same time. 
     In some embodiments, the computation component  226  of  FIG. 2  can use a predetermined randomness extraction algorithm such as, for example, a block parity extractor to process the initial random bit stream  36  and generate the secondary random bit stream  38 . In some embodiments, the computation component  226  can select one or more randomness extraction algorithms from a list of randomness extraction algorithms stored in the memory  214  according to the initial random bit streams  36 , and apply the selected one or more randomness extraction algorithms to process the initial random bit stream  36  and generate the secondary random bit stream  38 . 
     In the embodiment of  FIG. 3E , an exemplary initial random bit stream  37  is processed by applying a block parity extractor to generate a secondary bit stream  39 . The initial bit stream  37  is divided into k blocks (k being integers not less than one). Each of the blocks can include l bits or have a length l (l being integers not less than one). The parity of each block can be calculated to determine the corresponding bit of the secondary bit stream  39 . For example, for a specific block, if the number of 0 bits is greater than the number of 1 bits in the specific block, the corresponding bit of the secondary bit stream  39  can be determined to be 0. If the number of 0 bits is not greater than the number of 1 bits in the specific block, the corresponding bit of the secondary bit stream  39  can be determined to be 1. 
     In some embodiments, an initial random bit stream such as, for example, the initial random bit stream  36  or  37  of  FIG. 3D or 3E , can include n independent and biased (but not necessarily identically distributed) bits. The ith bit of the initial random bit stream can be distributed as a Bernoulli random variable with parameter δ≤p i ≤1−δ, for some 0&lt;δ&lt;1. For every constant δ&gt;0, every integer n and m, there is a function ƒ: {0, 1} n →{0, 1} m  that is an ∈-extractor for the initial random bit, with ∈=m·2 −Ω(n/m) . 
       FIGS. 4A to 4E  illustrates generating random numbers from optical images of a batch of material samples, according to one embodiment. “A batch of material samples” described herein refers to multiple material samples that include substantially the same composition, and/or produced by substantially the same process. Each of the material samples can exhibit similar properties having respective variability or inherent randomness. 
       FIG. 4A  shows optical images  41  and  42  measured for a first material sample and a second material sample, respectively. The first and second material samples are from the same batch of material samples. The optical images  41  and  42  each include pixels having various intensities which related to a substructure or a texture on the surface of the respective material samples. The various intensities of the pixels reflect the inherent randomness of the substructure or texture of the respective material sample surface. The intensity values of image pixels in the respective portions of the optical images  41  and  42  are shown in Tables  44  and  45 , respectively. The intensity values are normalized to be between 0 and 1 and are listed in arrays. The intensity values in Tables  44  and  45  vary with each other. Such variation reflects the randomness inherent in the properties of the material samples, and the randomness inherent in the processes for producing the batch of material samples. 
     The optical images  41  and  42  can be respectively converted to binary representations  41 ′ and  42 ′ shown in  FIG. 4B . Accordingly, the intensity values of image pixels in Tables  44  and  45  are converted into binary values 0 or 1 in Table  44 ′ and  45 ′. The arrays of binary values are then vectorized, via for example the computation component  226  of  FIG. 2 , to obtain respective initial bit streams  46   a  and  46   b  each including a series of bits  0  or  1  as shown in  FIG. 4C . The initial random bit streams  46   a  and  46   b  each can include the (m×n) number of binary bits which correspond to the binary representations  41 ′ and  42 ′ of the optical images  41  and  42  as shown in  FIG. 4B . 
     The initial random bit streams  46   a  and  46   b  that are respectively derived from the first and second material samples can be combined, via for example the computation component  226  of  FIG. 2 , to obtain a combined random bit stream. In the embodiment of  FIG. 4D , the initial random bit streams  46   a  and  46   b  are concatenated to be a combined initial random bit stream  46 . It is to be understood that the initial random bit streams  46   a  and  46   b  can be combined in various manners to arrive at the combined initial random bit stream. Instructions for obtaining the combined initial bit stream  46  from the initial random bit streams  46   a  and  46   b  can be stored in the memory  214  and executed by the processor  212  of the computation component  226 . 
     The combined initial random bit stream  46  is then processed, via for example the computation component  226 , by applying a randomness extraction algorithm to generate a secondary random bit stream  48  as shown in  FIG. 4E . The randomness extraction algorithm can include one or more of, for example, a block parity extractor, a von Neumann extractor, a random walk extractor, etc. It is to be understood that a randomness extraction algorithm can be iteratively applied to the initial bit stream and the secondary random bit stream to generate a final random bit stream. It is also to be understood that more than one randomness extraction algorithms can be combined to apply to the initial random bit stream. 
     The use of a batch of material samples may generate a random bit stream such as the secondary random bit stream  48  that is relatively longer that a random bit stream obtained from a single material sample. In addition, random bit streams obtained from a batch of material samples may be used for authentication of material samples. 
     As discussed above, an initial random bit stream can be processed by applying a randomness extraction algorithm to generate a secondary random bit stream. The performance of the randomness extractor can be justified by running the output of the randomness extractor through various tests for randomness. In some embodiments, a test suite by the National Institute for Standards in Technology can be used. The test suite can include tests of, for example, frequency, frequency within a block, runs, longest run of a block, spectral discrete Fourier transform (DFT), overlapping template match, approximate entropy, cumulative sums, etc. 
     In some embodiments, each of the above tests can be set up by taking a statistic of an initial or secondary random bit stream and fitting the statistic to a reference distribution. The reference distribution can be, for example, normal or chi-squared. The probability of failure under assumption of randomness (or a p-value) can be calculated. In some embodiments, if the p-value is below 0.01, a test failure is returned. In some embodiments, the secondary random bit streams described herein are capable of passing a test for independent, identically distributed (IID) random bits. 
     Unless otherwise indicated, all numbers expressing quantities or ingredients, measurement of properties and so forth used in the specification and embodiments are to be understood as being modified in all instances by the term “about.” Accordingly, unless indicated to the contrary, the numerical parameters set forth in the foregoing specification and attached listing of embodiments can vary depending upon the desired properties sought to be obtained by those skilled in the art utilizing the teachings of the present disclosure. At the very least, and not as an attempt to limit the application of the doctrine of equivalents to the scope of the claimed embodiments, each numerical parameter should at least be construed in light of the number of reported significant digits and by applying ordinary rounding techniques. 
     Exemplary embodiments of the present disclosure may take on various modifications and alterations without departing from the spirit and scope of the present disclosure. Accordingly, it is to be understood that the embodiments of the present disclosure are not to be limited to the following described exemplary embodiments, but is to be controlled by the limitations set forth in the claims and any equivalents thereof. 
     Various exemplary embodiments of the disclosure will now be described with particular reference to the Drawings. Exemplary embodiments of the present disclosure may take on various modifications and alterations without departing from the spirit and scope of the disclosure. Accordingly, it is to be understood that the embodiments of the present disclosure are not to be limited to the following described exemplary embodiments, but are to be controlled by the limitations set forth in the claims and any equivalents thereof. 
     The operation of the present disclosure will be further described with regard to the following detailed examples. These examples are offered to further illustrate the various specific and preferred embodiments and techniques. It should be understood, however, that many variations and modifications may be made while remaining within the scope of the present disclosure. 
     Examples 
     These Examples are merely for illustrative purposes and are not meant to be overly limiting on the scope of the appended claims. Notwithstanding that the numerical ranges and parameters setting forth the broad scope of the present disclosure are approximations, the numerical values set forth in the specific examples are reported as precisely as possible. Any numerical value, however, inherently contains certain errors necessarily resulting from the standard deviation found in their respective testing measurements. At the very least, and not as an attempt to limit the application of the doctrine of equivalents to the scope of the claims, each numerical parameter should at least be construed in light of the number of reported significant digits and by applying ordinary rounding techniques. 
     Testing Material Samples 
     Optical images for flame embossed films, blown microfiber (BMF) filtration materials, nonwoven materials, and resin materials commercially available from 3M Company, Saint Paul, Minn., are used to derive initial random bit streams.  FIGS. 5 and 6  illustrate optical images for an exemplary flame-embossed film sample and an exemplary non-woven sample, respectively. 
     Testing Randomness Extraction Algorithms 
     A various randomness extraction algorithms were applied to the above initial random bit streams to generate secondary random bit streams. In particular, a block parity extractor was applied to the initial random bit streams. 
     A test suite by the National Institute for Standards in Technology was used to evaluate the derived initial random bit streams and the respectively generated secondary random bit streams. The test suite includes tests of, for example, frequency, frequency within a block, runs, longest run of is in a block, spectral discrete Fourier transform (DFT), overlapping template match, approximate entropy, cumulative sums, etc. 
     Each of the above tests was set up by taking a statistic of the initial or secondary random bit stream and fitting the statistic to a reference distribution. The reference distribution can be, for example, normal or chi-squared. The probability of failure under assumption of randomness (or a p-value) can be calculated. If the p-value is below 0.01, a test failure is returned. 
     Table 1 lists test results in terms of randomness for initial random bit streams derived from the flame embossed film sample. Each fail rate was calculated by running tests on 1000 different bit strings derived from the material sample. 
     
       
         
           
               
               
               
             
               
                 TABLE 1 
               
               
                   
               
               
                   
                 Fail Rate  
                 Fail Rate  
               
               
                 Test 
                 (128 bits) 
                 (1024 bits) 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
            
               
                 Frequency (monobit) test 
                 0.097 
                 0.009 
               
               
                 Frequency within a block 
                 0.026 
                 0.103 
               
               
                 Runs test 
                 0.009 
                 0.007 
               
               
                 Longest run of 1 s in a block 
                 0.031 
                 0.007 
               
               
                 DFT (spectral) Test 
                 0.022 
                 0.012 
               
               
                 Overlapping Template Match 
                 0.087 
                 0.02 
               
               
                 Approximate Entropy 
                 0.045 
                 0.006 
               
               
                 Cumulative Sums 
                 0.011 
                 0.001 
               
               
                   
               
            
           
         
       
     
     Table 2 lists test results in terms of randomness for the secondary random bit streams derived from the initial random bit streams of Table 1 by applying a block parity extractor. Each fail rate was calculated by running tests on 1000 different bit strings. 128 bit strings were extracted with 1=45 initial random bits per secondary random bit. 1024 bit strings were extracted with l=5 initial random bits per secondary random bit. 
     
       
         
           
               
               
               
             
               
                 TABLE 2 
               
               
                   
               
               
                   
                 Fail Rate  
                 Fail Rate  
               
               
                 Test 
                 (128 bits) 
                 (1024 bits) 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
            
               
                 Frequency (monobit) test 
                 0.014 
                 0.014 
               
               
                 Frequency within a block 
                 0.012 
                 0.015 
               
               
                 Runs test 
                 0.013 
                 0.009 
               
               
                 Longest run of 1 s in a block 
                 0.009 
                 0.005 
               
               
                 DFT (spectral) Test 
                 0.021 
                 0.01 
               
               
                 Overlapping Template Match 
                 0.001 
                 0.028 
               
               
                 Approximate Entropy 
                 0.017 
                 0.008 
               
               
                 Cumulative Sums 
                 0.002 
                 0.001 
               
               
                   
               
            
           
         
       
     
     Table 3 lists test results in terms of randomness for initial random bit streams derived from the nonwoven material sample. Each fail rate was calculated by running tests on 1000 different bit strings derived from the material sample. 
     
       
         
           
               
               
               
             
               
                 TABLE 3 
               
               
                   
               
               
                   
                 Fail Rate  
                 Fail Rate  
               
               
                 Test 
                 (128 bits) 
                 (1024 bits) 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
            
               
                 Frequency (monobit) test 
                 0.998 
                 1 
               
               
                 Frequency within a block 
                 0.855 
                 1 
               
               
                 Runs test 
                 0.433 
                 0 
               
               
                 Longest run of 1 s in a block 
                 0.131 
                 1 
               
               
                 DFT (spectral) Test 
                 0.085 
                 0.004 
               
               
                 Overlapping Template Match 
                 0 
                 0.008 
               
               
                 Approximate Entropy 
                 0 
                 0.008 
               
               
                 Cumulative Sums 
                 0.083 
                 0.364 
               
               
                   
               
            
           
         
       
     
     Table 4 lists test results in terms of randomness for the secondary random bit streams derived from the initial random bit streams of Table 3 by applying a block parity extractor. Each fail rate was calculated by running tests on 1000 different bit strings. 128 bit strings were extracted with l=288 initial random bits per secondary random bit. 1024 bit strings were extracted with l=90 initial random bits per secondary random bit. 
     
       
         
           
               
               
               
             
               
                 TABLE 4 
               
               
                   
               
               
                   
                 Fail Rate  
                 Fail Rate  
               
               
                 Test 
                 (128 bits) 
                 (1024 bits) 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
            
               
                 Frequency (monobit) test 
                 0.01 
                 0.01 
               
               
                 Frequency within a block 
                 0.009 
                 0.0025 
               
               
                 Runs test 
                 0.007 
                 0.01 
               
               
                 Longest run of 1 s in a block 
                 0.01 
                 0.0025 
               
               
                 DFT (spectral) Test 
                 0.02 
                 0.01 
               
               
                 Overlapping Template Match 
                 0 
                 0.0325 
               
               
                 Approximate Entropy 
                 0.008 
                 0.005 
               
               
                 Cumulative Sums 
                 0.002 
                 0 
               
               
                   
               
            
           
         
       
     
     The application of the block parity extractor was benchmarked against state of the art pseudo random number generators (PRNGs), specifically the one built into MATLAB. The PRNG took a small random seed and generated a longer string which appears random. Table 5 lists test results in terms of randomness for the string generated by the MATLAB PRNG. Overall, the test results for the block parity extractor (e.g., Tables 2 and 4) has its performance be on par with the MATLAB PRNG (e.g., Table 5). 
     
       
         
           
               
               
               
             
               
                 TABLE 5 
               
               
                   
               
               
                   
                 Fail Rate  
                 Fail Rate  
               
               
                 Test 
                 (128 bits) 
                 (1024 bits) 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
            
               
                 Frequency (monobit) test 
                 0.014 
                 0.005 
               
               
                 Frequency within a block 
                 0.011 
                 0.011 
               
               
                 Runs test 
                 0.014 
                 0.01 
               
               
                 Longest run of 1 s in a block 
                 0.009 
                 0.007 
               
               
                 DFT (spectral) Test 
                 0.016 
                 0.01 
               
               
                 Overlapping Template Match 
                 0.027 
                 0 
               
               
                 Approximate Entropy 
                 0.015 
                 0.012 
               
               
                 Cumulative Sums 
                 0.004 
                 0.002 
               
               
                   
               
            
           
         
       
     
     Reference throughout this specification to “one embodiment,” “certain embodiments,” “one or more embodiments,” or “an embodiment,” whether or not including the term “exemplary” preceding the term “embodiment,” means that a particular feature, structure, material, or characteristic described in connection with the embodiment is included in at least one embodiment of the certain exemplary embodiments of the present disclosure. Thus, the appearances of the phrases such as “in one or more embodiments,” “in certain embodiments,” “in one embodiment,” or “in an embodiment” in various places throughout this specification are not necessarily referring to the same embodiment of the certain exemplary embodiments of the present disclosure. Furthermore, the particular features, structures, materials, or characteristics may be combined in any suitable manner in one or more embodiments. 
     While the specification has described in detail certain exemplary embodiments, it will be appreciated that those skilled in the art, upon attaining an understanding of the foregoing, may readily conceive of alterations to, variations of, and equivalents to these embodiments. Accordingly, it should be understood that this disclosure is not to be unduly limited to the illustrative embodiments set forth hereinabove. In particular, as used herein, the recitation of numerical ranges by endpoints is intended to include all numbers subsumed within that range (e.g., 1 to 5 includes 1, 1.5, 2, 2.75, 3, 3.80, 4, and 5). In addition, all numbers used herein are assumed to be modified by the term “about.” Furthermore, various exemplary embodiments have been described. These and other embodiments are within the scope of the following claims.