Technical Field

Examples of the present disclosure relate to methods and apparatuses for processing a series of bits generated at a computing device. Examples of the present disclosure also relate to methods and apparatuses for testing one or more computing devices.

Background

A random number, or a random symbol, is a mathematical object that cannot be predicted more successfully than by random chance. One example of a random chance is the outcome of flipping a coin. Another example of a random chance is the time at which a radioactive decay of a nucleus occurs. Random numbers have uses in several applications, such as Monte Carlo calculations, statistics, and cryptography.

Cryptography is a technique that may be used to secure communication between two parties in the presence of a third party. In cryptography, an encryption process converts an original representation of data into an alternative form that only authorized parties are able to decipher back to the original form from which the original information may be accessed. It will be appreciated that in order to protect confidentiality and integrity of sensitive data, data may be transmitted via an encrypted communication platform to ensure that the data does not traverse a network in plaintext.

It will be appreciated that many protocols for encryption make use of random numbers. One example of this is asymmetric cryptography (or public key cryptography). Asymmetric cryptography is a cryptosystem based on both public and private keys, which secure a communication channel between two parties. Generation of such keys depends on cryptographic algorithms based on mathematical problems to produce one-way functions. Two examples of asymmetric cryptographic algorithms are the RSA cryptosystem, and the Diffie-Hellman key exchange. The RSA algorithm is based on the difficulty of factoring a large number N that is a product of two (large) prime numbers p and q. The Diffie-Hellman key exchange is a method to exchange a secret key that is based on the difficultly of computing g ^ab (mod p) from the known values g ^a (mod p) and g ^b (mod p) where p is a prime number and g is a primitive root for F*p. Both the RSA algorithm, and the Diffie-Hellman key exchange rely on random numbers to generate the keys. Random numbers are required when generating the two prime numbers p and q for the RSA algorithm, and random numbers are used when two parties select their secret integers a and b for Diffie-Hellman key exchange.

As mentioned above, the underlying mathematical problems of the RSA algorithm and the Diffie-Hellman key exchange (as well as the Elliptic Curve Diffie-Hellman key exchange) are based on problems that are hard to solve on a classical computer. However, these problems have theoretically been shown to be easy to solve, given a large enough quantum computer. With the onset of quantum computers, asymmetric cryptography algorithms could be at a risk of being broken.

In view of this, quantum-safe cryptography as well as quantum key distribution (QKD) are being researched. One protocol of quantum key distribution, BB84, again relies on random numbers. The BB84 protocol defines that one party, Alice, prepares a photon with a random polarization. A second party, Bob, then measures the polarization of the photon and chooses which base he will randomly use for the measurement. If the preparation and the measurement are not random, the security of the protocol cannot be guaranteed.

In view of this, before deploying a random number generator, it is important that the random numbers that are produced by the random number generator have been verified to be random enough. There are a number of different statistical tests that can be used to test the randomness of the random numbers produced by a random number generator. Typically, these tests detect patterns, or absence of such. Since there are many different randomness tests that target different aspects of randomness, there is no complete set of randomness tests. However, theNational Institute of Standards and Technology (NIST) has published a test suite with 15 different randomness tests, where each test focuses on different areas where non-randomness could exist. The different tests they recommend are listed below:

1. The Frequency (Monobit) Test

2. Frequency Test within a Block

3. The Runs Test

4. Tests for the Longest-Run-of-Ones in a Block

5. The Binary Matrix Rank Test

6. The Discrete Fourier Transform (Spectral) Test 7. The Non-overlapping Template Matching Test

8. The Overlapping Template Matching Test

9. Maurer’s "Universal Statistical" Test

10. The Linear Complexity Test

11. The Serial Test

12. The Approximate Entropy Test

13. The Cumulative Sums (Cusums) Test

14. The Random Excursions Test

15. The Random Excursions Variant Test

The majority of the tests listed above check one or more statistical properties of the random numbers produced by a random number generator. For example, the frequency (monobit) test tests the number of incidences of the 0s and 1s present in a bit stream. Other tests may check properties such as bias, or serial autocorrelation. Compilation tests may be used to test the source of the random numbers itself, such as the Diehard test for pseudo-random number generators, or for generators based on hardware processes such as ENT.

Summary

One aspect of this disclosure provides a method for processing a series of bits generated at a computing device. The method comprises determining a first word size. The method also comprises determining a respective number of incidences in the series of bits of each possible value for a word of a size equal to the first word size, and determining a measure of randomness of the series of bits based on the determined numbers of incidences.

Another aspect of this disclosure provides a method of processing a series of bits generated at a computing device. The method comprises determining a plurality of different first word sizes. The method also comprises for each first word size, determining a respective number of incidences in the series of bits of each possible value for a word of a size equal to the first word size, and determining a measure of randomness of the series of bits based on the determined numbers of incidences for the first word sizes. A further aspect of this disclosure provides a method for testing one or more computing devices. The method comprises generating a plurality of series of bits at the one or more computing devices. The method also comprises, for each of the plurality of series of bits, determining a first word size, determining a respective number of incidences in the series of bits of each possible value for a word of a size equal to the first word size, and determining a measure of randomness of the series of bits based on the determined numbers of incidences. The method also comprises, based on the determined measures of randomness obtained for each of the plurality of series of bits, selecting one of the computing devices for use in a cryptographic method.

A still further aspect of this disclosure provides a method for testing one or more computing devices. The method comprises generating a plurality of series of bits at the one or more computing devices. The method also comprises, for each of the plurality of series of bits, determining a plurality of different first word sizes, for each first word size, determining a respective number of incidences in the series of bits of each possible value for a word of a size equal to the first word size, and determining the measure of randomness based on the determined numbers of incidences for the first word sizes. The method also comprises, based on the determined measures of randomness obtained for each of the plurality of series of bits, selecting one of the computing devices for use in a cryptographic method.

An additional aspect of this disclosure provides an apparatus for processing a series of bits generated at a computing device. The apparatus comprises a processor and a memory. The memory contains instructions executable by the processor such that the apparatus is operable to determine a first word size, determine a respective number of incidences in the series of bits of each possible value for a word of a size equal to the first word size, and determine a measure of randomness of the series of bits based on the determined numbers of incidences.

A further aspect of this disclosure provides an apparatus for processing a series of bits generated at a computing device. The apparatus comprises a processor and a memory. The memory contains instructions executable by the processor such that the apparatus is operable to determine a plurality of different first word sizes, for each first word size, determine a respective number of incidences in the series of bits of each possible value for a word of a size equal to the first word size, and determine a measure of randomness of the series of bits based on the determined numbers of incidences for the first word sizes.

A still further aspect of this disclosure provides an apparatus for testing one or more computing devices. The apparatus comprises a processor and a memory. The memory contains instructions executable by the processor such that the apparatus is operable to generate a plurality of series of bits at the one or more computing devices, and for each of the plurality of series of bits, determine a first word size, determine a respective number of incidences in the series of bits of each possible value for a word of a size equal to the first word size, determine a measure of randomness of the series of bits based on the determined numbers of incidences, and based on the determined measures of randomness obtained for each of the plurality of series of bits, select one of the computing devices for use in a cryptographic method.

A still further aspect of this disclosure provides an apparatus for testing one or more computing devices. The apparatus comprises a processor and a memory. The memory contains instructions executable by the processor such that the apparatus is operable to generate a plurality of series of bits at the one or more computing devices, and for each of the plurality of series of bits, determine a plurality of different first word sizes, for each first word size, determine a respective number of incidences in the series of bits of each possible value for a word of a size equal to the first word size, and determine the measure of randomness based on the determined numbers of incidences for the first word sizes, and based on the determined measures of randomness obtained for each of the plurality of series of bits, select one of the computing devices for use in a cryptographic method.

An additional aspect of this disclosure provides an apparatus for processing a series of bits generated at a computing device. The apparatus is configured to determine a first word size, determine a respective number of incidences in the series of bits of each possible value for a word of a size equal to the first word size, and determine a measure of randomness of the series of bits based on the determined numbers of incidences.

A further aspect of this disclosure provides an apparatus for processing a series of bits generated at a computing device. The apparatus is configured to determine a plurality of different first word sizes, for each first word size, determine a respective number of incidences in the series of bits of each possible value for a word of a size equal to the first word size, and determine a measure of randomness of the series of bits based on the determined numbers of incidences for the first word sizes.

A still further aspect of this disclosure provides an apparatus for testing one or more computing devices. The apparatus is configured to generate a plurality of series of bits at the one or more computing devices, and, for each of the plurality of series of bits, determine a first word size, determine a respective number of incidences in the series of bits of each possible value for a word of a size equal to the first word size, determine a measure of randomness of the series of bits based on the determined numbers of incidences, and based on the determined measures of randomness obtained for each of the plurality of series of bits, select one of the computing devices for use in a cryptographic method.

A still further aspect of this disclosure provides an apparatus for testing one or more computing devices. The apparatus is configured to generate a plurality of series of bits at the one or more computing devices, and, for each of the plurality of series of bits, determine a plurality of different first word sizes, for each first word size, determine a respective number of incidences in the series of bits of each possible value for a word of a size equal to the first word size, and determine the measure of randomness based on the determined numbers of incidences for the first word sizes, and based on the determined measures of randomness obtained for each of the plurality of series of bits, select one of the computing devices for use in a cryptographic method.

Brief Description of the Drawings

For a better understanding of examples of the present disclosure, and to show more clearly how the examples may be carried into effect, reference will now be made, by way of example only, to the following drawings in which:

Figure 1 is a flow chart of an example of a method for processing a series of bits generated at a computing device;

Figure 2 is a flow chart of another example of a method for processing a series of bits generated at a computing device; Figure 3 is a flow chart of an example of a method for testing one or more computing devices;

Figure 4 is a flow chart of another example of a method for testing one or more computing devices;

Figure 5 is a schematic of an example of apparatus for processing a series of bits generated at a computing device;

Figure 6 is a schematic of another example of apparatus for processing a series of bits generated at a computing device;

Figure 7 is a schematic of an example of apparatus for testing one or more computing devices; and

Figure 8 is a schematic of another example of apparatus for testing one or more computing devices.

Detailed Description

The following sets forth specific details, such as particular embodiments or examples for purposes of explanation and not limitation. It will be appreciated by one skilled in the art that other examples may be employed apart from these specific details. In some instances, detailed descriptions of well-known methods, nodes, interfaces, circuits, and devices are omitted so as not obscure the description with unnecessary detail. Those skilled in the art will appreciate that the functions described may be implemented in one or more nodes using hardware circuitry (e.g., analog and/or discrete logic gates interconnected to perform a specialized function, ASICs, PLAs, etc.) and/or using software programs and data in conjunction with one or more digital microprocessors or general purpose computers. Nodes that communicate using the air interface also have suitable radio communications circuitry. Moreover, where appropriate the technology can additionally be considered to be embodied entirely within any form of computer- readable memory, such as solid-state memory, magnetic disk, or optical disk containing an appropriate set of computer instructions that would cause a processor to carry out the techniques described herein. Hardware implementation may include or encompass, without limitation, digital signal processor (DSP) hardware, a reduced instruction set processor, hardware (e.g., digital or analogue) circuitry including but not limited to application specific integrated circuit(s) (ASIC) and/or field programmable gate array(s) (FPGA(s)), and (where appropriate) state machines capable of performing such functions.

In view of this, it will be appreciated that a perfectly random series of bits should be in compressible (in the same way that a perfectly random sequence should be incompressible). It will be appreciated that, equivalently, if a random series is to be divided into a series of words (where each word is of a certain word size), the source entropy (as a function of the expected probabilities for a value of a word taken from a perfectly random series of bits) per word in a perfectly random series of bits should be equal to 1 , regardless of how the words are formed. In view of this, it will be appreciated that a measure of randomness of a series of bits may be determined based on the incompressibility of that series of bits. It will be appreciated that methods described herein take advantage of this incompressibility of a random series of bits in order to determine a measure of randomness of a series of bits.

Figure 1 is a flow chart of an example of a method 100 for processing a series of bits generated at a computing device. In some embodiments, the series of bits may comprise a series of random and/or pseudorandom bits. In some embodiments, the computing device may comprise a quantum computing device.

The method 100 comprises, in step 102, determining a first word size. In some embodiments, the first word size is equal to one or more bits. As will be discussed in greater detail below, the use of a first word size in the methods described herein that comprises a larger number of bits may indicate that a series of bits is in fact more compressible than would be indicated by the use of a first word size that comprises a smaller number of bits (for example, a first word size equal to one bit) in the methods described herein. In other words, the use of a small first word size in the methods described herein may illustrate that on a smaller scale, a series of bits may appear to be relatively incompressible, whereas the use of a larger first word size may show that on a larger scale, a series of bits may appear to be more compressible than the compressibility highlighted by the use of a smaller first word size. An example of incompressibility of a series of bits for a small word, but compressibility of the series of bits for a large word, is now provided. The series of bits “1010101001” cannot be compressed when considering a word of a size equal to one bit (where the value of the word may comprise either “0”, or “1”). However, for a word of a size equal to two bits (where the value of the word may comprise either “00”, “01” “10” or “11”), and letting the word value “10” be represented as “0”, and the value “01” be represented as “1”, the series of bits may now be encoded as “00001”. Thus, a series of bits that appeared to be incompressible for a small word size can be shown to be compressible for a larger word size. Examples of this disclosure may use this observation and use larger word sizes (e.g. two or more bits), and/or multiple different word sizes, to determine a measure of randomness of a series of bits.

Step 104 of the method 100 comprises determining a respective number of incidences in the series of bits of each possible value for a word of a size equal to the first word size. In some embodiments, the step of determining a respective number of incidences in the series of bits of each possible value for a word of a size equal to the first word size may comprise forming a plurality of sequential words from the series of bits, wherein each sequential word is of a size equal to the first word size, and determining a respective number of incidences in the sequential words of each possible value of a sequential word. For example, in a series of bits comprising the bit values “0101101100”, and where the first word size is equal to 2 bits, forming a plurality of sequential words from the series of bits, wherein each sequential word is of a size equal to the first word size would comprise forming the 5 sequential words “01”, “01”, “ ” “ ” _anc| “go”. For a first word size is equal to 2 bits, the possible values of the word comprise the values “01”, “10”, “11” and “00”. Thus, in this example, the step of determining a respective number of incidences in the sequential words of each possible value of a sequential word would comprise determining that the value “01” occurs twice, the value “10” occurs once, the value “11” occurs once and the value “00” occurs once. It will be appreciated that, in some embodiments, the words formed from the series of bits may not be sequential. For example, the formed words may overlap with one another.

Step 106 of the method 100 comprises determining a measure of randomness of the series of bits based on the determined numbers of incidences. In some embodiments, the step of determining a measure of randomness of the series of bits based on the determined numbers of incidences may comprise for each possible value of a word of a size equal to the first word size, comparing the determined number of instances of the value in the series of bits to an expected number of instances of the value in the series of bits. It will be appreciated that, according to this embodiment, a measure of randomness of the series of bits may be determined based on said comparisons. For example, again referring to the series of bits comprising the bit values “0101101100”, and where the first word size is equal to 2 bits, a word of the first word size may take the value of “00”, “01”, “10” or “11”. In a random series of bits, it will be appreciated that each of these values of a word of the first word size should be equally probable within the series of bits, and that this probability will be equal to 0.25. It will be appreciated that, an expected number of instances of each possible value of a word of the first word size for a random series of bits may in some examples be calculated based on the probability of occurrence of the value, and the length of the series of bits. In this example, the expected number of instances of the value “00” would be calculated to be equal to the probability of the occurrence of the value (in this case, 0.25), and the length of the series of bits (in this case, 10 bits, or 5 words of that word size).

As discussed above, it will be appreciated that a perfectly random series of bits should be incompressible, and that if a random series were to be divided into a series of words (where each word is of a certain word size), the source entropy per word in a perfectly random series of bits should be equal to 1 (regardless of how the words are formed). In order to achieve maximum entropy in series of bits, this would require that each word should be equally probable within the series of bits.

For example, considering a word size equal to 1 bit, a word of this word size may either take the value of “0”, or of “1”, with equal probability. Thus, in a random series of bits, the probability of either values of the word occurring within a word within the series of bits will be equal to 0.5. In another example, considering a word size equal to 2 bits, a word of this word size may take the value of “00”, “01”, “10” or “11” with equal probability. Thus, in a random series of bits, the probability of each of these values of the word occurring within a word within the series of bits will be equal to 0.25. In another example, considering a word size equal to 3 bits, a word of this word size may take the value of “000”, “001”, “010”, “011”, “100”, “101”, “110”, or “111” with equal probability. Thus, in a random series of bits, the probability of each of these values of the word occurring within a word within the series of bits will be equal to 0.125. Hence, for a truly random series of bits, taking a word from this series of bits where the word size is equal to n bits, the probability of each possible value of the word occurring will be equal to 1/2 ⁿ . It will however be appreciated that, in practice, there will be some statistical fluctuations which may need to be taken into account. Therefore, in other words, the expected number of instances of a value (of a word) in the series of bits may be related to the reciprocal of the total number of possible values of a word of a size equal to the first word size.

For example, referring to the series of bits comprising the 10 bit values “0101101100”, and where the word size is equal to 1 bit, the expected number of incidences of the value “0” would be equal to 10 x 0.5 = 5. Similarly, the expected number of incidences of the value “1” would be equal to 10 x 0.5 = 5.

In some embodiments, the step of comparing the determined number of instances of the value in the series of bits to an expected number of instances of the value in the series of bits may comprise determining whether a difference between the determined number of instances of the value and the expected number of instances of the value falls within a predetermined non-zero range, and the step of determining a measure of randomness of the series of bits based on said comparisons may comprise determining a measure of randomness of the series of bits based on, for each possible value of a word of a size equal to the first word size, whether the difference falls within the predetermined non-zero range.

For example, the comparisons of the determined number of instances, and the expected number of instances, may indicate that the series of bits deviate less than what would be expected for a truly random and/or pseudorandom distribution. Such a situation may arise when a malicious party attempts to mimic a random distribution in the series of bits. In such a situation, the difference for one or more of the possible word values may fall outside the predetermined non-zero range. It will be appreciated that, in this example, the determined measure of randomness may indicate that the series of bits is “not random” or “insufficiently random”. In another example, the comparisons of the determined number of instances, and the expected number of instances, may indicate that the series of bits deviate less than what would be expected for a genuinely random distribution. It will be appreciated that, where the series of bits deviate less than what would be expected for a genuinely random distribution, a sequential word in the series of bit may become predictable. For example, considering a series of bits “000001010011100101110111”, and a word size equal to 3 bits, in a situation in which the series of bits deviate less than what would be expected for a genuinely random distribution, and 7 different values of the word have occurred sequentially in the series of bits, then occurrence of the value “111” following these 7 values can be predicted with certainty. In this situation, the 8th word, “111”, could be removed from the series of bits, and no information would be lost, thus, allowing the series of bits to become compressible, and the series of bits becomes predictable. In such a situation, the difference for one or more of the possible word values may fall outside the predetermined non-zero range. It will be appreciated that, in this example, the determined measure of randomness may indicate that the series of bits is “not random” or “insufficiently random”. It will be appreciated that, in the situation in which all of the differences fall inside the pre determined non-zero range, the determined measure of randomness may indicate that the series of bits is “random” or “sufficiently random”.

It will also be appreciated that, in some embodiments, the step of determining a measure of randomness of the series of bits based on the determined numbers of incidences may comprise performing a two-sided statistical test based on the determined number of instances, to obtain a result, and based on the result of the two-sided statistical test, determining a measure of randomness of the series of bits. In some embodiments, the two-sided statistical test may comprise a chi-squared test.

It will be appreciated that the skilled person will understand that a chi-squared test may be used to verify if an expected distribution is compatible with an observed distribution. Its probability density function may be written as follows: where k is the number of degrees of freedom, and G is a Gamma function.

For a discrete uniform distribution, the chi-squared (X ² ) statistic may be computed as: where v _t is the number of observations in bin i, N is the total number of observations and, in this example, m = 2 ⁿ is the number of possible values of a word of word size equal to n. As mentioned above, the assumed probability of each of these values occurring in a word (of a word size equal to n) taken from random series of bits will be equal to 1/2 ⁿ . The test statistic may then be compared with the cumulative distribution function.

Following this, the value p may then be denoted as the probability that a distribution was deemed not random, while in fact it was random (this may occur by chance due to statistical variance).

In this example, a lower critical bound c ² _/2 may be defined as the value of the test statistic c ² when the probability P(k,x £ c ² ) = p/2. An upper critical bound c - _r / ₂ may be defined as the value of the test statistic c ² when the probability P(k,x £ c ² ) = 1 -p/2.

In this example, the degrees of freedom k will be equal to 2 ⁿ -1, c _r/2 indicates the lower limit for the test statistic c ² when we accept the hypothesis that the data is random, and X _I -P/2 indicates the higher limit for the test statistic c ² when we accept the hypothesis that the data is random. If the value is lower than p/2 in this example, then we may reject that the data is random on the basis that the distribution (which in this example, is the series of bits) is too close to the expected probabilities described above. If the value if higher then 1 -p/2 in this example, then we may reject that the data is random on the basis that the distribution is not compatible with the expected distribution.

In some embodiments, the step of determining a measure of randomness of the series of bits based on the determined numbers of incidences may comprise, for each possible value of a word of a size equal to the first word size, comparing the determined number of instances of the value in the series of bits to an expected number of instances of the value in the series of bits, and performing the two-sided statistical test based on said comparisons. It will be appreciated that the step of comparing the determined number of instances of the value in the series of bits to an expected number of instances of the value in the series of bits may be performed in a substantially similar manner as described above.

It will be appreciated that a two-sided statistical test may be used to determine whether a statistically significant difference exists between the expected number of instances of a value of a word, and an observed number of instances of a value of a word in a series of bits. In some embodiments, the result of a two-sided statistical test may be used to confirm whether this statistically significant difference is smaller than would be predicted for a random series of bits (in other words, there is not enough deviation in the observed number of instances), but may also be used to confirm whether this statistically significant difference is larger than would be predicted for a random series of bits (in other words, there is too much deviation in the observed number of instances). It will be appreciated that the first situation, in which there is not enough deviation in the observed number of instances, may arise when a malicious party attempts to mimic a random distribution in the series of bits. Additionally, the second situation, in which there is too much deviation in the observed number of instances, may arise when the observed number of instances are deviating too greatly from what would be expected for a genuinely random distribution (for the reasons described above). It will be appreciated that the use of a two-sided statistical test may allow both this greater than expected deviation, and this less than expected deviation, to be tested for.

In some examples, where the result of the two-sided statistical test indicates that there is either insufficient deviation in the random series of bits, or too much deviation in the random series of bits, the step of, based on the result of the two-sided statistical test, determining a measure of randomness of the series of bits, may comprise determining that the series of bits is “not random” or “insufficiently random”.

That is, it will be appreciated that a two-sided statistical test (for example, a chi-squared test) may be used to determine whether a statistically significant difference exists between, for each possible value of a word of a size equal to a first word size, an expected number of instances of the value in the series of bits, and a determined number of instances of the value in the series of bits. Following this, it may then be determined whether the statistically significant difference is smaller than would be predicted for a random series of bits (in other words, that there is not enough deviation in the determined number of instances), and that therefore the series of bits is “not random” or “insufficiently random”.

It will be appreciated that the expected number of instances of the value in the series of bits may be formed as a predetermined sequence. It will be appreciated that the predetermined sequence may comprise a sequence that represents an amount of deviation that would be expected to occur in a random series of bits. In some embodiments, the result of the two-sided statistical test may comprise a p-value corresponding to the series of bits. It will be appreciated that a p-value may provide an indication of a measure of randomness of the series of bits. For example, a p-value exceeding a predetermined threshold may indicate that the series of bits can be considered “random” or “sufficiently random”, and a p-value failing to exceed a predetermined threshold may indicate that the series of bits can be considered “not random” or “insufficiently random”. It will be appreciated that such a predetermined threshold may be varied depending on the degree of randomness that is required by the series of bits (for example, the degree of randomness required may vary depending on a further use of the series of bits, or a further use of the computing device that has generated the series of bits). For example, the predetermined threshold may be equal to 0.01.

In some embodiments, the method may further comprises determining whether the measure of randomness meets a predetermined criterion, and in response to the measure of randomness meeting predetermined criterion, using the series of bits in a cryptographic method. In some embodiments, if the measure of randomness indicates that the series of bits is “random” or “sufficiently random”, the series of bits may be used in a cryptographic method. In another embodiment, if an obtained p-value indicates that indicates that the series of bits is “random” or “sufficiently random”, the series of bits may be used in a cryptographic method. It will be appreciated that the cryptographic method may comprise any suitable cryptographic method that relies on the use of random numbers, such as an asymmetric cryptographic method, an RSA algorithm, a Diffie- Hellman key exchange, or a quantum key distribution method.

It will be appreciated that, in some embodiments, the method 100 may further comprise determining one or more different second word sizes different to the first word size, and for each second word size, determining a respective number of incidences in the series of bits of each possible value for a word of a size equal to the second word size. It will be appreciated that, according to this embodiment, the step of determining the measure of randomness of the series of bits comprises determining the measure of randomness based on the determined numbers of incidences for the first word size and the one or more second word sizes. As mentioned above, the use of a first word size in the methods described herein that comprises a larger number of bits may indicate that a series of bits is in fact more compressible than would be indicated by the use of a first word size that comprises a smaller number of bits (for example, a first word size equal to one bit) in the methods described herein. In other words, the use of a small first word size in the methods described herein may illustrate that on a smaller scale, a series of bits may appear to be relatively incompressible, whereas the use of a larger first word size may show that on a larger scale, a series of bits may appear to be more compressible than the compressibility highlighted by the use of a smaller first word size. In other words, the use of a larger first word size in the methods described herein may allow a more accurate measure of randomness to be determined for a series of bits. Thus, the inclusion of these additional method steps may allow a more accurate measure of randomness to be determined for a series of bits.

Figure 2 is a flow chart of another example of a method 200 for processing a series of bits generated at a computing device. In some embodiments, the series of bits may comprise a series of random and/or pseudorandom bits. In some embodiments, the computing device may comprise a quantum computing device.

The method 200 comprises, in step 202, determining a plurality of different first word sizes. In some embodiments, the first word sizes are equal to one or more bits. As previously discussed, the use of a plurality of first word sizes in the methods described herein may allow a more accurate measure of randomness to be determined for a series of bits, as a larger word size as used in the methods described herein may highlight a degree of compressibility in the series of bits that may not be highlighted by the use of a smaller word size in the respective method. In some embodiments, the total number of the plurality of different first word sizes may be equal to a predetermined number. In some embodiments, this predetermined number may be based on the size of the series of bits. For example, the larger the series of bits, the larger total number of the plurality of different first word sizes that may be desired. This may allow any large scale compressibility in the series of bits to then be identified following the execution of the method 200.

Step 204 of the method 200 comprises, for each first word size, determining a respective number of incidences in the series of bits of each possible value for a word of a size equal to the first word size.

Step 206 of the method 200 comprises determining a measure of randomness of the series of bits based on the determined numbers of incidences for the first word sizes. In some embodiments, the step of determining a measure of randomness of the series of bits based on the determined numbers of incidences may comprise for each first word size, for each possible value of a word of a size equal to the first word size, comparing the determined number of instances of the value in the series of bits to an expected number of instances of the value in the series of bits, and determining a measure of randomness of the series of bits based on said comparisons. It will be appreciated that the step of comparing the determined number of instances of the value in the series of bits to an expected number of instances of the value in the series of bits for each possible value of a word of a size equal to the first word size may be performed in a substantially similar manner as described above.

For example, in some embodiments, the step of comparing the determined number of instances of the value in the series of bits to an expected number of instances of the value in the series of bits may comprise determining whether a difference between the determined number of instances of the value and the expected number of instances of the value falls within a predetermined non-zero range, and the step of determining a measure of randomness of the series of bits based on said comparisons may comprise determining a measure of randomness of the series of bits based on, for each possible value of a word of a size equal to the first word size, whether the difference falls within the predetermined non-zero range. Similarly these steps may also be implemented in a manner that is substantially similar to the process that is described above.

It will also be appreciated that the step of determining a measure of randomness of the series of bits based on said comparisons may be performed in a substantially similar manner as described above.

In some embodiments, the expected number of instances of the value in the series of bits may be related to the reciprocal of the total number of possible values of a word of a size equal to the first word size.

In some embodiments, the step of determining a measure of randomness of the series of bits based on the determined numbers of incidences for the first word sizes may comprise for each first word size, performing a two-sided statistical test based on the respective determined number of instances, to obtain a result, and based on the results of the two-sided statistical tests, determining a measure of randomness of the series of bits. In some embodiments, the two-sided statistical test may comprise a chi-squared test.

It will be appreciated that the step of performing a two-sided statistical test based on the respective determined number of instances, to obtain a result, may be performed in substantially the same manner as described above. For example, in some embodiments, the results of the two-sided statistical tests comprise p-values corresponding to the series of bits. Additionally or alternatively, the step of performing the two-sided statistical test may comprise, for each first word size, for each possible value of a word of a size equal to the first word size, comparing the determined number of instances of the value in the series of bits to an expected number of instances of the value in the series of bits, and performing the two-sided statistical test based on said comparisons.

It will be appreciated that, as the measure of randomness is determined from the result of more than one two-sided statistical test, a more accurate measure of randomness for the series of bits may be obtained. In some embodiments, if one or more of the obtained results indicate that the series of bits should be considered “not random” or “insufficiently random”, the measure of randomness of the series of bits may be determined to be “not random” or insufficiently random”.

That is, it will be appreciated that a two-sided statistical test (for example, a chi-squared test) may be used to determine whether a statistically significant difference exists between, for each first word size, and following this, for each possible value of a word of a size equal to the first word size, an expected number of instances of the value in the series of bits, and a determined number of instances of the value in the series of bits. Following this, it may then be determined whether the statistically significant difference is smaller than would be predicted for a random series of bits (in other words, that there is not enough deviation in the determined number of instances), and that therefore the series of bits is “not random” or “insufficiently random”.

It will be appreciated that the expected number of instances of the value in the series of bits may be formed as a predetermined sequence. It will be appreciated that the predetermined sequence may comprise a sequence that represents an amount of deviation that would be expected to occur in a random series of bits. A table containing the p-values corresponding to a plurality of different word sizes that were obtained as a result of the execution an embodiment of the method 200 on 3 respective series of bits generated by the 3 different random number generators (the QPU of a D-Wave quantum annealing machine, the LRNG /dev/urandom generator and the Python library NumPy’s function randint) is now presented. In this table, the presented p-values represent a probability that the series of bits as generated by the relevant random number generator represents a random series of bits: In this illustrated example, the threshold for the obtained p-value which the series of bits was determined to be sufficiently random was >0.01. It can be seen in this illustrated example that the series of bits generated by the QPU of a D-Wave quantum annealing machine was not found to represent a random series of bits as a result of the execution of an embodiment of the method 200 (as the obtained p-values always fail to exceed the value 0.01). Additionally, it can be seen that the series of bits generated by the Python library NumPy’s function randint was found to represent a random series of bits as a result of the execution of an embodiment of the method 200 (as the obtained p-values always exceed the value 0.01). Referring now to the p-values obtained for the series of bits by the LRNG /dev/urandom generator, for smaller word sizes (in this illustrated example, word sizes between 1 and 8 bits) indicate that the series of bits does represent a random and/or pseudorandom series of bits. However, the larger word sizes (in this case, word sizes between 9 and 12 bits) indicate that the series of bits does not represent a random and/or pseudorandom series of bits. This is an example of the large scale compressibility of a series of bits that may not be identified without the use of a larger word size in a method according to the method 200 described above. As a result, the series of bits has been more accurately identified to not represent a random and/or pseudorandom series.

A table containing the p-values corresponding to the 15 aforementioned NIST tests described above on 3 respective series of bits generated by the 3 different random number generators (the QPU of a D-Wave quantum annealing machine, the LRNG /dev/urandom generator and the Python library NumPy’s function randint) is shown below: In this table, the presented p-values represent a probability that the series of bits as generated by the relevant random number generator represents a random series of bits: Referring again to the series of bits corresponding to the LRNG /dev/urandom generator, each of the 15 NIST tests would have indicated that the series of bits does represent a random and/or pseudorandom series of bits. Again, the use of a larger word size in a method according to the method 200 described above may allow the series of bits to be more accurately identified to not represent a random and/or pseudorandom series.

In some embodiments, the step of determining a respective number of incidences in the series of bits of each possible value for a word of a size equal to the first word size may comprise, for each first word size forming a plurality of sequential words from the series of bits, wherein each sequential word is of a size equal to the first word size, and determining a respective number of incidences in the sequential words of each possible value of a sequential word. It will be appreciated that these steps may be performed in a substantially similar manner as described above.

In some embodiments, the method may further comprises determining whether the measure of randomness meets a predetermined criterion, and in response to the measure of randomness meeting predetermined criterion, using the series of bits in a cryptographic method. In some embodiments, if the measure of randomness indicates that the series of bits is “random” or “sufficiently random”, the series of bits may be used in a cryptographic method. In another embodiment, if an obtained p-value indicates that indicates that the series of bits is “random” or “sufficiently random”, the series of bits may be used in a cryptographic method. It will be appreciated that the cryptographic method may comprise any suitable cryptographic method that relies on the use of random numbers, such as an asymmetric cryptographic method, an RSA algorithm, a Diffie- Hellman key exchange, or a quantum key distribution method.

Figure 3 is a flow chart of an example of a method 300 for testing one or more computing devices. In some embodiments, one or more of the computing devices may comprise a quantum computing device.

The method 300 comprises, in step 302, generating a plurality of series of bits at the one or more computing devices. In some embodiments, one or more of the plurality of series of bits may comprise a random and/or pseudorandom series of bits. It will be appreciated that the method 300 comprises executing steps 304-308 for each of the plurality series of bits.

Step 304 of the method 300 comprises determining a first word size. In some embodiments, the first word size may be equal to one or more bits. As previously discussed, the use of a plurality of first word sizes in the methods described herein may allow a more accurate measure of randomness to be determined for a series of bits, as a larger word size as used in the methods described herein may highlight a degree of compressibility in the series of bits that may not be highlighted by the use of a smaller word size in the respective method. In some embodiments, the total number of the plurality of different first word sizes may be equal to a predetermined number. In some embodiments, this predetermined number may be based on the size of the series of bits. For example, the larger the series of bits, the larger total number of the plurality of different first word sizes that may be desired. This may allow any large scale compressibility in the series of bits to then be identified following the execution of the method 300.

Step 306 of the method 300 comprises determining a respective number of incidences in the series of bits of each possible value for a word of a size equal to the first word size.

Step 308 of the method 300 comprises determining a measure of randomness of the series of bits based on the determined numbers of incidences. In some embodiments, the step of determining a measure of randomness of the series of bits based on the determined numbers of incidences may comprise for each possible value of a word of a size equal to the first word size, comparing the determined number of instances of the value in the series of bits to an expected number of instances of the value in the series of bits and determining a measure of randomness of the series of bits based on said comparisons.

In some embodiments, the step of determining a respective number of incidences in the series of bits of each possible value for a word of a size equal to the first word size may comprise, for each first word size forming a plurality of sequential words from the series of bits, wherein each sequential word is of a size equal to the first word size, and determining a respective number of incidences in the sequential words of each possible value of a sequential word. It will be appreciated that these steps may be performed in a substantially similar manner as described above. It will be appreciated that the step of comparing the determined number of instances of the value in the series of bits to an expected number of instances of the value in the series of bits for each possible value of a word of a size equal to the first word size may be performed in a substantially similar manner as described above.

For example, in some embodiments, the step of comparing the determined number of instances of the value in the series of bits to an expected number of instances of the value in the series of bits may comprise determining whether a difference between the determined number of instances of the value and the expected number of instances of the value falls within a predetermined non-zero range, and the step of determining a measure of randomness of the series of bits based on said comparisons may comprise determining a measure of randomness of the series of bits based on, for each possible value of a word of a size equal to the first word size, whether the difference falls within the predetermined non-zero range. Similarly these steps may also be implemented in a manner that is substantially similar to the process that is described above.

It will also be appreciated that the step of determining a measure of randomness of the series of bits based on said comparisons may be performed in a substantially similar manner as described above.

In some embodiments, the expected number of instances of the value in the series of bits may be related to the reciprocal of the total number of possible values of a word of a size equal to the first word size.

In some embodiments, the step of determining a measure of randomness of the series of bits based on the determined numbers of incidences for the first word sizes may comprise performing a two-sided statistical test based on the respective determined number of instances, to obtain a result, and based on the results of the two-sided statistical tests, determining a measure of randomness of the series of bits. In some embodiments, the two-sided statistical test may comprise a chi-squared test.

It will be appreciated that the step of performing a two-sided statistical test based on the respective determined number of instances, to obtain a result, may be performed in substantially the same manner as described above. For example, in some embodiments, the obtained result comprises a p-value corresponding to the series of bits. Additionally or alternatively, the step of performing the two-sided statistical test may comprise, for each possible value of a word of a size equal to the first word size, comparing the determined number of instances of the value in the series of bits to an expected number of instances of the value in the series of bits, and performing the two- sided statistical test based on said comparisons.

It will be appreciated that, as the measure of randomness is determined from the result of more than one two-sided statistical test, a more accurate measure of randomness for the series of bits may be obtained. In some embodiments, if one or more of the obtained results indicate that the series of bits should be considered “not random” or “insufficiently random”, the measure of randomness of the series of bits may be determined to be “not random” or insufficiently random”.

That is, it will be appreciated that a two-sided statistical test (for example, a chi-squared test) may be used to determine whether a statistically significant difference exists between, for each possible value of a word of a size equal to the first word size, an expected number of instances of the value in the series of bits, and a determined number of instances of the value in the series of bits. Following this, it may then be determined whether the statistically significant difference is smaller than would be predicted for a random series of bits (in other words, that there is not enough deviation in the determined number of instances), and that therefore the series of bits is “not random” or “insufficiently random”.

It will be appreciated that the expected number of instances of the value in the series of bits may be formed as a predetermined sequence. It will be appreciated that the predetermined sequence may comprise a sequence that represents an amount of deviation that would be expected to occur in a random series of bits.

Step 310 of the method 300 comprises, based on the determined measures of randomness obtained for each of the plurality of series of bits, selecting one of the computing devices for use in a cryptographic method. In some embodiments, the step of selecting one of the computing devices for use in a cryptographic method may comprise determining if any of the determined measures of randomness meet a predetermined criterion. For example, in some embodiments, one or more computing devices corresponding to the series of bits with the highest determined measures of randomness may be selected for use in a cryptographic method. In another example, in some embodiments, the one or more computing devices corresponding to the series of bits with the largest determined p-values may be selected for use in a cryptographic method. It will be appreciated that the cryptographic method may comprise any suitable cryptographic method that relies on the use of random numbers, such as an asymmetric cryptographic method, an RSA algorithm, a Diffie-Hellman key exchange, or a quantum key distribution method.

It will be appreciated that, in some embodiments, the method 300 may further comprise, for each of the plurality of series of bits, determining one or more different second word sizes different to the first word size, and for each second word size, determining a respective number of incidences in the series of bits of each possible value for a word of a size equal to the second word size, and wherein the step of determining the measure of randomness of the series of bits may comprise determining the measure of randomness based on the determined numbers of incidences for the first word size and the one or more second word sizes. As mentioned above, the use of a first word size in the methods described herein that comprises a larger number of bits may indicate that a series of bits is in fact more compressible than would be indicated by the use of a first word size that comprises a smaller number of bits (for example, a first word size equal to one bit) in the methods described herein. In other words, the use of a small first word size in the methods described herein may illustrate that on a smaller scale, a series of bits may appear to be relatively incompressible, whereas the use of a larger first word size may show that on a larger scale, a series of bits may appear to be more compressible than the compressibility highlighted by the use of a smaller first word size. In other words, the use of a larger first word size in the methods described herein may allow a more accurate measure of randomness to be determined for a series of bits. Thus, the inclusion of these additional method steps may allow a more accurate measure of randomness to be determined for a series of bits.

Figure 4 is a flow chart of another example of a method 400 for testing one or more computing devices. In some embodiments, one or more of the computing devices may comprise a quantum computing device.

The method 400 comprises, in step 402, generating a plurality of series of bits at the one or more computing devices. In some embodiments, one or more of the plurality of series of bits may comprise a random and/or pseudorandom series of bits. It will be appreciated that the method 400 comprises executing steps 404-408 for each of the plurality series of bits.

Step 404 of the method 400 comprises determining a plurality of different first word sizes. In some embodiments, the first word sizes are equal to one or more bits. As previously discussed, the use of a plurality of first word sizes in the methods described herein may allow a more accurate measure of randomness to be determined for a series of bits, as a larger word size as used in the methods described herein may highlight a degree of compressibility in the series of bits that may not be highlighted by the use of a smaller word size in the respective method. In some embodiments, the total number of the plurality of different first word sizes may be equal to a predetermined number. In some embodiments, this predetermined number may be based on the size of the series of bits. For example, the larger the series of bits, the larger total number of the plurality of different first word sizes that may be desired. This may allow any large scale compressibility in the series of bits to then be identified following the execution of the method 400.

Step 406 of the method 400 comprises, for each first word size, determining a respective number of incidences in the series of bits of each possible value for a word of a size equal to the first word size.

Step 408 of the method 400 comprises determining the measure of randomness based on the determined numbers of incidences for the first word sizes.

In some embodiments, the step of determining a measure of randomness of the series of bits based on the determined numbers of incidences may comprise for each first word size, for each possible value of a word of a size equal to the first word size, comparing the determined number of instances of the value in the series of bits to an expected number of instances of the value in the series of bits, and determining a measure of randomness of the series of bits based on said comparisons. It will be appreciated that the step of comparing the determined number of instances of the value in the series of bits to an expected number of instances of the value in the series of bits for each possible value of a word of a size equal to the first word size may be performed in a substantially similar manner as described above. For example, in some embodiments, the step of comparing the determined number of instances of the value in the series of bits to an expected number of instances of the value in the series of bits may comprise determining whether a difference between the determined number of instances of the value and the expected number of instances of the value falls within a predetermined non-zero range, and the step of determining a measure of randomness of the series of bits based on said comparisons may comprise determining a measure of randomness of the series of bits based on, for each possible value of a word of a size equal to the first word size, whether the difference falls within the predetermined non-zero range. Similarly these steps may also be implemented in a manner that is substantially similar to the process that is described above.

It will also be appreciated that the step of determining a measure of randomness of the series of bits based on said comparisons may be performed in a substantially similar manner as described above.

In some embodiments, the expected number of instances of the value in the series of bits may be related to the reciprocal of the total number of possible values of a word of a size equal to the first word size.

In some embodiments, the step of determining a measure of randomness of the series of bits based on the determined numbers of incidences for the first word sizes may comprise for each first word size, performing a two-sided statistical test based on the respective determined number of instances, to obtain a result, and based on the results of the two-sided statistical tests, determining a measure of randomness of the series of bits. In some embodiments, the two-sided statistical test may comprise a chi-squared test.

It will be appreciated that the step of performing a two-sided statistical test based on the respective determined number of instances, to obtain a result, may be performed in substantially the same manner as described above. For example, in some embodiments, the results of the two-sided statistical tests comprise p-values corresponding to the series of bits. Additionally or alternatively, the step of performing the two-sided statistical test may comprise, for each first word size, for each possible value of a word of a size equal to the first word size, comparing the determined number of instances of the value in the series of bits to an expected number of instances of the value in the series of bits, and performing the two-sided statistical test based on said comparisons.

It will be appreciated that, as the measure of randomness is determined from the result of more than one two-sided statistical test, a more accurate measure of randomness for the series of bits may be obtained. In some embodiments, if one or more of the obtained results indicate that the series of bits should be considered “not random” or “insufficiently random”, the measure of randomness of the series of bits may be determined to be “not random” or insufficiently random”.

That is, it will be appreciated that a two-sided statistical test (for example, a chi-squared test) may be used to determine whether a statistically significant difference exists between, for each possible value of a word of a size equal to the first word size, an expected number of instances of the value in the series of bits, and a determined number of instances of the value in the series of bits. Following this, it may then be determined whether the statistically significant difference is smaller than would be predicted for a random series of bits (in other words, that there is not enough deviation in the determined number of instances), and that therefore the series of bits is “not random” or “insufficiently random”.

It will be appreciated that the expected number of instances of the value in the series of bits may be formed as a predetermined sequence. It will be appreciated that the predetermined sequence may comprise a sequence that represents an amount of deviation that would be expected to occur in a random series of bits.

In some embodiments, the step of determining a respective number of incidences in the series of bits of each possible value for a word of a size equal to the first word size may comprise, for each first word size forming a plurality of sequential words from the series of bits, wherein each sequential word is of a size equal to the first word size, and determining a respective number of incidences in the sequential words of each possible value of a sequential word. It will be appreciated that these steps may be performed in a substantially similar manner as described above.

Step 410 of the method 400 comprises, based on the determined measures of randomness obtained for each of the plurality of series of bits, selecting one of the computing devices for use in a cryptographic method. In some embodiments, the step of selecting one of the computing devices for use in a cryptographic method may comprise determining if any of the determined measures of randomness meet a predetermined criterion. For example, in some embodiments, one or more computing devices corresponding to the series of bits with the highest determined measures of randomness may be selected for use in a cryptographic method. In another example, in some embodiments, the one or more computing devices corresponding to the series of bits with the largest determined p-values may be selected for use in a cryptographic method. It will be appreciated that the cryptographic method may comprise any suitable cryptographic method that relies on the use of random numbers, such as an asymmetric cryptographic method, an RSA algorithm, a Diffie-Hellman key exchange, or a quantum key distribution method.

Figure 5 is a schematic of an example of apparatus 500 for processing a series of bits generated at a computing device. The apparatus 500 comprises processing circuitry 502 (e.g. one or more processors) and a memory 504 in communication with the processing circuitry 502. The memory 504 contains instructions executable by the processing circuitry 502. In one embodiment, the memory 504 contains instructions executable by the processing circuitry 502 such that the apparatus 500 is operable to determine a first word size, determine a respective number of incidences in the series of bits of each possible value for a word of a size equal to the first word size, and determine a measure of randomness of the series of bits based on the determined numbers of incidences In some examples, the memory 504 contains instructions executable by the processing circuitry 502 such that the apparatus 500 is operable to carry out the method 100 shown in Figure 1 and/or described above.

Figure 6 is a schematic of an example of apparatus 600 for processing a series of bits generated at a computing device. The apparatus 600 comprises processing circuitry 602 (e.g. one or more processors) and a memory 604 in communication with the processing circuitry 602. The memory 604 contains instructions executable by the processing circuitry 602. In one embodiment, the memory 604 contains instructions executable by the processing circuitry 602 such that the apparatus 600 is operable to determine a plurality of different first word sizes, for each first word size, determine a respective number of incidences in the series of bits of each possible value for a word of a size equal to the first word size, and determine a measure of randomness of the series of bits based on the determined numbers of incidences for the first word sizes. In some examples, the memory 604 contains instructions executable by the processing circuitry 602 such that the apparatus 600 is operable to carry out the method 200 shown in Figure 2 and/or described above.

Figure 7 is a schematic of an example of apparatus 700 for testing one or more computing devices. The apparatus 700 comprises processing circuitry 702 (e.g. one or more processors) and a memory 704 in communication with the processing circuitry 702. The memory 704 contains instructions executable by the processing circuitry 702. In one embodiment, the memory 704 contains instructions executable by the processing circuitry 702 such that the apparatus 700 is operable generate a plurality of series of bits at the one or more computing devices, and for each of the plurality of series of bits, determine a first word size, determine a respective number of incidences in the series of bits of each possible value for a word of a size equal to the first word size, determine a measure of randomness of the series of bits based on the determined numbers of incidences, and based on the determined measures of randomness obtained for each of the plurality of series of bits, select one of the computing devices for use in a cryptographic method. In some examples, the memory 704 contains instructions executable by the processing circuitry 702 such that the apparatus 700 is operable to carry out the method 300 shown in Figure 3 and/or described above.

Figure 8 is a schematic of an example of apparatus 800 for testing one or more computing devices. The apparatus 800 comprises processing circuitry 802 (e.g. one or more processors) and a memory 804 in communication with the processing circuitry 802. The memory 804 contains instructions executable by the processing circuitry 802. In one embodiment, the memory 804 contains instructions executable by the processing circuitry 802 such that the apparatus 800 is operable to generate a plurality of series of bits at the one or more computing devices, and, for each of the plurality of series of bits, determine a plurality of different first word sizes, for each first word size, determine a respective number of incidences in the series of bits of each possible value for a word of a size equal to the first word size, and determine the measure of randomness based on the determined numbers of incidences for the first word sizes, and, based on the determined measures of randomness obtained for each of the plurality of series of bits, select one of the computing devices for use in a cryptographic method. In some examples, the memory 804 contains instructions executable by the processing circuitry 802 such that the apparatus 800 is operable to carry out the method 400 shown in Figure 4 and/or described above. It should be noted that the above-mentioned examples illustrate rather than limit the invention, and that those skilled in the art will be able to design many alternative examples without departing from the scope of the appended statements. The word “comprising” does not exclude the presence of elements or steps other than those listed in a claim or embodiment, “a” or “an” does not exclude a plurality, and a single processor or other unit may fulfil the functions of several units recited in the statements below. Where the terms, “first”, “second” etc are used they are to be understood merely as labels for the convenient identification of a particular feature. In particular, they are not to be interpreted as describing the first or the second feature of a plurality of such features (i.e. the first or second of such features to occur in time or space) unless explicitly stated otherwise. Steps in the methods disclosed herein may be carried out in any order unless expressly otherwise stated. Any reference signs in the statements shall not be construed so as to limit their scope.