The Technique for Testing Short Sequences as a Component of Cryptography on the Internet of Things Svitlana Popereshnyak [0000-0002-0531-9809] Taras Shevchenko National University of Kyiv, 24, Bohdana Havrylyshyna str., Kyiv, 04116, Ukraine spopereshnyak@gmail.com Abstract. An article dedicated to on topical issues related to Internet of Things security. ІоТ dаtа prоteсtіоn sоlutіоns must spаn edge tо сlоud, prоvіde sсаlаble enсryptіоn аnd key mаnаgement, аnd nоt іmpede dаtа аnаlysіs. Тhe аvаіlаble аpprоасhes tо testіng rаndоm оr pseudоrаndоm sequenсes shоw lоw flexіbіlіty аnd versаtіlіty іn the meаns оf fіndіng hіdden pаtterns іn the dаtа. Іt іs reveаled thаt fоr sequenсes оf length up tо 100 bіts there аre nоt enоugh exіstіng stаtіstісаl pасkets. The classification of the main problems of information secu- rity in Internet of Things is given. The complexity of using classical crypto- graphic algorithms for information security in Internet of Things is considered. The paper proposed a methodology for testing pseudorandom sequences, ob- tained an explicit form of the joint distribution of numbers of 2-chains and numbers of 3-chains of various options random bit sequence of a given small length. Examples, tables, diagrams that can be used to test for randomness of the location of zeros and ones in the bit section are presented. As a result of the implementation of this technique, an information system will be created that will allow analyzing the pseudorandom sequence of a small length and choos- ing a quality pseudorandom sequence for use in a particular subject area. Keywords: Internet of Things, Algorithms, multidimensional Statistics, Ran- dom Sequence, s-chains, Cryptography, Pseudorandom Sequence, Statistical Testing. 1 Introduction Оrgаnіzаtіоns hаve оnly just begun dіsсоverіng аnd benefіtіng frоm the оppоrtunіtіes prоvіded by the Іnternet оf Тhіngs (IoT). Тhe аbіlіty tо саpture аnd аnаlyze dаtа frоm dіstrіbuted соnneсted devісes оffers the pоtentіаl tо оptіmіze prосesses, сreаte new revenue streаms, аnd іmprоve сustоmer servісe. Hоwever, the ІоТ аlsо expоses оrgаnіzаtіоns tо new seсurіty vulnerаbіlіtіes іntrоduсed by іnсreаsed netwоrk соnneсtіvіty аnd devісes thаt аre nоt seсured by desіgn. Аnd аdvаnсed аttасkers hаve demоnstrаted the аbіlіty tо pіvоt tо оther systems by leverаgіng vulnerаbіlіtіes іn ІоТ devісes. ІоТ devісes соlleсt sоme vоlumes оf dаtа, sоme оf whісh wіll requіre prоteсtіоn bаsed оn sensіtіvіty оr соmplіаnсe requіrements. ІоТ dаtа prоteсtіоn sоlutіоns must Copyright © 2019 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). spаn edge tо сlоud, prоvіde sсаlаble enсryptіоn аnd key mаnаgement, аnd nоt іmpede dаtа аnаlysіs. Тhe аvаіlаble аpprоасhes tо testіng rаndоm оr pseudоrаndоm sequenсes shоw lоw flexіbіlіty аnd versаtіlіty іn the meаns оf fіndіng hіdden pаtterns іn the dаtа. Random sequences have found the widest application from the gaming computer industry to mathematical modeling and cryptology. We list some areas of their usage: 1. Modeling. In computer simulation of physical phenomena. In addition, mathemati- cal modeling uses random numbers as one of the tools of numerical analysis. 2. Cryptography and information security. Random numbers can be used to test the correctness or effectiveness of algorithms and programs. Many algorithms use the generation of pseudo-random numbers to solve applied problems (for example, cryptographic encryption algorithms, the generation of unique identifiers, etc.). 3. Decision making in automated expert systems. The use of random numbers is part of decision-making strategies. For example, for the impartiality of the choice of examination paper by a student in an exam. Randomness is also used in the theory of matrix games. 4. Optimization of functional dependencies. Some mathematical optimization meth- ods use stochastic methods to search for extremums of functions. 5. Fun and games. Accident in games has a significant role. In computer or board games, chance helps to diversify the gameplay. There are various approaches to the formal definition of the term “randomness” based on the concepts of computability and algorithmic complexity [1]. By implementing some algorithm, software generators produce numbers (although not obvious) depending on the set of previous values, so the received numerical se- quences are not truly random and are called pseudo-random sequences (PRS). At the moment, more than a thousand software PRS generators are known, which differ in algorithms and values of parameters. Statistical properties are significantly different from the number sequences that are generated by them. The presented and not presented results allow us to characterize the state of modern technologies of designing the PRS (focusing on the most progressive of them by the following basic provisions [1-13]. 2 Features of Information Protection in the Internet of Things “Things” today are not only personal items of ordinary consumers, but also various equipment that is actively used in many fields of activity – trade, transport, medicine, construction, banking, sports, etc. It follows that the Internet of things is most often heterogeneous network, i.e. devices of various classes and types are combined and interact with each other. Recommendations for protecting information in the Internet of things are aimed at improving the security of devices, networks and data. First of all, IoT devices, as a rule, due to their portability and mobility, are physi- cally accessible to cybercriminals, and can be stolen to gain access to confidential data and establish communication with other network devices. To prevent this threat, it is necessary to provide physical protection, for example, by using protective covers on devices or cases that provide for restrictions on direct access to devices. In addition to direct access, devices can provide remote access to update configuration data or software. To protect against this, it is necessary to provide that the software ports to be closed and apply strong passwords at the level of downloading and updating firm- ware, which will prevent access to the device if it is compromised. At the same time, on the other hand, many IoT devices are becoming vulnerable to cyberattacks because their software is not updated in a timely manner. To minimize such risks, it is recommended to implement an automatic update by default, because, even if software updates are released on time, consumers do not always install them manually immediately after the release. Attention should also be paid to the organization of data storage on the devices themselves, because often this information is related to the user's personal data, finan- cial transaction data and data on critical objects of various fields of activity. Safety must be ensured both throughout the entire period of the product’s function- ing and after its decommissioning. Cryptographic keys must be stored in non-volatile memory of the device in not open form. In addition, disposal of decommissioned devices may be envisaged. To protect networks, first of all, methods of "strong authentication" should be pro- vided, including, for example, two-factor authentication, assignment of "hard" speci- fied unique identification and authentication data, as well as the use of modern secure protocols [14]. Cryptographic algorithms must be adapted to the Internet of things. In order to minimize the risks of denial of service attacks against devices, it is rec- ommended to provide bandwidth limits for the network of Internet of things devices, both at the software and hardware levels. In case of detection of suspicious traffic, devices should provide the ability to signal about that with the subsequent analysis of the identified threat. Data protection is primarily ensured through the use of cryptographic methods adapted to the features of devices with limited opportunities. If the device is compro- mised, it should be possible to urgently erase key information used in cryptographic operations. Devices of the Internet of things should transmit and process only the information that is necessary for the implementation of their main functions - as a rule, this is the collection of information about the state of their environment or about the user. It follows that it is necessary to pay attention to the information circulating in the net- work, minimizing the risk of leakage of confidential information. In addition to the heterogeneity of networks, a feature of the Internet of things is also that the devices have different computing resources, bandwidth and support dif- ferent technologies and protocols. The lack of common standards and protocols re- mains a serious problem in building a network of “things”. Also, many “things” have limited power capabilities and must support energy-saving modes. The listed features of the Internet of things impose restrictions when building a se- curity system in such a network. The usual methods of protecting information in wire- less networks may not be enough, or they may not be applicable due to the restrictions imposed by the Internet of things. The main methods for ensuring security, as in traditional networks, remain encryp- tion, identification / authentication, and the implantation of physical security measures. The security system should be designed to provide protection for devices and gateways, the transmission network, as well as applications that are deployed to en- sure the functioning of the devices. Encryption is a widely used, effective and quite flexible solution for ensuring in the confidentiality of information and for creating a security system. However, any en- cryption, and especially strong one, requires an increase in productivity and additional computing resources, which is not always possible in the conditions of the Internet of things. As for authentication, the researchers proposed a fairly large number of approaches that could be implemented to solve security problems [14, 15]. One common method is two-factor authentication. For example, one-time password authentication (OTP). With this approach, after providing the credentials, the user or device must also pre- sent a one-time password generated by the key distribution center, thereby confirming its authenticity. This method does not require additional computing resources or stor- age from the devices, but it is not applicable for devices that, for example, simply cannot support the ability to enter the received one-time password. The same problem is relevant for the authentication method, the second factor of which is the hardware identifier. Other studies suggest using the concept of “digital memories” for authentication, which would solve the problem of users remembering complex passwords. However, this method imposes resource limits on devices. The proposed methods also include authentication using cryptography based on el- liptic curves. Despite the fact that in this case the necessary basic parameters of ellip- tic curves are not calculated by the devices themselves, after the calculation, a suffi- ciently large amount of data must be transferred, which may be limited by the net- work bandwidth [14]. Thus, the various existing authentication methods are applicable to a single net- work and to a separate class of devices. The application of uniform methods and means is complicated by the lack of standardization and heterogeneity of such net- works. 3 Problem Statement Before responsible using in mathematical modeling and cryptology, PRS should be tested. Unfortunately, for many PRS tests, there are some limitations:  checked out only one of the probable ones properties that are characterize PRS;  not fix family alternatives;  do not have theoretical ones ratings power.  do not give a correct an estimate of chance sequences provided a little sample. Problems small and large samples refer to the main problems that arise in practical application methods analysis data. Let's be use the next classification samples by number [16], based on requirements presented in the program criteria:  very small sampling - from 5 to 12,  small sampling - from 13 to 40,  sampling average the number - from 41 to 100,  big ones sampling - from 101 and more. The minimum size of the sample limits not so much the algorithm of calculating the criterion, but the distribution of its statistics. For a row algorithms with too much small ones numbers sample normal approximation distribution of statistics criterion will be under question. During the research, the localization of the local sections of the bit sequence was conducted to detect the dependencies in the location of its elements by using the exact distributions of the corresponding statistics. In the work an explicit form of the joint distribution of the numbers of 2-chains and numbers of 3-chains of various variants in a random sequence was obtained. This joint distribution allows more accurate com- parison of the use of one-dimensional statistics, to analyze the bit sequence small length by chance. 4 Joint Distribution of number of 2-chains and number of 3- chains of a provided type in binary sequence Consider a sequence of random variables 𝛾1 , 𝛾2 , . . . , 𝛾𝑛 , (1) where 𝛾𝑖 = {0, 1}, i= 1, 2, . . . , 𝑛, 𝑛 > 0. Subsequences 𝛾𝑗 , 𝛾𝑗+1 , . . . , 𝛾𝑗+𝑠−1 , sequences (1) are called s-chains, 𝑗 = 1, 2, . . . , 𝑛 − 𝑠 + 1, 𝑠 = 1, 2, . . . , 𝑛. Denote 𝜂(𝑡1 , 𝑡2 , . . . , 𝑡𝑠 ) the number of s-chains in the sequence (1) that coincide with 𝑡1 , 𝑡2 , . . . , 𝑡𝑠 , where 𝑡𝑖 = {0, 1}, 𝑖 = 1, 2, . . . , 𝑠. Theorem. Let sequence (1) consist of n, 𝑛 > 0 independent identically distributed random variables; Ρ{𝛾𝑖 = 1} = 𝑝, Ρ{𝛾𝑖 = 0} = 𝑞, p + q = 1, i = 1, 2, . . . , n and 𝑘1 , 𝑘2 , 𝑘3 , 𝑡, – integer numbers such that 𝑘1 ≥ 0, 𝑘2 ≥ 0, 𝑛 ≥ 𝑘1 , 𝑘3 ≥ 0, 𝑡, 𝑡1 𝜖{0, 1}. Then 𝑚−𝑘 Ρ{𝜂(𝑡1∗ 𝑡1 ) = 𝑘1 , 𝜂( 𝑡 ∗ 0𝑡 ∗ ) + 𝜂( 𝑡 ∗ 1𝑡 ∗ ) = 𝑘2 } = ∑𝑚1=𝑘11 𝑝𝑚1 𝑞 𝑚0 𝜒(𝑚𝑡 , 𝑘1 , 𝑘2 ) (2) 1, 𝑖𝑓 𝑚𝑡 = 𝑘1 = 𝑘2 = 0, 𝜒(𝑚𝑡 , 𝑘1 , 𝑘2 ) = { , 𝜓(𝑚𝑡 , 𝑘1 , 𝑘2 ), elsewhere 𝛿 ∗ 𝛿 −𝑚𝑡 +2𝑖 𝑘 −𝛿 ∗ 𝜓(𝑚𝑡 , 𝑘1 , 𝑘2 ) = ∑ ∑ 𝑡 𝐶𝑖−1 𝐶𝑖 𝑡 Ζ(𝑚𝑡 − 𝑖; 𝑚𝑡 − 𝑖 − 𝛿𝑡 )𝐶𝑚1 ∗ −𝑖+1 𝑡 , 𝑡 𝑖∈{𝑘1 , 𝑘1 +1} where is the symbol ∑ denotes addition over all non-negative integers 𝛿𝑡 and 𝛿𝑡 ∗ such that 𝛿𝑡 + 𝛿𝑡 ∗ = 𝑘2 , 𝑏−1 𝐶𝑎−1 , if 𝑎 ≥ 𝑏 ≥ 1; Ζ(𝑎, 𝑏) ≝ { 1, if 𝑎 = 𝑏 = 0; 0, elsewhere. 𝑚−𝑘 𝑘 Ρ{𝜂( 𝑡1∗ 𝑡1 ) = 𝑘1 , 𝜂( 𝑡 ∗ 𝑡 ∗ 𝑡 ∗ ) = 𝑘2 } = ∑𝑚1=𝑘11 𝑝𝑚1 𝑞 𝑚0 𝐶𝑚1𝑡∗ 𝜑(𝑚𝑡 , 𝑘1 , 𝑘2 ), (3) 𝑚 −𝑘2 −𝑖 𝜑(𝑚𝑡 , 𝑘1 , 𝑘2 ) = ∑𝑖∈{𝑘1 , 𝑘1+1} 𝐶𝑖 𝑡 Ζ(𝑚𝑡 − 𝑖, 𝑚𝑡 − 𝑖 − 𝑘2 ), 𝑚−𝑘 Ρ{𝜂(𝑡1∗ 𝑡1 ) = 𝑘1 , 𝜂( 𝑡 ∗ 𝑡𝑡 ∗ ) = 𝑘2 } = ∑𝑚1 =𝑘11 𝑝 𝑚1 𝑞 𝑚0 𝜙(𝑚𝑡 , 𝑘1 , 𝑘2 ) (4) 1, 𝑖𝑓 𝑚𝑡 = 𝑘1 = 𝑘2 = 0, 𝜙(𝑚𝑡 , 𝑘1 , 𝑘2 ) = { . ∑𝑖∈{𝑘1, 𝑘1−1} 𝐶𝑖𝑘2 Ζ(𝑚𝑡 ; 𝑖 + 1)𝐶𝑚 𝑘1 −𝑘2 ∗ −𝑖 , elsewhere 𝑡 5 Results and Discussion As a result of applying this technique for testing pseudo-random sequences for two- dimensional statistics (relations (2) - (4)), you can build a bubble diagram with which you can get the probability of the distribution of zeros and ones in a given sequence. Consider examples of bubble diagrams for a bit sequence of small length n, n = 16. 5.1 Graphic Illustration of the Use of Equality (2) Fig. 1 gives a bubble chart in which the first parameter (horizontal axis) is the value 𝑘1 , the second parameter (vertical axis) is the value 𝑘2 , and the third parameter (the bubble size) is the probability of the event occurring {𝜂(𝑡1∗ 𝑡1 ) = 𝑘1 , 𝜂( 𝑡 ∗ 0𝑡 ∗ ) + 𝜂( 𝑡 ∗ 1𝑡 ∗ ) = 𝑘2 }, presented in percent. After analyzing Fig. 1 it can be concluded that for the analysis of the sequence of chains of small and medium length (from 13 to 100 elements), one-dimensional statis- tics do not always give the correct result. For example, if we consider the sequence where the parameter 𝑘1 = 4, then we can draw a conclusion with a high degree of probability of randomness of the sequence with these characteristics, however, if we pay attention when 𝑘1 = 4 and 𝑘2 = 0 it can be argued that this sequence is non- random, therefore as shown in Fig. 1 we have Ρ{𝜂(𝑡1∗ 𝑡1 ) = 𝑘1 , 𝜂( 𝑡 ∗ 0𝑡 ∗ ) + 𝜂( 𝑡 ∗ 1𝑡 ∗ ) = 𝑘2 } = 1,30%. What also shows the lack of use of one-dimensional sta- tistics for the analysis of short and medium bit sequences. Fig. 1. Bubble chart of sequence with the length 13 for (2). An approach to testing using n-dimensional statistics allows us to rely on a deeper justification of the randomness of generated sequences. 5.2 Graphic Illustration of the Use of Equality (3) In Fig. 2 shows the use of the relation (3) for a small sample. 𝑛, 𝑛 = 16 , and some values 𝑘1 and 𝑘2 . Fig. 2. Bubble chart of sequence with the length 16 for formula (3). Fig. 2 gives a bubble chart in which the first parameter (horizontal axis) is the value 𝑘1 , the second parameter ( vertical axis) is the value 𝑘2 , and the third parameter ( bubble size) is the probability of the event occurring {𝜂( 𝑡1∗ 𝑡1 ) = 𝑘1 , 𝜂( 𝑡 ∗ 𝑡 ∗ 𝑡 ∗ ) = 𝑘2 }, which is represented as a percentage. 5.3 Graphic Illustration of the Use of Equality (4) In Fig. 3 shows the use of relation (4) for a small sample 𝑛, 𝑛 = 16, and some values 𝑘1 and 𝑘2 . Fig. 3 gives a bubble chart in which the first parameter (horizontal axis) is the val- ue 𝑘1 , the second parameter (vertical axis) is the value 𝑘2 , and the third parameter (bubble size) is the probability of the event occurring {𝜂(𝑡1∗ 𝑡1 ) = 𝑘1 , 𝜂( 𝑡 ∗ 𝑡𝑡 ∗ ) = 𝑘2 }, which is represented as a percentage. Fig. 3. Bubble chart of sequence with the length 16 for formula (4). In this paper, the exact compatible distributions of some statistics (0, 1) -sequences of length 1 < 𝑛 < ∞ are given. For a bit sequence of small length n, n = 16, the tables containing the numerical values of the corresponding distribution are given. These tables, as well as the proposed graphic representations, can be used to test the hypoth- esis of the randomness of the arrangement of zeros and units. 6 The Results of the Comparison the NIST Statistical Test Suite and Test of PRS of Small Length using Multidimensional Statistics Consider the well-known examples that are given in [17, 18]. Let us analyze the submitted sequences for the corresponding tests, where:  P is the probability of sequence randomness according to the selected criterion from the first column,  P1 is the probability obtained using relation (2),  P2 is the probability obtained using relation (3),  P3 is this is the probability obtained using relation (4). Table 1. The results of the comparison. Input Size Test Recomme length Sequences Р P1 P2 P3 ndation Frequency (Monobit) n>=100 10 1011010101 0,527 0,021 0,049 0,021 Test Frequency Test within a n>=100 10 0110011010 0,801 0,097 0,212 0,129 Block Runs test n>=100 10 1001101011 0,147 0,097 0,212 0,129 Binary N=20 01011001001 Matrix Rank n>= 38000 0,741 0,112 0,289 0,245 M=Q=3 010101101 Test Discrete Fourier Transform n>=1000 N=10 0001010011 0,109 0,106 0,212 0,129 (Spectral) Test Non- overlapping N=20, 2 10100100101 Template N >= 200 blocks of 0,344 0,098 0,176 0,105 110010110 Matching length 10 Test Maurer’s “Universal n>= 01011010011 N=20 0.767 0,112 0,289 0,245 Statistical” 380000 101010111 Test Serial test n>=100 N=10 0011011101 0,907 0,025 0,212 0,028 Approximate n>=100 N=10 0100110101 0,261 0,021 0,049 0,021 Entropy test Cumulative Sums n>=100 N=10 1011010111 0,411 0,097 0,212 0,129 (Cusum) Test Input Size Test Recomme length Sequences Р P1 P2 P3 ndation Random Excursions n>106 N=10 0110110101 0,502 0,003 0,049 0,003 Test Random Excursions n>106 N=10 0110110101 0,683 0,003 0,049 0,003 Variant Test As can be seen from the table, the use of two-dimensional statics gives a more accu- rate result for short sequences. And also, according to [16], the recommended mini- mum sequence length n is greater than 100 bits. 7 Conclusions An analysis of the effectiveness of pseudorandom sequence generators is an urgent issue of cybersecurity in the use of more advanced methods of encryption and infor- mation security. The available techniques show low flexibility and versatility in the means of finding hidden patterns in the data. To solve this problem, it is suggested to use algorithms based on multidimensional statistics. The approach to testing using multidimensional statistics allows you to rely on a deeper justification of the randomness of the generated sequences. This area is prom- ising for scientific research. The paper proposed a methodology for testing a sequence and obtained a correct view of the joint distribution of the numbers of 2-chains and the numbers of 3-chains of various variants in a random bit sequence of a given small length. These algorithms and scheme of work for verification statistical tests of random- ness sequences (proposed in chapter II) combine all the advantages of statistical methods and are the only alternative for the analysis of sequences of small and medi- um length. To implement the proposed approach, a PRS software test package is being devel- oped, which will include tests using multidimensional statistics, which are well rec- ommended for testing a small length PRS. The complex is based on software products developed in C++, Python, for analyzing PRS. Currently, more than 20 PRS tests have been implemented, and the test database is being updated. As a result of the implementation of this technique, an information system will be created that will allow analyzing the PRS of a small length and choosing a quality PRS for use in a particular subject area. References 1. Popereshnyak, S.: Analysis of pseudorandom small sequences using multidimensional statis- tics. 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