On the Development of an Optimal Structure of Tree Parity

Research ArticleOn the Development of an Optimal Structure of Tree ParityMachine for the Establishment of a Cryptographic Key

Eacutedgar Salguero Dorokhin Walter Fuertes and Edison Lascano

Department of Computer Sciences Universidad de las Fuerzas Armadas ESPE Sangolquı PO Box 171-5-231-B Ecuador

Correspondence should be addressed to Walter Fuertes wmfuertesespeeduec

Received 30 December 2018 Revised 20 February 2019 Accepted 26 February 2019 Published 18 March 2019

Academic Editor Kuo-Hui Yeh

Copyright copy 2019 Edgar SalgueroDorokhin et alThis is an open access article distributed under theCreativeCommonsAttributionLicensewhichpermits unrestricteduse distribution and reproduction in anymedium provided the original work is properly cited

When establishing a cryptographic key between two users the asymmetric cryptography scheme is generally used to send it throughan insecure channel However given that the algorithms that use this scheme such as RSA have already been compromised it isimperative to research for new methods of establishing a cryptographic key that provide security when they are sent To solvethis problem a new branch known as neural cryptography was born using a modified artificial neural network called Tree ParityMachine or TPM Its purpose is to establish a private key through an insecure channel This article proposes the analysis of anoptimal structure of a TPM network that allows generating and establishing a private cryptographic key of 512-bit length betweentwo authorized parties To achieve this the combinations that make possible to generate a key of that length were determinedIn more than 15 million simulations that were executed we measured synchronization times the number of steps required andthe number of times in which an attacking TPM network manages to imitate the behaviour of the two networks The simulationsresulted in the optimal combination minimizing the synchronization time and prioritizing security against the attacking networkFinally the model was validated by applying a heuristic rule

1 Introduction

Cryptography is the mathematical science and the disci-pline of writing messages in encoded text Its purpose isto protect secrets from adversaries interceptors intrudersopponents and attackers [1] It also pays special attention onmechanisms that guarantee information integrity and dealswith techniques for exchanging users authentication keys andprotocols [2] The most known method is the combinationof the symmetric and asymmetric encryption and decryp-tion algorithms The asymmetric algorithm is used for theexchange of cryptographic keys and the symmetric algorithmis used for information encryption and decryption

The RSA public key cryptosystem and the systems basedon elliptic curves are the most common forms for public-key cryptography in the encryption and digital signaturestandards [3] However the security of the RSA algorithmdepends on the length of the prime numbers used forfactoring [4] Thus one of the main concerns of RSA is thedemand for large keys in todayrsquos cryptographic algorithmssince the product of two long prime numbers must be

factored This generates a bigger value than the originalIncreasing the length of prime numbers increases securityhowever the computational cost of factoring these numbers isalso increased For this reason it is important to look for newmethods of exchanging cryptographic keys in a secure wayand with a relatively low computational cost Hence neuralcryptography is born this method uses a neural networkcalled Tree Parity Machine (TPM) In this way through twoTPM networks with exactly the same structure two userscan establish a cryptographic key by exchanging the inputsand outputs of these networks keeping the synaptic weightssecret

The aim of our study is to search for an optimal structurethat minimizes the time and number of steps that two TPMnetworks require to establish a cryptographic key of 512-bit length In addition we also emphasize on an attackingnetwork whose behaviour is difficult to imitate For this wecreated an algorithm using Python this algorithm allowsperforming the simulations to test and find the optimalstructure A total of 15rsquo041100 simulations were conductedwith all the possible structures of the network Additionally

HindawiSecurity and Communication NetworksVolume 2019 Article ID 8214681 10 pageshttpsdoiorg10115520198214681

2 Security and Communication Networks

we used R and GNU Octave for statistical analysis Finallya heuristic rule was applied to validate the computed values(proposed by [5 6]) As a result it was possible to determinethe optimal structure of the TPM network with an averageamount of 211 steps and a percentage of 000004 of successof a passive attack network and with a successful 0 of ageometric attack

The main contribution of this study is that two userswill be able to generate a cryptographic key based on certainprobability according to the number of steps established atthe beginning of the synchronization and with a very lowprobability of a successful passive attack

The remainder of this document is structured as followsSection 2 mentions the related works that supports thepresent study Section 3 describes the procedure that helpsdetermine the best combination Section 4 shows the resultsobtained from the experiment Finally Section 5 detailsconclusions and proposes future work

2 Related Works

Concerning the studies related to this research Kanter etal [7] and Rosen-Zvi et al [8] demonstrated that whentwo artificial neural networks are trained on their outputsaccording to a learning rule they are able to developequivalent states of their internal synaptic weights Kinzeland Kanter [9] and Ruttor et al [10] revealed that thepossibility of a successful attack decreases as the synapticdepth of the network increases and the computational costof the attacker increases since its effort grows exponentiallywhile the effort required by the users grows polynomiallySimilarly Sarkar and Mandal [11] and Sarkar [12] mentionthat the performance is improved by increasing the synap-tic depth of the TPM networks henceforth counteractingthe brute force attacks with the current computation Incomparison with the present study we used the structureof the TPM network proposed by the latter However wedetermined that the security of the TPM network can alsobe increased by increasing the number of neurons in thehidden layer and also the number of entries of each neuronIn addition the range for the synaptic depth value hasbeen modified to adapt the need to generate a key of 512bits

With respect to the proposed algorithms for the designof the TPM network Lei et al [13] developed a two-layerprepowered TPMnetworkmodel A fast synchronization canbe achieved by increasing the minimum value of the internalrepresentationsHamming distance and by reducing the prob-ability of a step that does notmodify networks weights Allamet al [14] proposed an algorithm that increases the securityof neural cryptography by authenticating communicationsusing previously shared secrets in this way it increases thesecurity of neural cryptography As a result it shows that thealgorithm reaches a very high security level without increas-ing synchronization time Ruttor [15] mentions that the valueof K is 3 since lower values have negative consequencesfrom a safety point of view and higher values have negativeconsequences in terms of synchronization time Klimov et al[16] calculate the probability that two networks take to either

synchronize their weights or not Although the probabilityrises with a low value of K the success of the attackingnetwork also increases and consequently the value of Kmustbe greater In comparison with our study the initial structureof the TPMnetworks is random and different and has a singleoutput In addition because the goal is to generate a 512-bitkey it was not arithmetically adequate to use odd numbersfor K N and L values

In relation to measurements on TPM networks per-formance Dolecki and Kozera [17] present a method offrequency analysis that allows evaluating the synchronizationlevel of two TPM networks before they finish up with thecalculated value not related to the difference of their synapticweights As a result the selection of the appropriate rangefor the count frequency and the threshold allow them tospecify whether it is a short or a long synchronizationSanthanalakshmi et al [18] and Dolecki and Kozera [19]analyse and compare the performance of synchronization byemploying respectively a genetic algorithm and a Gaussiandistribution instead of uniform randomvalues for theweightsof the TPM networks As a result they found that replacingthe random weights with optimal weights helps reduce thesynchronization time In addition increasing the numberof hidden and input neurons accelerates convergence andalso reduces the probability of success of the ldquoMajorityFlipping Attackrdquo attack Dolecki et al [20] perform anadjustment of the timing distribution of two TPM networksto a Poisson distribution Pu et al [21] perform an algo-rithm that combines ldquotrue random sequencesrdquo (generatedby artificial and validated circuits with randomness tests)with TPM networks demonstrating more complex dynamicbehaviours that offer better performance as an encryptiontool and resistance to attacks In comparison with ourstudy the synaptic weights values of the TPM networksare generated according to a discrete uniform distributionAdditionally the Poisson distribution adjustment was madewith the results in the number of steps in each of thesimulations

With reference to rules that contribute in the designof TPM networks Mu and Liao [5] and Mu et al [6]define the following heuristic rule ldquoKeeping the equationsof motion constant a high value in the state classifier withrespect to the minimum values of the smallest Hammingdistances between the state vectors of the networks has ahigh probability of fulfilling the condition that the averagechange in the percentage difference between synaptic weightsis greater than zero improving the security of neuronalcryptographyrdquo For our study we used the proposed heuristicrule to determine the level of security of the final structure ofthe TPM network

In regard to modification of initial structures of aTPM network Gomez et al [22] state that with an initialmisalignment in the weights between 15 and 20 thesynchronization time is reduced from 125119898119904 to less than07119898119904 This also reduces the number of steps from 220 toless than 100 Within this context Niemiec [23] presentsa new idea for a key reconciliation method in quantumcryptography using TPM networks correcting errors thatoccur during transmission in the quantum channel The

Security and Communication Networks 3

Table 1 Possible values of 119871Decimal value 119871 Binary length 11987120 121 223 427 8215 16231 32263 642127 1282255 2562511 512

number of steps necessary to establish the key is significantlyreduced with a low value in the error rate of the quantum bitand a high value in the initial percentage of synchronizationof the two networks In comparison with our study we didnot use initial structures that presented an initial alignmentwith an established percentage This initial alignment shouldbe delivered and shared by the two users which implies thatan attacker has more initial information about the structureof the TPM networks

3 Materials and Methods

31 Background A TPM is a neural network that is formedby a hidden layer and a single output The general structureconsists of the triad of values 119870 119873 and 119871 where 119870 is thenumber of neurons in the hidden layer 119873 is the number ofentries of each neuron in the hidden layer and 119871 sets the limitof the range of possible integer values related to the synapticweights

To generate and establish a key of 512 bits some vari-ations of the TPM structure whose synaptic weights allowthis key length were tested To determine it the value of 119871was taken as a base which in its binary notation establishesthe final length of the key To establish pair values in the keylength the limits of the range of values of [minus119871 119871] to [minus119871 119871minus1]whose maximum value in decimal notation is 2119871 minus 1 weremodified and the length of its binary value is a multiple of512Thanks to this there are 10 possible values of 119871 that allowgenerating keys of 512 bits see Table 1

For each value of 119871 there are values of119870 and119873 such that119897119890119899(119871119861119868119873) lowast119870lowast119873 =512 considering the restriction 119870 lt= 119873as shown in Table 2

The restriction of119870 lt= 119873 is due to the fact that if119870 gt 119873the neural network will be able to reach values that make thesynchronizationnot to be possibleThen there are 30 possiblecombinations of 119870119873 and 119871 values that allow us to generatea key of 512-bit length

To find the optimal combination that allows rapid syn-chronization in a minimum number of steps which preventsan attacker from being able to copy their behaviour 500000simulations were performed for each combination Eachsimulation consisted of three TPM networks two networkssynchronized their weights in an authoritative way (AliceandBob) and another unauthorized attacking network (Eve)

Table 2 Possible combinations with the values of 119871119871 119870 119873

201 5122 2564 1288 6416 32

211 2562 1284 648 3216 16

231 1282 644 328 16

271 642 324 168 8

215 1 322 164 8

231 1 162 84 4

263 1 82 4

2127 1 42 2

2255 1 22511 1 1

they tried to imitate the behaviour of the other two networksto discover the key they tried to establish For simulationthe values of 119871 greater than 231 (263 2127 2255 y 2511) werenot considered due to its length and complexity in thecalculation

32 Implementation of the Algorithm and the Attack ModelWe used Python to implement an algorithm to measure thetime and the number of steps needed for synchronization inaddition to calculating the cumulative values of the numberof steps during simulations (see Algorithm 1)

As can be seen in Algorithm 1 the second line definesthe total of simulations that will be executed Lines 3 to 5initialize the three TPM networks with 119870 119873 and 119871 Line 6initializes the step counter to 0 Line 7 establishes that thesimulation will not finish while the weights of both networksare not equal In line 8 we create a random vector that will bethe input for the networks Lines 9 to 11 compute the outputsof the three networks Line 12 analyses whether the outputsof Alice and Bob networks are the same In this case lines13 and 14 update weights of both networks according to the


Data119870119873 119871 number of simulationsResult none1 initialization2 for 119894 larr997888 0 to 119899119906119898119887119890119903119878119894119898119906119897119886119905119894119900119899119904 do3 Alicelarr997888 TPM (119870119873 119871)4 Boblarr997888 TPM (119870119873 119871)5 Evelarr997888 TPM (119870119873 119871)6 stepslarr997888 07 while 119860119897119894119888119890119908119890119894119892ℎ119905119904 = 119861119900119887119908119890119894119892ℎ119905119904 do8 inputlarr997888 randomVector ()9 outAlarr997888 Alice(input)10 outBlarr997888 Bob(input)11 outElarr997888 Eve(input)12 if 119900119906119905119860 = 119900119906119905119861 then13 Aliceupdate(outB)14 Bobupdate(outA)15 end16 if 119900119906119905119860 = 119900119906119905119861 = 119900119906119905119864 then17 Eveupdate(outA)18 end19 stepslarr997888 steps+120 end21 saveToFile (steps)22 end

Algorithm 1 Pseudocode of the simulations

previously computed outputs Line 16 allows Eve to updateher weights as shown by line 17 but only when the outputsof the three networks are equal Line 19 increases the stepcounter by 1 Finally line 21 saves the number of steps in afile

At a glance from the algorithm attacks occur when theoutputs of the three TPMnetworks match Eve can only listento messages that are sent between Alice and Bob and itslearning process is slower since it only updates her weightsif its output matches the other two outputs According to[24] a geometric attack has better results when an attackeruses a single TPM network to mimic the behaviour ofthe other two authorized networks Therefore additional180000 simulations were conducted in a geometric attackThe condition is taken into account when the outputs ofthe two authorized networks are equal and the output ofthe attacking network is different In this case the attackingnetwork can modify its internal representation by adjustingthe partial output with the lowest absolute value by itscounterpart

For this purpose we designed and included Algorithm 2In this algorithm the third line calculates the absolute valuesof the partial outputs of each hidden neuron using the inputvalues and the weights of the attacking network Line 4calculates the sign of each of the partial values of each hiddenneuron In line 5 the sign corresponding to the position ofthe minimum absolute value of the partial output is changedLine 6 calculates the value of the output of the network withthe new partial outputs Finally line 7 updates the networkweights

Data outA outB outE input weightEResult none1 initialization2 if 119900119906119905119860 = 119900119906119905119861 = 119900119906119905119864 then3 sigmaAbslarr997888 abs(sum(input lowast weightE))4 sigmaSignlarr997888 sign(sum(input lowast weightE))5 sigmaSign[sigmaAbsmin()]larr997888

-sigmaSign[sigmaAbsmin()]6 taularr997888 prod(sigmaSign)7 update(tau)8 end

Algorithm 2 Pseudocode geometric attack

4 Results and Discussion

41 Results of the Simulations To perform and obtain theresults of the simulations a DELL Inspiron 5759 computerwith a 250GHz sixth generation Intel Core i7-6500U CPU(model 78) was used It has four CPUs one socket twocores per socket and two processing threads per core cachememory L1d of 32K L1i of 32K L2 of 256K and L3 of 4096K

Table 3 shows the results obtained from the simulationsperformed for each combination of 119870 119873 and 119871 ColumnsMin Steps and Max Steps show the minimum and themaximum number of steps that took for the two networksto synchronize their weights The Average Steps columnshows the average of steps of all the executed simulationsColumns Min Time and Max Time show the minimum andmaximum time (in seconds) that took for the two networksto synchronize their weights The Average Time columnshows the average time (in seconds) of all the executedsimulations ColumnEve sync successfully shows the numberof occurrences in which the attacking network managed toimitate the behaviour of the other two networks of the total ofexecuted simulations Finally columnEve sync successfullyshows the percentage represented by the previous columnwith respect to the total of simulations

As can be seen in Table 3 items 14 and 18 correspondingto the combinations (8 16 23) and (8 8 27) respectively gotthe best results from the point of view of security against theattacking network (Eve) None of the 500000 simulationsallowed Eve to imitate the behaviour of the other twonetworks with the same number of steps of Alice and Bobrsquosnetworks However two combinations took a greater numberof steps to achieve synchronization (211659 and 2182211respectively) than the other combinations which involves alonger average time

For synchronization time item 22 corresponding to(1 16 231) combination produced the best result with anaverage synchronization time of 00549 seconds Howeverit got an attacking network sync percentage of 92488Given that security was prioritized in the present studywe considered the combinations mentioned above as goodresults It should also be noted that the combinations where119871 = 20 = 1 despite having the best times in synchronizationproduced the worst results because the attacking network Eve


Table 3 Results obtained from 500000 simulations for each combination

Item Combination(K N L)

Minimumsteps

Maximumsteps Average steps Minimum

time (sec)Maximumtime (sec)

Average time(sec)

Eversquossuccessful

sync

Eversquossuccessful

sync1 (1 512 20) 5 28 93367 00657 22718 04259 368680 737362 (2 256 20) 5 25 93312 00596 15588 04253 368229 7364583 (4 128 20) 5 32 93374 00680 21275 04263 368471 7369424 (8 64 20) 4 27 93370 00656 19089 04233 368652 7373045 (16 32 20) 5 32 93356 00661 14625 04249 368389 7367786 (1 256 21) 8 45 155450 00573 14665 03824 355397 7107947 (2 128 21) 8 25371 157774 00523 5375299 03857 349495 698998 (4 64 21) 8 4104 180226 00491 958512 03965 308691 6173829 (8 32 21) 9 814 316046 00403 196915 04597 161148 32229610 (16 16 21) 15 587 734867 00289 73628 05381 8102 1620411 (1 128 23) 11 78 253284 00472 11114 02983 132715 2654312 (2 64 23) 15 289 652706 00411 27924 04699 41475 829513 (4 32 23) 26 616 1281582 00309 45353 05325 487 0097414 (8 16 23) 38 769 2116590 00262 60476 05741 0 0015 (1 64 27) 10 71 250318 00180 06389 01679 132666 26533216 (2 32 27) 15 372 749943 00362 19875 03599 39111 7822217 (4 16 27) 23 712 1379569 00193 27179 04572 759 0151818 (8 8 27) 30 955 2182211 00342 34490 05109 0 0019 (1 32 215) 7 68 213921 00105 02913 00847 144461 28892220 (2 16 215) 10 451 698188 00196 12301 02161 61628 12325621 (4 8 215) 14 777 1320805 00181 27993 03200 4080 081622 (1 16 231) 6 84 249428 00020 02366 00549 46244 9248823 (2 8 231) 10 804 1005999 00121 14193 01842 25006 5001224 (4 4 231) 9 837 1342786 00032 20644 02281 9168 18336

Table 4 Results obtained from 1rsquo000000 simulations for the two combinations


Minimumsteps



Average time(sec)

Eversquossuccessful

sync

Eversquossuccessful

sync1 (8 16 23) 40 785 2115888 00780 15155 04058 0 02 (8 8 27) 27 861 2183696 00312 09229 02385 2 00002

managed to imitate the behaviour of the other two networksin approximately 737 of the total simulations

To determine the best combination of the two previoussituations specifically with better results from the securitypoint of view we performed onemillion simulations for eachand we got the following results see Table 4

From Table 4 we can determine the best combinationfor the triad (119870 119873 119871) is (8 16 23) From Table 4 it wasdetermined that the best combination for the trio of values(119870 119873 119871) is (8 16 23) Simulations were carried out usinggeometric attack and the results were obtained as shown inTable 5

More simulations with the combination of (8 16 23)wereconducted to determine the evolution of the distribution inthe number of steps Of all simulations we totalled the num-ber of steps that were necessary for the two TPM networks tosynchronize See the histograms shown in Figure 1

Table 5 Success probabilities of an attacking TPM using geometri-cal attack

Combination(K N L)

Number ofsimulations

Eversquos successfulsync

Eversquossuccessful sync

(8 16 23) 180000 0 00

During additional simulations on a single occasion Eversquosattacking network managed to synchronize its weights withthe other networks Table 6 shows the probabilities of asuccessful attack of both combinations compared to the totalof performed simulations

42 Adjustment of Probabilities Distribution and CalculationAs can be seen in Figure 1 the number of steps usedin all simulations follows a positive asymmetric discrete


100 200 300 400Number of steps

00

05

10

15

20

25

30

Freq

uenc

y

(a) 100 simulationsrsquo histogram

100 200 300 400 500Number of steps

0

2

4

6

8

10

12

Freq

uenc

y

(b) 1000 simulationsrsquo histogram

100 200 300 400 500 600Number of steps

0

10

20

30

40

50

60

70

80

Freq

uenc

y

(c) 10000 simulationsrsquo histogram

100 200 300 400 500 600 700Number of steps

0

100

200

300

400

500

600

700

Freq

uenc

y

(d) 100000 simulationsrsquo histogram

100 200 300 400 500 600 700Number of steps

0

200

400

600

800

1000

1200

1400

1600

Freq

uenc

y

(e) 250000 simulationsrsquo histogram

100 200 300 400 500 600 700Number of steps

0

500

1000

1500

2000

2500

3000

3500

Freq

uenc

y

(f) 500000 simulationsrsquo histogram

100 200 300 400 500 600 700Number of steps

0

1000

2000

3000

4000

5000

6000

7000

Freq

uenc

y

(g) 1rsquo000000 simulationsrsquo histogram

Figure 1 Illustrations derived from the evolution of the distribution in the number of steps


Table 6 Success probabilities of an attacking TPM of the twocombinations

Combination(K N L)




(8 16 23) 2rsquo361100 1 000004(8 8 27) 1rsquo500000 2 00001

Table 7 Poisson adjust results


Max limit Delta Probability Function

1000 515 05150 04787 11989110000 634 00634 08147 119892100000 709 000709 09625 ℎ250000 750 00030 09814 119894500000 718 00014 09902 1198951rsquo000000 785 00007 09946 119896distribution or to the right For the present work the Poissondistribution was considered by the characteristics of themathematical parameters such as lambda and delta wheredelta was considered the maximum value of the limit as afunction of the total simulations Delta is the probability ofoccurrences of the event Poisson has the formula as shownby the equation ref Poisson

119891 (120582 119889) = 119890minus120582120582119889119889 (1)

Then for the first test we used the results obtained fromthe 1000 simulations Themaximum value of the limit is 515where

119889119890119897119905119886 = 5151 000 = 0515 515 (2)

Using (1) taking as delta value = 0515 of the equationref delta 1 (lambda equal to delta to obtain the highest prob-ability) a probability of 04787 was obtained Similarly theprobabilities were calculated for the rest of the simulationsFor the 10 thousand simulations a delta value = 00634(634 ) was obtained and its probability is 08147Thereforelambda varies by 34 In an analogous way the same test wascarried out with the results of the other simulations Table 7shows the obtained values

As the number of simulations increases the probabilityhas a tendency towards 1 (maximum probability value)which implies that the tests performed are correct The samecan be seen in Figure 2 As can be seen the data tend to adjustmathematically to a Poisson distribution

43 Generation of theCryptographic Key of 512 Bits When thenetworks finish their synchronization their synaptic weightswill have values in the range [minus8 7] To obtain the key of512 bits the value of 119871 is added to each synaptic weight thisproduces that the range is [0 15] Then each synaptic weightis transformed to its binary equivalent of 4 bits in length Eachof these values is concatenated and the key of 512-bit lengthis obtained (8 lowast 16 lowast 4 = 512)

X

Y

+10 +25 +55+01+01+02+02+03+03+03+04+04+05+05+06+07+07+08+08+09+09+10+10+11

f(x) = (exp(minusx)∙(x)^(05150))05150

g(x) = (exp(minusx)∙(x)^(00634))00634

h(x) = (exp(minusx)∙(x)^(000709))000709

i(x) = (exp(minusx)∙(x)^(00030))00030

j(x) = (exp(minusx)∙(x)^(00014))00014

k(x) = (exp(minusx)∙(x)^(00007))00007

+40

Figure 2 Poisson distribution for all simulations

44 Process of Validation To validate the security of theproposed TPM network structure with the values of 119870 = 8119873 = 16 and 119871 = 23 = 8 and according to the heuristic ruleproposed by Mu and Liao [5] and Mu et al [6] the value ofthe state classifier ℎ was computed For this the Hammingdistances were calculated between all the possible vectorswhose outputs are equal (1 and minus1) Since 119870 = 8 there are28 possible vectors (27 vectors with output 1 and 27 vectorswith output minus1) The combinations for each group accordingto their output are given by the binomial coefficient as shownby

(272 ) = (1282 ) = 1282 (128 minus 2) = 8128 (3)

Therefore there are 8128 combinations for each groupThe value of the state classifier was computed as follows FirstHamming distances were calculated between each pair ofvectors whose output is equal to

11986711986311988811= 119867119863((1 1 1 1 1 1 1 1) (1 1 1 1 1 1 minus1 minus1))= 2

11986711986311988812= 119867119863((1 1 1 1 1 1 1 1) (1 1 1 1 1 minus1 1 minus1))= 2

sdot sdot sdot


Data 119870119873 119871 119901119903119900119887119886119887119894119897119894119905119894119890119904Result Graph1 initialization2 scatter3 (119870()119873() 119871() [16] 119901119903119900119887119886119887119894119897119894119905119894119890119904() lsquorsquo)3 set(gcalsquozscalersquolsquologrsquo)4 xlabel ldquoKrdquo5 ylabel ldquoNrdquo6 zlabel ldquoLrdquo7 colorbar

Algorithm 3 Code in GNU Octave to graph a scatter plot

11986711986311988821= 119867119863((1 1 1 1 1 1 1 minus1) (1 1 1 1 1 1 minus1 1))= 2

11986711986311988822= 119867119863((1 1 1 1 1 1 1 minus1) (1 1 1 1 1 minus1 1 1))= 2

sdot sdot sdot(4)

Then for each group the minimum value of the saiddistances is calculated

1198781198671198631198881 = min 11986711986311988811 11986711986311988812 = 21198781198671198631198882 = min 11986711986311988821 11986711986311988822 = 2 (5)

Finally the value of the state classifier is calculated as thesmallest value between theminimum values of the Hammingdistances

ℎ119879119875119872(119870=8) = min 1198781198671198631198881 1198781198671198631198882 = 2 (6)

The value of ℎ = 2 shows that it meets the probabilitythat the average change in the percentage difference betweensynaptic weights is greater than zero This implies that thesynchronization time is increased polynomially by increasingthe value of 119871 This means that the proposed structure issecure

45 Process of Analysis of the Values of 119870 119873 and 119871 Theprobability of success of an attacking network has a verynoticeable dependencewith respect to the values of119870119873 and119871 as shown in Table 3 Therefore an analysis of the values of119870119873 and 119871 was conducted with respect to their probabilityAll possible combinations were plotted with their respectiveprobabilities To do this we used the GNUOctave scatter3function to draw a scatter diagram Algorithm 3 presents thecode that generated this graph

Algorithm 3 starts with the values of 119870 119873 119871 and theprobabilities of Eve in each combination Line 2 draws a 3Dscatter diagram with the values of 119870 119873 and 119871 The value of

N 400500

600

300200

1000

20

1e+00

K

1e+011e+021e+031e+04

L 1e+051e+061e+071e+081e+091e+10

04

06

1510

50

0

02

Figure 3 Scatter plot of the values of 119870119873 and 119871

16 indicates the size of the points to be plotted The value of119901119903119900119887119886119887119894119897119894119905119894119890119904() reveals the value of the probabilities that willbe every point colour Line 3 changes the scale of the 119911 axis(value of 119871) to a logarithmic scale to improve visualizationLines 4 to 6 change the labels of the axes to their respectivevalues Finally line 7 discloses a colour bar Figure 3 displaysthe result of Algorithm 3

Figure 3 shows how the probabilities change according tothe values of 119870 119873 and 119871 On the 119909 axis are the values of 119870on the 119910 axis the values of119873 and on the 119911 axis the values of 119871The colour of each point displays the probability of success ofthe attacking network A low probability is better (blue) anda high probability is worse (red)

As can be seen in Figure 3 the value of 119870 does influencemuch on probability On the other hand a high value of 119873has a negative impact on the result Finally a very low valueof 119871 has a negative impact For this reason to propose valuesof 119870 119873 and 119871 a relatively low value of 119873 is recommendedbut it has to be higher than 119870 The value of 119871 should not betoo low These conditions ensure a very low probability of asuccessful passive attack

5 Conclusions

This article has been developed in the context of the designof a TPM neural network that allows generating a 512-bit keybetween two authorized parties To do this the optimal valuesof 119870 119873 and 119871 were computed through simulations thatallowed prioritizing security against an attacking networkThis attacking network through a passive attack tried toimitate the behaviour of the other two networks achievingthe synchronization with the other two networks on asingle occasion This demonstrates that a secret key canbe exchanged between two authorized parties in a networkwith a probability of 000004 so that an attacking networkcan mimic the original network behaviour In addition ageometric attack was designed and executed with successfulresults of 0 Furthermore if the values of 119870 119873 and 119871 ofa TPM neural network are modified cryptographic keys ofvarious lengths can be obtained These values also determine


how easy or how difficult it might be for an attacking networktomimic the behaviour of the other two networks Accordingto our results it is recommended that the values of119870119873 and119871meet the following conditions 119870 lt= 119873 and 119871 gt 1

As future work we plan the analysis of L values that werenot considered in this study due to software and hardwarelimitations (263 2127 2255 and 2511) in which security is alsoprioritized in front of an attacking network Also we planstudying the combination of neural cryptography using thevalues calculated in the study along with symmetric cryp-tographic systems that need a cryptographic key establishedbetween two parties (such as DNA-based cryptography)Thisin order to perform secure communications and without theneed to have sent its keys through insecure means Finallyit is recommended to perform a more exhaustive securityanalysis to the values found for the structure of the TPMneural network namely specific attack models designed tomanage the cryptographic key especially in its distribution

Data Availability

The data used to support the findings of this study (CSVTXT and PNG) have been deposited in the Google Driverepository In the following link you can find the data usedin our manuscript httpsdrivegooglecomdrivefolders1MOf7CfXFrFB-gtK6ue4JkHURzga1o8zA

Conflicts of Interest

The authors declare that there are no conflicts of interest

Acknowledgments

The funding of this scientific research is provided by theEcuadorian Corporation for the Development of Researchand the Academy (RED CEDIA) and also from the MobilityRegulation of the Universidad de las Fuerzas Armadas ESPEfrom Sangolquı Ecuador The authors would like to thankthe financial support of the Ecuadorian Corporation for theDevelopment of Research and the Academy (REDCEDIA) inthe development of this study within the Project Grant GT-Cybersecurity

References

[1] F L Bauer ldquoCryptologyrdquo in Encyclopedia of Cryptography andSecurity pp 283-284 Springer 2011

[2] Y Lindell and J Katz ldquoIntroduction to Modern CryptographyrdquoChapman and HallCRC 2014

[3] X Zhou and X Tang ldquoResearch and implementation of RSAalgorithm for encryption and decryptionrdquo in Proceedings of the6th International Forumon Strategic Technology IFOST 2011 pp1118ndash1121 IEEE China August 2011

[4] F Meneses W Fuertes J Sancho et al ldquoRSA EncryptionAlgorithm Optimization to Improve Performance and SecurityLevel of Network Messagesrdquo IJCSNS vol 16 no 8 p 55 2016

[5] N Mu and X Liao ldquoAn approach for designing neural cryp-tographyrdquo in International Symposium on Neural Networks pp99ndash108 Springer Heidelberg Berlin Germany 2013

[6] N Mu X Liao and T Huang ldquoApproach to design neuralcryptography A generalized architecture and a heuristic rulerdquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 87 no 6 Article ID 062804 2013

[7] I Kanter W Kinzel and E Kanter ldquoSecure exchange ofinformation by synchronization of neural networksrdquo EPL(Europhysics Letters) vol 57 no 1 pp 141ndash147 2002

[8] M Rosen-Zvi I Kanter and W Kinzel ldquoCryptography basedon neural networksmdashanalytical resultsrdquo Journal of Physics AMathematical and General vol 35 no 47 pp L707ndashL713 2002

[9] W Kinzel and I Kanter ldquoInteracting neural networks andcryptographyrdquo in Advances in Solid State Physics pp 383ndash391Springer 2002

[10] A Ruttor W Kinzel R Naeh and I Kanter ldquoGenetic attack onneural cryptographyrdquo Physical Review E Statistical Nonlinearand Soft Matter Physics vol 73 no 3 Article ID 036121 2006

[11] A Sarkar and J K Mandal ldquoCryptanalysis of key exchangemethod in wireless communicationrdquo Compare vol 1 2015

[12] A Sarkar ldquoGenetic Key Guided Neural Deep Learning basedEncryption for Online Wireless Communication (GKNDLE)rdquoInternational Journal of Applied Engineering Research vol 13 no6 pp 3631ndash3637 2018

[13] X Lei X Liao F Chen and T Huang ldquoTwo-layer tree-connected feed-forward neural networkmodel for neural cryp-tographyrdquo Physical Review E Statistical Nonlinear and SoftMatter Physics vol 87 no 3 Article ID 032811 2013

[14] A M Allam H M Abbas and M W El-Kharashi ldquoAuthen-ticated key exchange protocol using neural cryptography withsecret boundariesrdquo in Proceedings of the 2013 International JointConference on Neural Networks IJCNN 2013 pp 1ndash8 USAAugust 2013

[15] A Ruttor ldquoNeural synchronization and cryptographyrdquo Disor-dered Systems and Neural Networks 2007

[16] A Klimov A Mityagin and A Shamir ldquoAnalysis of neuralcryptographyrdquo in Proceedings of the International Conferenceon the Theory and Application of Cryptology and InformationSecurity pp 288ndash298 Springer 2002

[17] M Dolecki and R Kozera ldquoThreshold method of detectinglong-time TPM synchronizationrdquo in Computer InformationSystems and Industrial Management vol 8104 of Lecture Notesin Computer Science pp 241ndash252 Springer Berlin HeidelbergBerlin Heidelberg 2013

[18] S Santhanalakshmi K Sangeeta and G K Patra ldquoAnalysisof neural synchronization using genetic approach for securekey generationrdquoCommunications in Computer and InformationScience vol 536 pp 207ndash216 2015

[19] M Dolecki and R Kozera ldquoThe Impact of the TPM WeightsDistribution on Network Synchronization Timerdquo in ComputerInformation Systems and Industrial Management vol 9339of Lecture Notes in Computer Science pp 451ndash460 SpringerInternational Publishing Cham Swizerland 2015

[20] M Dolecki and R Kozera ldquoDistribution of the tree paritymachine synchronization timerdquo Advances in Science and Tech-nology ndash Research Journal vol 7 no 18 pp 20ndash27 2013

[21] X Pu X-J Tian J Zhang C-Y Liu and J Yin ldquoChaoticmultimedia stream cipher scheme based on true randomsequence combinedwith tree paritymachinerdquoMultimedia Toolsand Applications vol 76 no 19 pp 19881ndash19895 2017

[22] H Gomez O Reyes and E Roa ldquoA 65 nm CMOS keyestablishment core based on tree parity machinesrdquo Integrationthe VLSI Journal vol 58 pp 430ndash437 2017


[23] M Niemiec ldquoError correction in quantum cryptography basedon artificial neural networksrdquo Cryptography and Security 2018

[24] A Ruttor W Kinzel and I Kanter ldquoDynamics of neuralcryptographyrdquo Physical Review E Statistical Nonlinear and SoftMatter Physics vol 75 no 5 056104 9 pages 2007

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

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VLSI Design



Shock and Vibration


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Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

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Control Scienceand Engineering

Journal of



Journal ofEngineeringVolume 2018

SensorsJournal of



RotatingMachinery


Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018


Chemical EngineeringInternational Journal of Antennas and

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Navigation and Observation


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wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom


we used R and GNU Octave for statistical analysis Finallya heuristic rule was applied to validate the computed values(proposed by [5 6]) As a result it was possible to determinethe optimal structure of the TPM network with an averageamount of 211 steps and a percentage of 000004 of successof a passive attack network and with a successful 0 of ageometric attack

The main contribution of this study is that two userswill be able to generate a cryptographic key based on certainprobability according to the number of steps established atthe beginning of the synchronization and with a very lowprobability of a successful passive attack

The remainder of this document is structured as followsSection 2 mentions the related works that supports thepresent study Section 3 describes the procedure that helpsdetermine the best combination Section 4 shows the resultsobtained from the experiment Finally Section 5 detailsconclusions and proposes future work

2 Related Works

Concerning the studies related to this research Kanter etal [7] and Rosen-Zvi et al [8] demonstrated that whentwo artificial neural networks are trained on their outputsaccording to a learning rule they are able to developequivalent states of their internal synaptic weights Kinzeland Kanter [9] and Ruttor et al [10] revealed that thepossibility of a successful attack decreases as the synapticdepth of the network increases and the computational costof the attacker increases since its effort grows exponentiallywhile the effort required by the users grows polynomiallySimilarly Sarkar and Mandal [11] and Sarkar [12] mentionthat the performance is improved by increasing the synap-tic depth of the TPM networks henceforth counteractingthe brute force attacks with the current computation Incomparison with the present study we used the structureof the TPM network proposed by the latter However wedetermined that the security of the TPM network can alsobe increased by increasing the number of neurons in thehidden layer and also the number of entries of each neuronIn addition the range for the synaptic depth value hasbeen modified to adapt the need to generate a key of 512bits

With respect to the proposed algorithms for the designof the TPM network Lei et al [13] developed a two-layerprepowered TPMnetworkmodel A fast synchronization canbe achieved by increasing the minimum value of the internalrepresentationsHamming distance and by reducing the prob-ability of a step that does notmodify networks weights Allamet al [14] proposed an algorithm that increases the securityof neural cryptography by authenticating communicationsusing previously shared secrets in this way it increases thesecurity of neural cryptography As a result it shows that thealgorithm reaches a very high security level without increas-ing synchronization time Ruttor [15] mentions that the valueof K is 3 since lower values have negative consequencesfrom a safety point of view and higher values have negativeconsequences in terms of synchronization time Klimov et al[16] calculate the probability that two networks take to either

synchronize their weights or not Although the probabilityrises with a low value of K the success of the attackingnetwork also increases and consequently the value of Kmustbe greater In comparison with our study the initial structureof the TPMnetworks is random and different and has a singleoutput In addition because the goal is to generate a 512-bitkey it was not arithmetically adequate to use odd numbersfor K N and L values

In relation to measurements on TPM networks per-formance Dolecki and Kozera [17] present a method offrequency analysis that allows evaluating the synchronizationlevel of two TPM networks before they finish up with thecalculated value not related to the difference of their synapticweights As a result the selection of the appropriate rangefor the count frequency and the threshold allow them tospecify whether it is a short or a long synchronizationSanthanalakshmi et al [18] and Dolecki and Kozera [19]analyse and compare the performance of synchronization byemploying respectively a genetic algorithm and a Gaussiandistribution instead of uniform randomvalues for theweightsof the TPM networks As a result they found that replacingthe random weights with optimal weights helps reduce thesynchronization time In addition increasing the numberof hidden and input neurons accelerates convergence andalso reduces the probability of success of the ldquoMajorityFlipping Attackrdquo attack Dolecki et al [20] perform anadjustment of the timing distribution of two TPM networksto a Poisson distribution Pu et al [21] perform an algo-rithm that combines ldquotrue random sequencesrdquo (generatedby artificial and validated circuits with randomness tests)with TPM networks demonstrating more complex dynamicbehaviours that offer better performance as an encryptiontool and resistance to attacks In comparison with ourstudy the synaptic weights values of the TPM networksare generated according to a discrete uniform distributionAdditionally the Poisson distribution adjustment was madewith the results in the number of steps in each of thesimulations

With reference to rules that contribute in the designof TPM networks Mu and Liao [5] and Mu et al [6]define the following heuristic rule ldquoKeeping the equationsof motion constant a high value in the state classifier withrespect to the minimum values of the smallest Hammingdistances between the state vectors of the networks has ahigh probability of fulfilling the condition that the averagechange in the percentage difference between synaptic weightsis greater than zero improving the security of neuronalcryptographyrdquo For our study we used the proposed heuristicrule to determine the level of security of the final structure ofthe TPM network

In regard to modification of initial structures of aTPM network Gomez et al [22] state that with an initialmisalignment in the weights between 15 and 20 thesynchronization time is reduced from 125119898119904 to less than07119898119904 This also reduces the number of steps from 220 toless than 100 Within this context Niemiec [23] presentsa new idea for a key reconciliation method in quantumcryptography using TPM networks correcting errors thatoccur during transmission in the quantum channel The











201 5122 2564 1288 6416 32

211 2562 1284 648 3216 16

231 1282 644 328 16

271 642 324 168 8

215 1 322 164 8

231 1 162 84 4

263 1 82 4

2127 1 42 2

2255 1 22511 1 1





















Minimumsteps



Average time(sec)

Eversquossuccessful

sync

Eversquossuccessful

sync1 (1 512 20) 5 28 93367 00657 22718 04259 368680 737362 (2 256 20) 5 25 93312 00596 15588 04253 368229 7364583 (4 128 20) 5 32 93374 00680 21275 04263 368471 7369424 (8 64 20) 4 27 93370 00656 19089 04233 368652 7373045 (16 32 20) 5 32 93356 00661 14625 04249 368389 7367786 (1 256 21) 8 45 155450 00573 14665 03824 355397 7107947 (2 128 21) 8 25371 157774 00523 5375299 03857 349495 698998 (4 64 21) 8 4104 180226 00491 958512 03965 308691 6173829 (8 32 21) 9 814 316046 00403 196915 04597 161148 32229610 (16 16 21) 15 587 734867 00289 73628 05381 8102 1620411 (1 128 23) 11 78 253284 00472 11114 02983 132715 2654312 (2 64 23) 15 289 652706 00411 27924 04699 41475 829513 (4 32 23) 26 616 1281582 00309 45353 05325 487 0097414 (8 16 23) 38 769 2116590 00262 60476 05741 0 0015 (1 64 27) 10 71 250318 00180 06389 01679 132666 26533216 (2 32 27) 15 372 749943 00362 19875 03599 39111 7822217 (4 16 27) 23 712 1379569 00193 27179 04572 759 0151818 (8 8 27) 30 955 2182211 00342 34490 05109 0 0019 (1 32 215) 7 68 213921 00105 02913 00847 144461 28892220 (2 16 215) 10 451 698188 00196 12301 02161 61628 12325621 (4 8 215) 14 777 1320805 00181 27993 03200 4080 081622 (1 16 231) 6 84 249428 00020 02366 00549 46244 9248823 (2 8 231) 10 804 1005999 00121 14193 01842 25006 5001224 (4 4 231) 9 837 1342786 00032 20644 02281 9168 18336



Minimumsteps



Average time(sec)

Eversquossuccessful

sync

Eversquossuccessful

sync1 (8 16 23) 40 785 2115888 00780 15155 04058 0 02 (8 8 27) 27 861 2183696 00312 09229 02385 2 00002






Combination(K N L)




(8 16 23) 180000 0 00





00

05

10

15

20

25

30

Freq

uenc

y


100 200 300 400 500Number of steps

0

2

4

6

8

10

12

Freq

uenc

y


100 200 300 400 500 600Number of steps

0

10

20

30

40

50

60

70

80

Freq

uenc

y


100 200 300 400 500 600 700Number of steps

0

100

200

300

400

500

600

700

Freq

uenc

y


100 200 300 400 500 600 700Number of steps

0

200

400

600

800

1000

1200

1400

1600

Freq

uenc

y


100 200 300 400 500 600 700Number of steps

0

500

1000

1500

2000

2500

3000

3500

Freq

uenc

y


100 200 300 400 500 600 700Number of steps

0

1000

2000

3000

4000

5000

6000

7000

Freq

uenc

y





Combination(K N L)




(8 16 23) 2rsquo361100 1 000004(8 8 27) 1rsquo500000 2 00001





119891 (120582 119889) = 119890minus120582120582119889119889 (1)


119889119890119897119905119886 = 5151 000 = 0515 515 (2)




X

Y

+10 +25 +55+01+01+02+02+03+03+03+04+04+05+05+06+07+07+08+08+09+09+10+10+11

f(x) = (exp(minusx)∙(x)^(05150))05150

g(x) = (exp(minusx)∙(x)^(00634))00634

h(x) = (exp(minusx)∙(x)^(000709))000709

i(x) = (exp(minusx)∙(x)^(00030))00030

j(x) = (exp(minusx)∙(x)^(00014))00014

k(x) = (exp(minusx)∙(x)^(00007))00007

+40



(272 ) = (1282 ) = 1282 (128 minus 2) = 8128 (3)


11986711986311988811= 119867119863((1 1 1 1 1 1 1 1) (1 1 1 1 1 1 minus1 minus1))= 2

11986711986311988812= 119867119863((1 1 1 1 1 1 1 1) (1 1 1 1 1 minus1 1 minus1))= 2

sdot sdot sdot




11986711986311988821= 119867119863((1 1 1 1 1 1 1 minus1) (1 1 1 1 1 1 minus1 1))= 2

11986711986311988822= 119867119863((1 1 1 1 1 1 1 minus1) (1 1 1 1 1 minus1 1 1))= 2

sdot sdot sdot(4)


1198781198671198631198881 = min 11986711986311988811 11986711986311988812 = 21198781198671198631198882 = min 11986711986311988821 11986711986311988822 = 2 (5)


ℎ119879119875119872(119870=8) = min 1198781198671198631198881 1198781198671198631198882 = 2 (6)




N 400500

600

300200

1000

20

1e+00

K

1e+011e+021e+031e+04

L 1e+051e+061e+071e+081e+091e+10

04

06

1510

50

0

02





5 Conclusions





Data Availability




Acknowledgments


References




























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia












201 5122 2564 1288 6416 32

211 2562 1284 648 3216 16

231 1282 644 328 16

271 642 324 168 8

215 1 322 164 8

231 1 162 84 4

263 1 82 4

2127 1 42 2

2255 1 22511 1 1





















Minimumsteps



Average time(sec)

Eversquossuccessful

sync

Eversquossuccessful

sync1 (1 512 20) 5 28 93367 00657 22718 04259 368680 737362 (2 256 20) 5 25 93312 00596 15588 04253 368229 7364583 (4 128 20) 5 32 93374 00680 21275 04263 368471 7369424 (8 64 20) 4 27 93370 00656 19089 04233 368652 7373045 (16 32 20) 5 32 93356 00661 14625 04249 368389 7367786 (1 256 21) 8 45 155450 00573 14665 03824 355397 7107947 (2 128 21) 8 25371 157774 00523 5375299 03857 349495 698998 (4 64 21) 8 4104 180226 00491 958512 03965 308691 6173829 (8 32 21) 9 814 316046 00403 196915 04597 161148 32229610 (16 16 21) 15 587 734867 00289 73628 05381 8102 1620411 (1 128 23) 11 78 253284 00472 11114 02983 132715 2654312 (2 64 23) 15 289 652706 00411 27924 04699 41475 829513 (4 32 23) 26 616 1281582 00309 45353 05325 487 0097414 (8 16 23) 38 769 2116590 00262 60476 05741 0 0015 (1 64 27) 10 71 250318 00180 06389 01679 132666 26533216 (2 32 27) 15 372 749943 00362 19875 03599 39111 7822217 (4 16 27) 23 712 1379569 00193 27179 04572 759 0151818 (8 8 27) 30 955 2182211 00342 34490 05109 0 0019 (1 32 215) 7 68 213921 00105 02913 00847 144461 28892220 (2 16 215) 10 451 698188 00196 12301 02161 61628 12325621 (4 8 215) 14 777 1320805 00181 27993 03200 4080 081622 (1 16 231) 6 84 249428 00020 02366 00549 46244 9248823 (2 8 231) 10 804 1005999 00121 14193 01842 25006 5001224 (4 4 231) 9 837 1342786 00032 20644 02281 9168 18336



Minimumsteps



Average time(sec)

Eversquossuccessful

sync

Eversquossuccessful

sync1 (8 16 23) 40 785 2115888 00780 15155 04058 0 02 (8 8 27) 27 861 2183696 00312 09229 02385 2 00002






Combination(K N L)




(8 16 23) 180000 0 00





00

05

10

15

20

25

30

Freq

uenc

y


100 200 300 400 500Number of steps

0

2

4

6

8

10

12

Freq

uenc

y


100 200 300 400 500 600Number of steps

0

10

20

30

40

50

60

70

80

Freq

uenc

y


100 200 300 400 500 600 700Number of steps

0

100

200

300

400

500

600

700

Freq

uenc

y


100 200 300 400 500 600 700Number of steps

0

200

400

600

800

1000

1200

1400

1600

Freq

uenc

y


100 200 300 400 500 600 700Number of steps

0

500

1000

1500

2000

2500

3000

3500

Freq

uenc

y


100 200 300 400 500 600 700Number of steps

0

1000

2000

3000

4000

5000

6000

7000

Freq

uenc

y





Combination(K N L)




(8 16 23) 2rsquo361100 1 000004(8 8 27) 1rsquo500000 2 00001





119891 (120582 119889) = 119890minus120582120582119889119889 (1)


119889119890119897119905119886 = 5151 000 = 0515 515 (2)




X

Y

+10 +25 +55+01+01+02+02+03+03+03+04+04+05+05+06+07+07+08+08+09+09+10+10+11

f(x) = (exp(minusx)∙(x)^(05150))05150

g(x) = (exp(minusx)∙(x)^(00634))00634

h(x) = (exp(minusx)∙(x)^(000709))000709

i(x) = (exp(minusx)∙(x)^(00030))00030

j(x) = (exp(minusx)∙(x)^(00014))00014

k(x) = (exp(minusx)∙(x)^(00007))00007

+40



(272 ) = (1282 ) = 1282 (128 minus 2) = 8128 (3)


11986711986311988811= 119867119863((1 1 1 1 1 1 1 1) (1 1 1 1 1 1 minus1 minus1))= 2

11986711986311988812= 119867119863((1 1 1 1 1 1 1 1) (1 1 1 1 1 minus1 1 minus1))= 2

sdot sdot sdot




11986711986311988821= 119867119863((1 1 1 1 1 1 1 minus1) (1 1 1 1 1 1 minus1 1))= 2

11986711986311988822= 119867119863((1 1 1 1 1 1 1 minus1) (1 1 1 1 1 minus1 1 1))= 2

sdot sdot sdot(4)


1198781198671198631198881 = min 11986711986311988811 11986711986311988812 = 21198781198671198631198882 = min 11986711986311988821 11986711986311988822 = 2 (5)


ℎ119879119875119872(119870=8) = min 1198781198671198631198881 1198781198671198631198882 = 2 (6)




N 400500

600

300200

1000

20

1e+00

K

1e+011e+021e+031e+04

L 1e+051e+061e+071e+081e+091e+10

04

06

1510

50

0

02





5 Conclusions





Data Availability




Acknowledgments


References




























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



















Minimumsteps



Average time(sec)

Eversquossuccessful

sync

Eversquossuccessful

sync1 (1 512 20) 5 28 93367 00657 22718 04259 368680 737362 (2 256 20) 5 25 93312 00596 15588 04253 368229 7364583 (4 128 20) 5 32 93374 00680 21275 04263 368471 7369424 (8 64 20) 4 27 93370 00656 19089 04233 368652 7373045 (16 32 20) 5 32 93356 00661 14625 04249 368389 7367786 (1 256 21) 8 45 155450 00573 14665 03824 355397 7107947 (2 128 21) 8 25371 157774 00523 5375299 03857 349495 698998 (4 64 21) 8 4104 180226 00491 958512 03965 308691 6173829 (8 32 21) 9 814 316046 00403 196915 04597 161148 32229610 (16 16 21) 15 587 734867 00289 73628 05381 8102 1620411 (1 128 23) 11 78 253284 00472 11114 02983 132715 2654312 (2 64 23) 15 289 652706 00411 27924 04699 41475 829513 (4 32 23) 26 616 1281582 00309 45353 05325 487 0097414 (8 16 23) 38 769 2116590 00262 60476 05741 0 0015 (1 64 27) 10 71 250318 00180 06389 01679 132666 26533216 (2 32 27) 15 372 749943 00362 19875 03599 39111 7822217 (4 16 27) 23 712 1379569 00193 27179 04572 759 0151818 (8 8 27) 30 955 2182211 00342 34490 05109 0 0019 (1 32 215) 7 68 213921 00105 02913 00847 144461 28892220 (2 16 215) 10 451 698188 00196 12301 02161 61628 12325621 (4 8 215) 14 777 1320805 00181 27993 03200 4080 081622 (1 16 231) 6 84 249428 00020 02366 00549 46244 9248823 (2 8 231) 10 804 1005999 00121 14193 01842 25006 5001224 (4 4 231) 9 837 1342786 00032 20644 02281 9168 18336



Minimumsteps



Average time(sec)

Eversquossuccessful

sync

Eversquossuccessful

sync1 (8 16 23) 40 785 2115888 00780 15155 04058 0 02 (8 8 27) 27 861 2183696 00312 09229 02385 2 00002






Combination(K N L)




(8 16 23) 180000 0 00





00

05

10

15

20

25

30

Freq

uenc

y


100 200 300 400 500Number of steps

0

2

4

6

8

10

12

Freq

uenc

y


100 200 300 400 500 600Number of steps

0

10

20

30

40

50

60

70

80

Freq

uenc

y


100 200 300 400 500 600 700Number of steps

0

100

200

300

400

500

600

700

Freq

uenc

y


100 200 300 400 500 600 700Number of steps

0

200

400

600

800

1000

1200

1400

1600

Freq

uenc

y


100 200 300 400 500 600 700Number of steps

0

500

1000

1500

2000

2500

3000

3500

Freq

uenc

y


100 200 300 400 500 600 700Number of steps

0

1000

2000

3000

4000

5000

6000

7000

Freq

uenc

y





Combination(K N L)




(8 16 23) 2rsquo361100 1 000004(8 8 27) 1rsquo500000 2 00001





119891 (120582 119889) = 119890minus120582120582119889119889 (1)


119889119890119897119905119886 = 5151 000 = 0515 515 (2)




X

Y

+10 +25 +55+01+01+02+02+03+03+03+04+04+05+05+06+07+07+08+08+09+09+10+10+11

f(x) = (exp(minusx)∙(x)^(05150))05150

g(x) = (exp(minusx)∙(x)^(00634))00634

h(x) = (exp(minusx)∙(x)^(000709))000709

i(x) = (exp(minusx)∙(x)^(00030))00030

j(x) = (exp(minusx)∙(x)^(00014))00014

k(x) = (exp(minusx)∙(x)^(00007))00007

+40



(272 ) = (1282 ) = 1282 (128 minus 2) = 8128 (3)


11986711986311988811= 119867119863((1 1 1 1 1 1 1 1) (1 1 1 1 1 1 minus1 minus1))= 2

11986711986311988812= 119867119863((1 1 1 1 1 1 1 1) (1 1 1 1 1 minus1 1 minus1))= 2

sdot sdot sdot




11986711986311988821= 119867119863((1 1 1 1 1 1 1 minus1) (1 1 1 1 1 1 minus1 1))= 2

11986711986311988822= 119867119863((1 1 1 1 1 1 1 minus1) (1 1 1 1 1 minus1 1 1))= 2

sdot sdot sdot(4)


1198781198671198631198881 = min 11986711986311988811 11986711986311988812 = 21198781198671198631198882 = min 11986711986311988821 11986711986311988822 = 2 (5)


ℎ119879119875119872(119870=8) = min 1198781198671198631198881 1198781198671198631198882 = 2 (6)




N 400500

600

300200

1000

20

1e+00

K

1e+011e+021e+031e+04

L 1e+051e+061e+071e+081e+091e+10

04

06

1510

50

0

02





5 Conclusions





Data Availability




Acknowledgments


References




























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





Minimumsteps



Average time(sec)

Eversquossuccessful

sync

Eversquossuccessful

sync1 (1 512 20) 5 28 93367 00657 22718 04259 368680 737362 (2 256 20) 5 25 93312 00596 15588 04253 368229 7364583 (4 128 20) 5 32 93374 00680 21275 04263 368471 7369424 (8 64 20) 4 27 93370 00656 19089 04233 368652 7373045 (16 32 20) 5 32 93356 00661 14625 04249 368389 7367786 (1 256 21) 8 45 155450 00573 14665 03824 355397 7107947 (2 128 21) 8 25371 157774 00523 5375299 03857 349495 698998 (4 64 21) 8 4104 180226 00491 958512 03965 308691 6173829 (8 32 21) 9 814 316046 00403 196915 04597 161148 32229610 (16 16 21) 15 587 734867 00289 73628 05381 8102 1620411 (1 128 23) 11 78 253284 00472 11114 02983 132715 2654312 (2 64 23) 15 289 652706 00411 27924 04699 41475 829513 (4 32 23) 26 616 1281582 00309 45353 05325 487 0097414 (8 16 23) 38 769 2116590 00262 60476 05741 0 0015 (1 64 27) 10 71 250318 00180 06389 01679 132666 26533216 (2 32 27) 15 372 749943 00362 19875 03599 39111 7822217 (4 16 27) 23 712 1379569 00193 27179 04572 759 0151818 (8 8 27) 30 955 2182211 00342 34490 05109 0 0019 (1 32 215) 7 68 213921 00105 02913 00847 144461 28892220 (2 16 215) 10 451 698188 00196 12301 02161 61628 12325621 (4 8 215) 14 777 1320805 00181 27993 03200 4080 081622 (1 16 231) 6 84 249428 00020 02366 00549 46244 9248823 (2 8 231) 10 804 1005999 00121 14193 01842 25006 5001224 (4 4 231) 9 837 1342786 00032 20644 02281 9168 18336



Minimumsteps



Average time(sec)

Eversquossuccessful

sync

Eversquossuccessful

sync1 (8 16 23) 40 785 2115888 00780 15155 04058 0 02 (8 8 27) 27 861 2183696 00312 09229 02385 2 00002






Combination(K N L)




(8 16 23) 180000 0 00





00

05

10

15

20

25

30

Freq

uenc

y


100 200 300 400 500Number of steps

0

2

4

6

8

10

12

Freq

uenc

y


100 200 300 400 500 600Number of steps

0

10

20

30

40

50

60

70

80

Freq

uenc

y


100 200 300 400 500 600 700Number of steps

0

100

200

300

400

500

600

700

Freq

uenc

y


100 200 300 400 500 600 700Number of steps

0

200

400

600

800

1000

1200

1400

1600

Freq

uenc

y


100 200 300 400 500 600 700Number of steps

0

500

1000

1500

2000

2500

3000

3500

Freq

uenc

y


100 200 300 400 500 600 700Number of steps

0

1000

2000

3000

4000

5000

6000

7000

Freq

uenc

y





Combination(K N L)




(8 16 23) 2rsquo361100 1 000004(8 8 27) 1rsquo500000 2 00001





119891 (120582 119889) = 119890minus120582120582119889119889 (1)


119889119890119897119905119886 = 5151 000 = 0515 515 (2)




X

Y

+10 +25 +55+01+01+02+02+03+03+03+04+04+05+05+06+07+07+08+08+09+09+10+10+11

f(x) = (exp(minusx)∙(x)^(05150))05150

g(x) = (exp(minusx)∙(x)^(00634))00634

h(x) = (exp(minusx)∙(x)^(000709))000709

i(x) = (exp(minusx)∙(x)^(00030))00030

j(x) = (exp(minusx)∙(x)^(00014))00014

k(x) = (exp(minusx)∙(x)^(00007))00007

+40



(272 ) = (1282 ) = 1282 (128 minus 2) = 8128 (3)


11986711986311988811= 119867119863((1 1 1 1 1 1 1 1) (1 1 1 1 1 1 minus1 minus1))= 2

11986711986311988812= 119867119863((1 1 1 1 1 1 1 1) (1 1 1 1 1 minus1 1 minus1))= 2

sdot sdot sdot




11986711986311988821= 119867119863((1 1 1 1 1 1 1 minus1) (1 1 1 1 1 1 minus1 1))= 2

11986711986311988822= 119867119863((1 1 1 1 1 1 1 minus1) (1 1 1 1 1 minus1 1 1))= 2

sdot sdot sdot(4)


1198781198671198631198881 = min 11986711986311988811 11986711986311988812 = 21198781198671198631198882 = min 11986711986311988821 11986711986311988822 = 2 (5)


ℎ119879119875119872(119870=8) = min 1198781198671198631198881 1198781198671198631198882 = 2 (6)




N 400500

600

300200

1000

20

1e+00

K

1e+011e+021e+031e+04

L 1e+051e+061e+071e+081e+091e+10

04

06

1510

50

0

02





5 Conclusions





Data Availability




Acknowledgments


References




























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




00

05

10

15

20

25

30

Freq

uenc

y


100 200 300 400 500Number of steps

0

2

4

6

8

10

12

Freq

uenc

y


100 200 300 400 500 600Number of steps

0

10

20

30

40

50

60

70

80

Freq

uenc

y


100 200 300 400 500 600 700Number of steps

0

100

200

300

400

500

600

700

Freq

uenc

y


100 200 300 400 500 600 700Number of steps

0

200

400

600

800

1000

1200

1400

1600

Freq

uenc

y


100 200 300 400 500 600 700Number of steps

0

500

1000

1500

2000

2500

3000

3500

Freq

uenc

y


100 200 300 400 500 600 700Number of steps

0

1000

2000

3000

4000

5000

6000

7000

Freq

uenc

y





Combination(K N L)




(8 16 23) 2rsquo361100 1 000004(8 8 27) 1rsquo500000 2 00001





119891 (120582 119889) = 119890minus120582120582119889119889 (1)


119889119890119897119905119886 = 5151 000 = 0515 515 (2)




X

Y

+10 +25 +55+01+01+02+02+03+03+03+04+04+05+05+06+07+07+08+08+09+09+10+10+11

f(x) = (exp(minusx)∙(x)^(05150))05150

g(x) = (exp(minusx)∙(x)^(00634))00634

h(x) = (exp(minusx)∙(x)^(000709))000709

i(x) = (exp(minusx)∙(x)^(00030))00030

j(x) = (exp(minusx)∙(x)^(00014))00014

k(x) = (exp(minusx)∙(x)^(00007))00007

+40



(272 ) = (1282 ) = 1282 (128 minus 2) = 8128 (3)


11986711986311988811= 119867119863((1 1 1 1 1 1 1 1) (1 1 1 1 1 1 minus1 minus1))= 2

11986711986311988812= 119867119863((1 1 1 1 1 1 1 1) (1 1 1 1 1 minus1 1 minus1))= 2

sdot sdot sdot




11986711986311988821= 119867119863((1 1 1 1 1 1 1 minus1) (1 1 1 1 1 1 minus1 1))= 2

11986711986311988822= 119867119863((1 1 1 1 1 1 1 minus1) (1 1 1 1 1 minus1 1 1))= 2

sdot sdot sdot(4)


1198781198671198631198881 = min 11986711986311988811 11986711986311988812 = 21198781198671198631198882 = min 11986711986311988821 11986711986311988822 = 2 (5)


ℎ119879119875119872(119870=8) = min 1198781198671198631198881 1198781198671198631198882 = 2 (6)




N 400500

600

300200

1000

20

1e+00

K

1e+011e+021e+031e+04

L 1e+051e+061e+071e+081e+091e+10

04

06

1510

50

0

02





5 Conclusions





Data Availability




Acknowledgments


References




























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




Combination(K N L)




(8 16 23) 2rsquo361100 1 000004(8 8 27) 1rsquo500000 2 00001





119891 (120582 119889) = 119890minus120582120582119889119889 (1)


119889119890119897119905119886 = 5151 000 = 0515 515 (2)




X

Y

+10 +25 +55+01+01+02+02+03+03+03+04+04+05+05+06+07+07+08+08+09+09+10+10+11

f(x) = (exp(minusx)∙(x)^(05150))05150

g(x) = (exp(minusx)∙(x)^(00634))00634

h(x) = (exp(minusx)∙(x)^(000709))000709

i(x) = (exp(minusx)∙(x)^(00030))00030

j(x) = (exp(minusx)∙(x)^(00014))00014

k(x) = (exp(minusx)∙(x)^(00007))00007

+40



(272 ) = (1282 ) = 1282 (128 minus 2) = 8128 (3)


11986711986311988811= 119867119863((1 1 1 1 1 1 1 1) (1 1 1 1 1 1 minus1 minus1))= 2

11986711986311988812= 119867119863((1 1 1 1 1 1 1 1) (1 1 1 1 1 minus1 1 minus1))= 2

sdot sdot sdot




11986711986311988821= 119867119863((1 1 1 1 1 1 1 minus1) (1 1 1 1 1 1 minus1 1))= 2

11986711986311988822= 119867119863((1 1 1 1 1 1 1 minus1) (1 1 1 1 1 minus1 1 1))= 2

sdot sdot sdot(4)


1198781198671198631198881 = min 11986711986311988811 11986711986311988812 = 21198781198671198631198882 = min 11986711986311988821 11986711986311988822 = 2 (5)


ℎ119879119875119872(119870=8) = min 1198781198671198631198881 1198781198671198631198882 = 2 (6)




N 400500

600

300200

1000

20

1e+00

K

1e+011e+021e+031e+04

L 1e+051e+061e+071e+081e+091e+10

04

06

1510

50

0

02





5 Conclusions





Data Availability




Acknowledgments


References




























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





11986711986311988821= 119867119863((1 1 1 1 1 1 1 minus1) (1 1 1 1 1 1 minus1 1))= 2

11986711986311988822= 119867119863((1 1 1 1 1 1 1 minus1) (1 1 1 1 1 minus1 1 1))= 2

sdot sdot sdot(4)


1198781198671198631198881 = min 11986711986311988811 11986711986311988812 = 21198781198671198631198882 = min 11986711986311988821 11986711986311988822 = 2 (5)


ℎ119879119875119872(119870=8) = min 1198781198671198631198881 1198781198671198631198882 = 2 (6)




N 400500

600

300200

1000

20

1e+00

K

1e+011e+021e+031e+04

L 1e+051e+061e+071e+081e+091e+10

04

06

1510

50

0

02





5 Conclusions





Data Availability




Acknowledgments


References




























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





Data Availability




Acknowledgments


References




























RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


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