Applying Cuckoo Search for analysis of LFSR based cryptosystem

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Availableonlineatwww.sciencedirect.com

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j ou rn a l h o m e p a g e :w w w . e l s e v i e r . c o m / p i s c

Applying

Cuckoo

Search

for

analysis

of

LFSR

based

cryptosystem

夽

Maiya

Din

a,∗

,

Saibal

K. Pal

a

,

S.K.

Muttoo

b

,

Anjali

Jain

c

a

DRDO,Delhi,India

b_Delhi_University,_Delhi,_India c

BanasthaliUniversity,Jaipur,India

Received26January2016;accepted9April2016 Availableonline 28April2016

KEYWORDS LFSR;

Swarmintelligence; CuckooSearch; Cryptanalysis

Summary Cryptographictechniquesareemployedforminimizingsecurityhazardstosensitive information.Tomakethesystemsmorerobust,cyphersorcryptsbeingusedneedtobeanalysed forwhichcryptanalystsrequirewaystoautomatetheprocess,sothatcryptographicsystems canbetestedmoreefﬁciently.Evolutionaryalgorithmsprovideonesuch resortastheseare capable ofsearchingglobaloptimalsolutionveryquickly. CuckooSearch(CS)Algorithmhas beenusedeffectivelyincryptanalysisofconventionalsystemslikeVigenereandTransposition cyphers.LinearFeedbackShiftRegister(LFSR)isacryptoprimitiveusedextensivelyindesign ofcryptosystems.Inthispaper,we analyseLFSRbasedcryptosystem usingCuckooSearchto ﬁndcorrectinitialstatesofusedLFSR.Primitivepolynomialsofdegree11,13,17and19are consideredtoanalysetextcryptsoflength200,300and400characters.Optimalsolutionswere obtainedforthefollowingCSparameters:Levydistributionparameter(ˇ)=1.5andAlieneggs discoveringprobability(pa)=0.25.

Introduction

Inpresentworldofcommunication,conﬁdentiality,integrity and availability of a message can be ensured only if the

夽 _This_article_belongs_to_the_special_issue_on_Engineering_and

Mate-rialSciences.

∗_{Corresponding}_author._Tel.:₊₉₁_8585943774;_fax:₊₉₁_23812683.

E-mailaddresses:[email protected](M.Din), [email protected](S.K.Pal),[email protected] (S.K.Muttoo),[email protected](A.Jain).

content is encrypted before transmission using crypto-graphictechniques.Ononehandwherecryptographydeals with the encryption of messages, cryptanalysis (Bhateja etal., 2015; Heydari and Senejani, 2014) is another side ofcryptographyused,toﬁndplaintextwithoutknowingthe key.Cryptanalysisrequireslargenumberofpossiblekeysto betested for most of thecryptographic algorithms which makesitatimeconsumingprocess,thusmakingtheresults obsoleteanduselessifnotobtainedintime.Cryptanalysis thusformsasubsetofhardoptimizationproblemsasthese cannotbesolvedwithinreasonabletimebyusingstandard, mathematicalanddeterministicmethods.Researchisbeing

http://dx.doi.org/10.1016/j.pisc.2016.04.098

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carriedoutinﬁndingautomatedwaystocomputeoptimal solutionstotheseproblems andit hasbeenobservedthat meta-heuristicsinspiredbynatureprovidepromisingresults inthisﬁeld.

In this paper, we use Linear Feedback Shift Register (LFSR) (Som and Ghosh, 2012) based PN sequence for encrypting the plaintext and an evolutionary algorithm (CuckooSearchAlgorithm)for findingthekeyfor decrypt-ingthecyphertext.WeuseLFSRwithprimitivepolynomials forencryptionasthisapproachmakesthekeymoresecure anddifficult toguessbytheintruder.CuckooSearch Algo-rithmdevelopedbyYangandDeb(2010a)isanatureinspired techniquewhichusestheLevybehaviourshownbyvarious animals.Newsolutionsaregeneratedaroundthebest solu-tionobtainedsofarwhichspeedupthelocalsearchbutalso employsfarfieldrandomizationwhichpreventsthetrapping ofsystem in a localoptimum. This inspires toinvestigate theapplicabilityofCuckooSearchincryptanalysisofLFSR basedcryptosystems.Literaturereviewsectiongivesabrief literaturereviewandLinearFeedbackShiftRegister(LFSR) sectionprovidesadescriptionofLFSR.CuckooSearch(CS) Algorithm section describes the Cuckoo Search Algorithm withthe sub-sections giving a briefinsight in the mathe-maticalfoundationsandtheapproachusedintheanalysis. Fitnessor objectivefunction section describes the fitness orobjectivefunction usedforexperimentation, resultsof whicharepresentedinExperimentalresultssection.

Literature

review

Swarmintelligence has emerged asan important subfield of artificial intelligence withpromising research opportu-nitiesandappreciableresultswhenappliedtothefieldof computationalproblems. Millonas(1994) putsforwardthe basic idea behind swarm intelligence as employing many simple agents which arepreferably present in the nature which in turn lead to an emergent global behaviour as no rule is applied to them. The general principles that provideachannelizedapproachtowardsswarmintelligence explainedinthispaperhelpsinunderstandingthecommon behaviourexhibitedbythevariousswarmslikeantcolonies, fishschools,beecoloniesetc.andaidsindrawinga mathe-maticalformulationforthebehaviour.Thesemathematical formulations help in drawing meta-heuristic algorithms derivedfromnaturelikeAntColonyOptimization(ACO),Bee Colony Optimization, Particle Swarm Optimization (PSO), Cuckoo Search Algorithm (CSA), etc. Meta-heuristic algo-rithms (Yangand Deb,2010b) aregeneral purposesearch algorithms that enable complexsearch spaces to be tra-versed in search of optimal or high quality solutions toa givenproblem.

DanzigerandHenriques(2011)givesan insightintothe factthattheconceptofswarmintelligenceorbio-inspired computingalgorithms can beappliedtothe ﬁeldof cryp-tography.Khan etal.(2013)propose anovelswarmbased attackcalledAntColonyOptimizationtothecryptanalysisof DataEncryptionStandard (DES).DadhichandYadav(2014)

givea cryptanalytic approach towards 4 round DES using swarmintelligenceand evolutionarycomputation. Geetha andGeorgeAmalarethinam(2015)haveproposedABCRNG, i.e.ArtiﬁcialBeeColonyRandomNumberGenerationwhich

can be applied to the Public Key cryptography. Yang and DebproposedanewalgorithmcalledCuckooSearch(Tuba, 2013)whichwasinspiredbythebroodparasitismexhibited bycuckoos.Thisapproachwasmathematicalsoundwitha greatamountofrandomnesswhichmakesitquiteidealto beusedintheﬁeldofcryptographyasrandomnessprovides strengthtothecodethusincreasingthesecurity.

Linear

Feedback

Shift

Register

(LFSR)

Stream cyphers (Stallings, 2006) are type of Symmetric-Key cryptosystemsthatencrypts plaintext onebit/byteat a time.Pseudorandom numbersequences(PNS) (Som and Ghosh,2012)aresequenceswhosepropertiesapproximate thepropertiesofsequencesofrandomnumbers.Theseare nottrulyrandom,becauseitiscompletelydeterminedbya relativelysmallsetofinitialvalues.

LFSRs are used in a stream cypher to generate linear sequencesofpseudorandomnumbers.Theyrequireveryless hardwareandhavehighspeedofoperations.An-stageLFSR is of maximum length if initial states repeat after every (2n₋₁₎_bits._The_contents_of_the_registers_are_shifted_by_one

positionateachclock.Theleft-mostbitfedtotheregisteris theresultofmod-2additionofbitscorrespondingtothe non-zero coefﬁcients of considered primitivepolynomial. The right most bit is used toformthe pseudorandom number sequence.Allinitialstatesshouldnotbe‘‘0’’sbecausethe LFSRwouldremainlocked-upinthesestates.

Cuckoo

Search

(CS)

Algorithm

Cuckoo Search Algorithmwasdeveloped by Yangand Deb (2010a)in2009inspiredbytheuniqueaggressive reproduc-tionstrategyofcuckoobirds.Cuckoobirdsengageinbrood parasitisminwhichabirdlaysandabandonsitseggsinthe nestofanotherspecies,whichactassurrogateparentsand unwillinglyraiseherbrood.Someeggscanevenmimicthe colourofthehost bird’seggsandcanalsoproducesounds similartoitschicksinordertodupethehostbird.Somehost birdsdonotbehavefriendlywiththese‘intruders’andget involvedinadirectconﬂict,inwhichtheythrowawaythe alieneggs.Otherwise,thehostbirdsjustabandontheeggs andnestandbuildanewnestsomewhereelse.

DescriptionoforiginalCSAlgorithm

Cuckoo Search Algorithm is one of the powerful nature-inspired population based stochastic global search meta-heuristictechnique.Forallsuchnatureinspiredalgorithms fundamentalissueisthebalancebetweentheuseoffound good solutions, i.e. exploitation and investigation of new areasofsearchspace,i.e.farfieldrandomizationinorder toavoidbeingtrappedinlocalminima,i.e.exploration.This algorithmfindsthisbalancebyemployinglevyflight accord-ingtolevydistributionwithinfinitemeanandvariance.The threemainidealizedapproximationrulesonwhichthe algo-rithmisbasedareasfollows:

• Eachartiﬁcialcuckoocanlayonlyoneeggatatimeand dumpsitinarandomlychosennest.

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• Thebestnestswithhighqualityeggswillbepassedtothe nextgeneration.

• Hostnestnumberisnotadjustable,i.e.itisﬁxedandthe egglaidbythecuckooisdiscoveredbythehostbirdwith a probability,pa∈[0,1].In thiscase, thehost bird can eitherthrowawaytheeggorabandonthenestandbuild anewnest.

In this algorithm, potential solutions correspond to cuckooeggs.Tomakethingsevensimpler,thelast assump-tioncanbeapproximatedbythefractionpaof‘n’nestsin current iteration which needs tobe replacedby the new nests,i.e.newrandomsolutionsinthenextiteration.

Mathematicalformulation

Whileﬂying, someinsects and animalsfollow the pathof long trajectories withsudden right angle turns combined withshort,randommovements.Thisrandomwalkiscalled Levy ﬂight (Yang et al., 2013), named after the French mathematicianPaul PierreLevy.This behaviour is usedin generatingnewsolutionforacuckooinCSalgorithm accord-ingtothefollowingequation:

x(t+1) i =x

(t)

i +˛ˆLevy() (1)

where˛(˛>0)isthestepsizewhichisadjustedaccordingto thescaleoftheproblemofinterest.Inmostcases,˛canbe unity or someother constant.This equationsuggests that therandomwalkis aMarkov chain,i.e. thenextlocation dependsonpresent locationgivenbytheﬁrsttermofthe equationandthetransitionprobabilityrepresentedbythe secondterm.

The transition probability advocates entry wise multi-plicationwhichmakestheexplorationofthesearchspace moreefficientasstepsizeincreasesexponentially.The ran-domsteplengthfollowstheLevydistributionwhichhasan infinitevariancewithaninfinitemean:

Levy∼u=t−_, _where₁_<_≤_3. ₍₂₎

CuckooSearchforcryptanalysisofLFSR

LFSRisusedtogeneratekeystreambitsbyprimitive poly-nomials.GeneratedPNsequenceisusedforencryptingthe plain textmessages. The plain messageis converted into binarymessageusingMurraycode.Thebinaryplainmessage is XORed withthe keystream bits toget encrypted mes-sage.Togetplainmessage,wegeneratekeystreambitsand XORthemwithencryptedmessage.Toﬁndthekeyfor deci-pheringencryptedmessageCuckooSearchbasedapproach isused.Inthisapproachfollowingstepsareinvolved:

Generatingthehabitat:Thedegreeoftheprimitive poly-nomialisenteredusingwhichthetotalnumberofsolutions is calculated using the formula 2ˆn−1 where n is the degreeofthepolynomial.Asthenumberofprobable solu-tionsincreaseswiththedegreeofthepolynomial,weuse clusteringmethodforﬁndingouttheoptimalsolution(s). Anapproximateclustersizeiscalculatedbyroundingoff thevalueobtainedby2ˆn/n.

Initialization of CuckooSearch parameters: Number of nestswillbeequaltotheclustersizecalculated.Valuesfor alieneggsdiscoveringprobability(pa),stepsizeparameter

(˛),Levydistributionparameter(ˇ)andsigma()areset experimentally.

Initializationofdiscretenestsoreggsofhostbirds:From the clustersformed out oftotal possible nests,the nest withthemaximumscorefortheobjectiveorﬁtness func-tionisusedastheinitialbestnest.Usingthisinitialvalue thestepsizeiscalculated.

GenerationofnewnestsusingLevyflights:Newnestsare generatedusingLevyflightssimulatedbyMantegna’s algo-rithm(Mantegna,1994).Thebestnestisreplacedwiththe currentbestifthevalueofthefitnessfunctionisgreater thanthecalculatedfitnessfunctionvalue.

Discoveryofalieneggsorworsenests:Afractionofworst nests are discovered with the probability pa. A random number is generated and ifthe probabilitypa is greater thantherandomnumberthenanewsolutionisgenerated bybiasedorselectiverandomwalks.Theexistingbestnest isreplacedwiththenewsolutionifitisbetterinquality.

Criterionforstopping:Newnestsaregeneratedandalien eggsorworstnestsarediscoveredusingtheproposedsteps until the clusters reachtheir maximum value. The best nests from each cluster arewritten toa ﬁle for further analysis.

Fitness

or

objective

function

Fitnessfunctionisusedtoobtainthebestsolution(s)within alargesolutionspace.Itindicatestheproximityofthe pos-siblesolution(s)tothedesiredsolution.ForanalysisofLFSR, afitnessfunctionisusedwhichgivehighscoreforapieceof textsimilartoEnglishplaintextwhilelowforrandomtext. The fitness function is based on15 most frequent mono-grams, bigrams and trigrams as found by Norvig through experimentation.Theformulausedforfindingfitnessvalue is: Fitness= ⎛ ⎝ 15 i=1Fi×Wi ×100 L ⎞ ⎠

whereFiisthefrequencyoffeaturesasmonograms,bigrams andtrigrams,Listhemessagelength,Wiistheweightsof ﬁrstmostfrequent15monograms,bigramsandtrigrams.

Usingthisformula,anaveragefitnessvalueiscalculated whichisusedasathresholdvalueforcomparingthequality oftheobtainedprobablesolutions.Asolutiongeneratinga decryptedtextofhigher fitnessvaluethan theaverage is consideredoneoftheoptimalsolutions.Forthisatextfile (size59KB)ofnormalEnglishtextfromvariousbooksand articleswascreated.Textfileafterremovingofallspecial charactersandnumeralsisusedforcalculatingathreshold averagefitnessfunctionvalue.

Experimental

results

TheschemeofanalysingLFSRbasedcryptosystemis imple-mentedinMATLAB®_on_2.8_GHz_PC._English_text_of_size_200,

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Table1 Textlength=200characters,ˇ=1.5.

Poly.degree n=10 n=15

Pa=0.2 Pa=0.25 Pa=0.4 Pa=0.2 Pa=0.25 Pa=0.4

11 CPUtime(insec) 41.51 40.86 41.20 41.24 40.70 40.83

Bestﬁtnessvalue 2.64 2.64

13 CPUtime(insec) 483.67 480.57 482.51 482.18 480.10 483.21

17 CPUtime(insec) 69,390 69,134 68,890 69,402 68,102 69,256

19 CPUtime(insec) 276,543 277,534 277,560 277,542 266,545 276,459

Table2 Textlength=300characters,_ˇ=1.5.

Poly.degree n=10 n=15

Pa=0.2 Pa=0.25 Pa=0.4 Pa=0.2 Pa=0.25 Pa=0.4

11 CPUtime(insec) 57.05 57.94 57.30 57.25 57.56 57.27

13 CPUtime(insec) 673.30 670.01 673.26 673.29 669.30 673.18

17 CPUtime(insec) 92,520 91,562 92,678 91,672 91,529 92,561

19 CPUtime(insec) 360,799 368,800 360,800 360,695 359,765 360,799

Figure1 Fortextlength:200characters.

300and400charactersareusedfor analysingresults.The polynomialsusedforencryptingthetextareofdegree11, 13, 17 and 19. Murray code is used for obtaining binary text.Experimentationwascarriedoutwithdifferent param-etersof theCSalgorithm.Probabilityofdiscoveringworst nests (pa) are assigned three different values, i.e. 0.2, 0.25 and 0.4. Value of ˇ is varied between three values (1.1,1.5 and1.8). The valuesof searchdomainsare also variedand assignedthreevalues,i.e. 10,15and 20. Cor-rectinitialstatesofLFSRhavebeenobtainedforprimitive

Figure2 Fortextlength:300characters.

polynomialofdegree11,13,17and19.Accordingto plot-tedgraphs(Figs.1and2)andresultsgiveninTables1and2

fortextlength200and300characters,computationaltime increasesasdegreeofprimitivepolynomialincreases.

Conclusion

We have developed novel approach for analysis of LFSR based cryptosystemusingCuckoo Search. Developedcode hasbeen tested for varying parametersasdeﬁnedabove.

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The optimumsolutions wereobtainedwhen ˇ=1.5, n=15 andpa=0.25.Cuckoo Searchmayalsobe appliedin anal-ysis of Non-LinearFeedback ShiftRegister (N-LFSR)based Cryptosystems.

Conﬂict

of

interest

Theauthorsdeclarethatthereisnoconﬂictofinterest.

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