A recursive non-linear pre-encryption for opto-digital double random phase encoding

random

phase

encoding

Toufik

Bekkouche

a

_,

_Saad

_Bouguezel

b,∗

a_{ETALaboratory,DepartmentofElectronics,FacultyofTechnology,BordjBouarreridj,34000,Algeria}

b_{LCCNSLaboratory,DepartmentofElectronics,FacultyofTechnology,UniversityFerhatAbbasSetif-1,Setif,19000,Algeria}

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received29August2016

Receivedinrevisedform20May2017 Accepted28December2017 Keywords:

Chaoticmaps

Doublerandomphaseencoding FractionalFouriertransforms Imageencryption

Recursivenon-linearpre-encryption

a

b

s

t

r

a

c

t

Inthispaper,weproposearecursivenon-linearpre-encryptiontobecarriedoutdigitallyon theinputimageinthespatialdomainbeforeapplyinganydoublerandomphaseencoding (DRPE).Itconsistsoffirstlyscramblingtheinputimagechaoticallyandthenperforming thebit-wiseXORoperationrecursivelybetweentwoconsecutivepixelsfollowingagiven pattern.Thestartingtwoconsecutivepixelsareformedbytheinitialpixelinthepattern andarandomeight-bitinteger.Theproposedpre-encryption-basedDRPEimageencryption systemandexistingDRPEcryptosystemsarecomparedintermsofthesensitivityofthe variousencryptionkeystodecryptionattacks.

©2017ElsevierGmbH.Allrightsreserved.

1. Introduction

Moderncommunicationsystemsandtheirdifferentapplicationsinmilitary,commercial,medicalandsocial sectors necessitatehighlyrobustandfastinformationsecuritytools.Encryptionisoneofthemostpowerfultoolsininformation security.Specifically,opticalimageencryptionsystemsareofgreatimportancefortheirparallelprocessingandhighspeed [1].Oneofsuchsystemsisthewell-knownopticaldoublerandomphaseencoding(DRPE)introducedbyRefregierandJavidi in[2].Itconsistsof(1)multiplyingtheinputimagebyarandomphasemask(RPM1)inthespatialdomain,(2)transforming

theresultobtainedfrom(1)usingthetwo-dimensionalFouriertransform(FT),(3)multiplyingtheresultobtainedfrom(2) byanotherrandomphasemask(RPM2)inthefrequencydomain,andfinally(4)transformingtheresultobtainedfrom(3)

usingthetwo-dimensionalFTtoobtaintheencryptedimage.ThetwomasksRPM1andRPM2arestatisticallyindependent,

wherethefirstisusedtowhitentheimage,whereasthesecondisemployedasasecretkeyforencryptionanddecryption processes.ThistechniqueiscalledhereFT-DRPE.

InordertoincreasethesecurityoftheFT-DRPE,parametrictransformssuchasthefractionalFouriertransform(FrFT) [3],Fresneltransform[4],multiple-parameterdiscretefractionalFouriertransform(MPDFrFT)[5],Gyratortransform[6], reciprocal-orthogonalparametrictransforms[7,8]andinvolutoryparametrictransform[9]havebeenusedinsteadofthe FT,wheretheirindependentparametershavebeenexploitedasanadditionalsecretkey.Specifically,theoptical FrFT-basedDRPE(FrFT-DRPE)andMPDFrFT-basedDRPE(MPDFrFT-DRPE)areofgreatimportanceandhaveextensivelybeen consideredintheliteraturetodevelopmoresecuredversionsbyintroducingthereinscramblingschemes[10–13].The resultingDRPEversionsgenerallyneedacomputertoperformthescramblingandhenceareconsideredasopto-digital encryptiontechniques.

∗ Correspondingauthor.

E-mailaddress:bouguezelsaad@yahoo.com(S.Bouguezel).

https://doi.org/10.1016/j.ijleo.2017.12.142

Althoughtheaboveopticalandopto-digitalDRPEversionsareefficient,theymaynotresisttosomeattacks[14–16].This ismainlyduetothefactthatthetransformsemployedarelinearoperatorsandtheassociatedscramblingschemescanalso beconsideredaslineartransformations,whichmaketheentireresultingDRPEalinearimageencryptiontechniqueand hencefragiletosomeattacks.Inordertoovercomethislinearityproblem,anon-linearpre-processinghasrecentlybeen proposedin[17].Itconsistsofobtainingapre-processedimagebyperformingtheXORoperationinthespatialdomain betweeneachpixeloftheinputimageanditscorrespondingpixelinthesamepositionofarandomlycreatedimagebefore applyingaDRPE.Ithasbeenshownin[17]thatthisnon-linearpre-processingcoupledwithanopto-digitalMPDFrFT-DRPE leadstoanewopto-digitalDRPEthatoutperformstheotherexistingDRPEversions,specificallyintermsofthesecretkey sensitivity.Eventhoughthepre-processingproposedin[17]isveryefficientandattractive,thenon-linearityofferedby theXORoperationhasbeenintroducedthereinforeachpixelseparatelyandindependently.However,fromtheencryption pointofview,itishighlydesirabletoconstructamorecomplexnon-linearpre-processing.Thisisachievedinthispaper byintroducinganewrecursivenon-linearpre-encryptiontobeappliedpriortoanyDRPE.Itconsistsof(1)resizingthe inputimageintoavector,(2)chaoticallyscramblingtheresultingvectorusingthepiecewiselinearchaoticmap(PLCM),(3) performingthebit-wiseXORoperationrecursivelybetweentwoadjacentelementsoftheresultingscrambledvector,where thefirstelementisbit-XORedwitharandomeight-bitinteger,and(4)reshapingtheresultingXORedvectorintoamatrix toobtainthepre-encryptedimage.Therecursivepropertyoftheproposedpre-encryptionintroducessomedependency betweenthepixelsofthepre-processedimageandhenceensurestheaccumulationandpropagationoftheerrortoall pixelsinthecaseofanywrongnessinthedecryptionkey.Therefore,anyopto-digitalcryptosystemobtainedbycoupling theproposedpre-encryptionwithanopticalDRPEcryptosystemincludingtheFT-DRPEwouldresisttodecryptionattacks. Asmentionedabove,inordertofurtherincreasethesecurityofanopto-digitalcryptosystem,itishighlydesirabletouse transformshavingopticalimplementationandincreasednumberofindependentparameterssuchasFrFTandMPDFrFT.

Thispaperisorganizedasfollows.InSection2,weintroduceanewrecursivenon-linearpre-encryptionandreviewthe PLCMusedtoscrambleimages.Theproposedopto-digitalimageencryptionanddecryptionsystemsaredescribedinSection 3.Section4presentsperformanceanalysisandcomparison.

2. Proposedrecursivenon-linearpre-encryption

Itiswell-knowninthenumericalmethodsthatthemainseriousdrawbackofanyrecursiveapproachistheaccumulation andpropagationoftheerror.Incontradictionwiththesemethods,encryptiontechniquesstronglysearchforanyapproach havingthisdrawbackorproperty.Therefore,webeneficiallyexploitthispropertyofarecursiveapproachinimageencryption toachievethedesireddependencybetweenthepixelsofthepre-processedimagebyintroducinginthis papera new recursivenon-linearpre-processing.ItconsistsoffirstlyscramblingtheinputimageandthenapplyingthebitXORoperation recursivelyoneachpairofconsecutivepixelsfollowingagivenpattern.Thestartingorinitialpairisformedbytheinitial pixelinthepatternandarandomeight-bitinteger.Thesecretkeyfortheproposeddigitalpre-processing(pre-encryption) isthisrandomintegerthatcanbechosenarbitrarilyfrom0to255,andthescrambling,whichhas (N×M)!possibilities, whereN×Misthesizeoftheinputimage.Forcomputersimulations,weconsiderheresquareinputimages,i.e.M=N.

Inordertoefficientlyaccomplishtheaboverequiredscrambling,weexploitchaosmaps,which areknowntohave attractivecryptographicpropertiessuchashighsensitivitytotheirinitialparameters,ergodicity,andpseudo-randomness. Specifically,weconsiderthepiecewiselinearchaoticmap(PLCM)proposedin[18]andexpressediterativelyas[17]

z_k+1=F (zk,)=

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

zk , 0≤zk< zk− 0.5−, ≤zk<0.5 F (1−zk,) , 0.5≤zk<1 (1)

wherez0istheinitialconditionparameterand∈(0,0.5)isthecontrolparameter.Therefore,oneschemefordigitally

implementingtheproposedpre-encryptioncanbeachievedbythefollowingsteps: 1Resizetheinputimageintoavectorioflength1×(N×N)

2Generateachaoticvectorz,{zk,k=1,2,3,...,N×N},usingthePLCMgivenbyEq.(1)withtheparameters



z0,



3Sortthevectorzintoanascendingordertoformavectoryandthenformapermutationmapvectormsuchasmkisthe

positionoftheelementy_kinthevectorz,i.e.,_{yk=zmk,k=1,2,3,...,N×N}

4Scramblethevectoriusingthepermutationmapvectormtoformavectorssuchasskisthe(mk)thelementofi,i.e.,

{sk=imk,k=1,2,3,...,N×N}

5Performthebit-wiseXORoperationrecursivelyontheadjacentelementsofstoformavectorxas

xk=



sk⊕r, k=1

sk⊕xk−1, k=2,3,...,N×N

Fig.1. Anopto-digitalsetupfortheproposedFrFT-DRPE/MPDFrFT-DRPEencryption/decryption.

Fig.2. Proposedencryptionsystem.

Fig.3. Proposeddecryptionsystem.

wherer=round

255∗z(N×N)

,i.e.,therandomintegerrischosentobethelastelementofthechaoticvectorzconverted toaneight-bitinteger

6Finally,thepre-encryptedimageisobtainedbyreshapingthevectorxintoanN×Nmatrix.

TherandomintegerrischoseninStep5tobeascalarandthepre-encryptedpixelxkin(2)isobtainedbyonlyone

XORoperation.Thisschemeisadoptedinthecomputersimulationsbelowforitssimplicity.However,anotherscheme canbeestablishedbychoosingrasaneight-bitintegerrandom1×(N×N)vectorrandthenperformingXORoperation element-by-elementbetweenthevectorssandrbeforeperformingXORoperationrecursivelyontheadjacentelementsof theresultingvector.Itisworthtomentionthatforanyselectedscheme,arecursiveapplicationoftheXORoperationinthe pre-encryptionismandatorytoachievehighkeysensitivityfortheentireimageencryptionsystem.

3. Proposedpre-encryption-basedDRPEtechnique

Theproposeddigitalpre-encryption,whichcombinesascramblingschemewitharecursivenon-linearapproachand hasasecretkeyconstitutedoftheparameters



z0,



,canbefollowedbyanyoftheexistingDRPEversionstoconstruct ahighlysecureimageencryptiontechnique.Thisconstructioncanefficientlyexploitanyoftheexistingwell-established DRPEsystemswithoutanyalteration.Withoutloseofgenerality,theproposedpre-encryptioncanbeusedattheinputofthe existingopticalFrFT-orMPDFrFT-DRPEcryptosystem.Therefore,anopto-digitalimplementationsimilartotheonereported in[1]canbesuggestedfortheproposedFrFT-DRPE/MPDFrFT-DRPEencryption/decryptionasshowninFig.1,inwhichthe computerisusedtodigitallyperformtheproposedrecursivenon-linearpre-encryptionandPLCM-basedpermutations. Thisimplementationhasawell-known4-fopticalsetupforimplementingtheFrFT/MPDFrFT.Thespatiallightmodulators SLM1andSLM2areusedtodisplaythecomplex-valuedsignalduringtheencryption/decryptionsteps.TheCCDcamerais employedtodigitallyrecordthecomplex-valuedsignalusingdigitalholographictechniquesandareferencebeam.

Theresultingproposedpre-encryption-basedDRPEsignificantlydiffersfromtheexistingscrambling-basedDRPE ver-sionsnotonlybecauseofitsnon-linearityproperty,butalsoforitsrecursivepropertythatensurestheaccumulationand propagationoftheerrortoallpixelsinthecaseofanywrongnessinthedecryptionkey.

Figs.2and3showtheproposedopto-digitalimageencryptionanddecryptionsystems,respectively.Thedecryption systemtakesthestepsoftheencryptionsysteminaninversemanner.Fortheencryptionsystem,theinputimage,which isgenerallyareal-valuedeight-bitgray-scaleimage,ispassedthroughtwomainblocks.Thefirstblockconsistsofdigitally pre-encryptingtheinputimagetoobtainauniformlydistributedimagewiththesameformattotheinput.Forinstance,we takethe256×256gray-scaleLenaimagegivenbyFig.4astheinput.Itspre-encryptedimageisgivenbyFig.5,whichis clearlyauniformlydistributedimage.Thepre-encryptedimageisfedtothesecondblock,whichcanbeanyoftheexisting opticaloropto-digitalDRPEsystems,toproducetheencryptedimage.IfthesecondblockischosentobetheFrFT-DRPE

Fig.4.InputLenaimageanditshistogram.

Fig.5.Pre-encryptedLenaimageanditshistogram.

system,whichhasfourindependentfractionalorderparameters



a,b,c,d



[3],ortheMPDFrFT-DRPEsystem,whichhas fourindependent1×Nvectors



−a,−b,−c,−d



ofindependentfractionalorderparametersthatcanbechosenrandomly fromtheinterval[0,2][5],thentheencryptedimageisgivenbyFigs.6or7.

4. Performanceanalysisandcomparison 4.1. Histogramanalysis

Toshowtherobustnessoftheproposedtechniqueagainsthistogramanalysis,weencryptthreedifferentimages,namely Lena,BarbaraandBaboonimagesshowninFigs. 4,8and10,respectively,andthencomparethehistogramsgivenin Figs.7,9and11,respectively,oftheresultingencryptedimages.Itisclearfromthesefiguresthateventhoughthehistograms oftheoriginalimagesarecompletelydifferent,thehistogramsoftheamplitude(orthephase)ofthecorrespondingencrypted imagesareverysimilar.Theseresultsconfirmthatnoinformationleakageabouttheoriginalimagecanbelearnedbyan attackerfromhistogramanalysisofencryptedimages,thus,theproposedschemeisrobustagainsthistogramanalysis. 4.2. Noiseattack

Anyencryptedimagecanbecorruptedbythenoiseduringtransmissionorstorage.Toverifytheresistanceagainstthe additivenoiseofagivenimageencryptiontechnique,weaddawhiteGaussiannoisewithzeromeanandstandarddeviation equalstounitytotheencryptedLenaimage.ThedecryptionoftheresultingnoisyencryptedLenaimageisgiveninFig.12 fordifferenttechniquesintermsofthePSNRbetweentheoriginalanddecryptedimages.Itisseenfromthisfigurethat thethreetechniqueshavesimilarperformanceinthecaseofnoiseattackandtheoriginalimagecanbeobtainedfromthe decryptednoisyimageafterusingsomeknowndenoisingtechniques.

Fig.6.EncryptedLenaimageusingtheproposedpre-encryption-basedFrFT-DRPEsystem:(a)amplitudeanditshistogram,(b)phaseanditshistogram.

4.3. Keysensitivityanalysis

It can be seen from Section 2 that the secret key constitutes of



z0,,a,b,RPM2,c,d



in the case of the proposed pre-encryption-based FrFT-DRPE system and



z0,,−a,−b,RPM2,−c,−d



in the case of the pro-posedpre-encryption-basedMPDFrFT-DRPEsystem.Thecorrespondingdecryptionkeysare



z₀,,a,b,RPM₂,c,d



and



z₀,,−a,−b,RPM₂,−c_,−d



_{, respectively. If}



_z 0=z0,  =,a=a,b=b,RPM₂ =RPM2,c=c,d=d



or



z₀ =z0,=,−a=−a,−b  =−b,RPM₂=RPM2,−c=−c,−d 

=−d



, then the corresponding decrypted image is exactly the original image given by Fig. 4. To verify the key sensitivity of the proposed system, the encrypted image is decrypted by introducing small errors in one or some of the parameters that constitute the secret key. If the decryption key is set to be



z₀=z0+10−16,=,a=a,b=b,RPM

 2=RPM2,c=c,d=d



,



z₀=z0,=+10−16,a=a,b=b,RPM  2=RPM2,c=c,d=d



,{z0=z0+10−16,  =,−a=−a,−b=−b,RPM₂ =RPM2,−c=−c,−d  =−d}or



z₀=z0,=+10−16,−a=−a,−b  =−b,RPM₂=RPM2,−c=−c,−d  =−d



,then thecorrespondingdecryptedimageremainstotallyencrypted.Thisshowsthattheproposedsystemishighlysensitiveto thepre-encryptionkey



z0,



.

Wenowcomputethemeansquareerror(MSE)betweentheinputimageandtheimagedecryptedbytheproposed pre-encryption-basedFrFT-DRPEsystemusing



z₀ =z0,=,a=a,b=b,RPM



2=RPM2,c=c+ı1,d=d+ı2



,where theerrorsı1andı2areindependentanduniformlydistributedontheset



−ı,ı



.Fordifferentvaluesofı,theMSEobtained bytheproposedsystemisplottedinFig.13andcomparedwiththecorrespondingMSEobtainedbyusingonlytheFrFT-DRPE systemconsideredin[5],forwhichtheemployeddecryptionkeyis



a=a,b=b,RPM₂=RPM2,c=c+ı1,d=d+ı2



. Itisclearfromthisfigurethattheproposedpre-encryption-basedFrFT-DRPEsystemsignificantlyimprovesthekey sen-sitivityofFrFT-DRPEsystem.Inordertofurtherimprovethekeysensitivityofproposedpre-encryption-basedFrFT-DRPE

Fig.7.EncryptedLenaimageusingtheproposedpre-encryption-basedMPDFrFT-DRPEsystem:(a)amplitudeanditshistogram,(b)phaseanditshistogram.

Fig.8.InputBarbaraimageanditshistogram.

system,weintroduceadependencybetweentheencryptionkeysofthefirstandsecondblocks.Forinstance,wereplace z0byz0+0.1a+0.1cintheencryptionanddecryptionkeys.TheresultingMSEcurveisdepictedinFig.13,whichclearly

showstheimportanceoftheintroduceddependency.

Let us now perform computer simulations on the proposed pre-encryption-based MPDFrFT-DRPE system similar to the above ones carried out on the proposed pre-encryption-based FrFT-DRPE system. The MSE obtained using



z₀=z0,=,−a=−a,−b  =−b,RPM₂=RPM2,−c=−c+−ı1,−d  =−d+−ı2



Fig.9.Histogramofthe(a)amplitudeand(b)phaseoftheencryptedBarbaraimageusingtheproposedpre-encryption-basedMPDFrFT-DRPEsystem.

Fig.10.InputBaboonimageanditshistogram.

Fig.11.Histogramofthe(a)amplitudeand(b)phaseoftheencryptedBaboonimageusingtheproposedpre-encryption-basedMPDFrFT-DRPEsystem.

−ı2areindependentandtheirelementsarealsoindependentanduniformlydistributedontheset



−ı,ı



,isplottedin Fig.14andcomparedwiththecorrespondingMSEobtainedbyusingonlytheMPDFrFT-DRPEsystemreportedin[5],for whichtheuseddecryptionkeyis



−a=−a,−b=−b,RPM₂=RPM2,−c=−c+−ı1,−d  =−d+−ı2



.Itisclearfrom thisfigurethattheproposedsystemsignificantlyimprovesthekeysensitivitycomparedtothesystemin[5].Wealsoinclude inFig.14theMSEobtainedin[17]toshowthattheproposedrecursivenon-linearpre-encryptionleadstokeysensitivity betterthanthatreachedbythenon-linearpre-processingintroducedin[17],whichhasbeenshowntobebetterthanthose ofalltheotherexistingDRPE-basedsystems.Tofurtherimprovethekeysensitivityoftheproposedpre-encryption-based

Fig.12.Resultsofnoiseattackinthecaseofthe(a)proposedpre-encryption-basedFrFT-DRPEtechnique,(b)proposedpre-encryption-based MPDFrFT-DRPEtechnique,and(c)techniquein[17].

Fig.13.MSEintermsofthedeviationerrorıfordifferentFrFT-DRPEsystems.

Fig.14.MSEintermsofthedeviationerrorıfordifferentMPDFrFT-DRPEsystems.

Fig.15.MSEintermsofthedeviationerrorıfortheproposedpre-encryption-basedFrFT-andMPDFrFT-DRPEsystems.

MPDFrFT-DRPEsystem,weintroduceadependencybetweentheencryptionkeysofthefirstandsecondblocks.Forinstance, wereplacez0intheencryptionanddecryptionkeysbyz0+0.1−a(N/2)+0.1−c(N/2),where−a(N/2)and−c(N/2)arethe

(N/2)thelementsofthevectors−aand−c,respectively.TheresultingMSEcurveisdepictedinFig.14,whichclearlyconfirms againtheimportanceoftheintroduceddependency.

Figs.13and14compareseparatelythekeysensitivityofthesystemsthatarebasedontheFrFTandMPDFrFT,respectively, intheinterval[-0.04,0.04]ofthedeviationerrorı.Thisistoclearlyshowtheimprovementsthatcanbeachievedby theproposedsystemeitherintheFrFTdomainorintheMPDFrFTdomain.ThecomparisonbetweentheclassicalDPREs employingthesetwodomainshasalreadybeencarriedoutin[5],whichshowsthatthelatteroutperformstheformerin termsofthekeysensitivity.Inordertocloselycomparethesetwodomainsinthecaseoftheproposedsystem,werestrict

Fig.16.MSEintermsofthedeviationerrorıfortheproposedpre-encryption-basedMPDFrFT-DRPEsystemfordifferentnumbersofpre-encryption iterations.

Fig.17.MSEintermsofthedeviationerrorıfortheproposedpre-encryption-basedFrFT-DRPEsystemfordifferentnumbersofpre-encryptioniterations.

inFig.15theintervalofthedeviationerrorıto[-0.01,0.01]andthendepicttheMSEcurvesobtainedbytheproposed pre-encryption-basedFrFT-DRPEandMPDFrFT-DRPEsystems.Itisseenfromthisfigurethatthelatter,whichcanreach anMSEgreaterthan9000intheinterval

ı

≥6×10−3,outperformstheformerforwhichthecorrespondingintervalis smallerandis

ı

≥8×10−3.ThisismainlyduetothefactthattheMPDFrFTdomainpossesses4Nindependentparameters, whereastheFrFTdomainhasonlyfour.

Thekeysensitivityoftheproposedpre-encryption-basedtechniquecansignificantlybeimprovedbyrepeatingthe proposedrecursivenon-linearpre-encryptionforafixednumberoftimesbeforeapplyingtheDRPE.Thesimulationresults correspondingtotheexperimentscarriedoutforthispurposearegiveninFigs.16and17fordifferentnumbersofrepetitions oriterations.Itisclearfromthesefiguresthat,forboththeproposedpre-encryption-basedFrFT-DRPEandMPDFrFT-DRPE systems,theMSEcanbeincreasedfromabout9500foroneiterationtoabout13,700for11or16iterations

5. Conclusion

Wehaveshownthatbyintroducinganewdigitalpre-encryptionbasedonarecursiveXORoperation,thekeysensitivity oftheexistingDRPE-basedcryptosystemscansignificantlybeimproved.Wehavealsoshownthatfartherimprovements canbeachievedbytheproposedpre-encryption-basedDRPEsystembyintroducingadependencybetweenthekeysofthe pre-encryptionandDRPEblocksand/orbyrepeatingtheproposedrecursivenon-linearpre-encryptionanumberoftimes beforeapplyingtheDRPE.Computersimulationsconfirmthattheproposedpre-encryption-basedFrFT-orMPDFrFT-DRPE systemoutperformstheexistingDRPEsystems.Moreover,theproposedpre-encryptioncaneasilybeincorporatedatthe inputoftheexistingwell-establishedopticaloropto-digitalDRPEsystemswithoutanyalteration.Therefore,theopticalor opto-digitalimplementationoftheproposedpre-encryption-basedDRPEsystemisstraightforward.

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