DataHidingCapacityinthePresenceofanImperfectlyKnown
Channel
R.Chandramouli
DepartmentofElectricalandComputerEngineering
StevensInstituteofTechnology
ABSTRACT
Weconsideradatahidingchannelinthispaperthatisnotperfectlyknownbytheencoderandthedecoder.The
imperfectknowledgecouldbeduetothechannelestimationerror,time-varyingactiveadversaryetc.Amathematical
modelforthisscenarioisproposed.Manyimportantattackssuchasscaling,geometricaltransformationsetc.fall
undertheproposedmathematicalmodel.Minimalassumptionsaremaderegardingtheprobabilitydistributionsof
thedata-hidingchannel.Lowerandupperboundsonthedatahidingcapacityarederived.Itisshownthatthe
popularadditiveGaussiannoisechannelmodelmaynotsufficeinreal-worldscenarios;thecapacityestimatesusing
theadditiveGaussianchannelmodeltendtoeitherover-orunder-estimatethecapacityunderdifferentscenarios.
Asymptoticvalueofthecapacityasthesignaltonoiseratiobecomesarbitrarilylargeisalsogiven.Manyexisting
datahidingcapacityestimatesareobservedtobeaspecialcaseoftheformulasderivedinthispaper.Wealsoobserve
thattheproposedmathematicalmodelcanbeappliedtoreal-lifeapplicationssuchasdatahidinginimage/video.
Theoreticalresultsarefurtherexplainedusingnumericalvalues.
Keywords:Datahidingchannel,capacity,informationtheory
1.INTRODUCTION
Datahidingisemergingasanimportantareaofresearchwiththeadventofdigitaltechnologies.Watermarking,
authentication,andsteganographyaresomeoftheimportantapplicationsofdatahidingtechniquestovariousreal-life
scenarios.1Astheapplicabilityofdatahidingmethodsincreasesitbecomesessentialtostudythemmorerigorously.
Athoroughmathematicalanalysisofgeneraldatahidingprinciplesunderbroadassumptionswillproveveryusefulin
theiranalysisandoptimization.Moreover,ageneralmathematicalmodelwillbeusefulinpredictingtheperformance
ofawiderangeofalgorithms.Therehasbeensomepreviousattemptstowardsthisgoal.2–7Mathematicaltools
frominformationtheoryhaveprovidedsomeinsightsonthedatahidingproblem.2–4,6,8–12Thenumberofbitsthat
canbeenhiddeninagivenhostsignal,namely,thedata-hidingcapacitywhenthehostsignalundergoesspecific
kindsofprocessing/attackswithknownprobabilitydistributionshavebeenstudiedinsomeoftheseworks.The
Gaussianprobabilitydistributionisapopularmodelforthedata-hidingchannel.Thismodelgivesrisetoclosed-
formsolutionsforthedata-hidingcapacity.Acommonalitybetweenthemajorityofthestudiesondata-hidingis
thatthetypeofprocessingthehiddendataundergoesisassumedtobeknownatthereceiver.Theprocessing/attack
isusuallymodeledasadditivenoise.But,theattackbyanactiveadversaryisnotguaranteedtobeknownatthe
receiverandneednotbeonlyadditive;e.g.scalingandrotationoperationsarenotadditive.Therefore,amore
generalmathematicalmodelisneed.Differingfromtheinformationtheoreticanalysis,adecisiontheoreticmethod
tocomputethewatermarklengthcapacityforaspecificnoisedistributionisintroducedin.7Thelengthcapacityis
theminimumnumberofsignalsamplesthathavetobewatermarkedsothatthewatermarkcanbedetectedwitha
givenprobabilityoferror.
Weconsideranactiveadversaryinthispaperwhosestrategycouldchangewithtime.Thetime-varyingattack
strategyoftheactiveadversarymeansthattheestimateoftheattackchannelbythereceiverwillalwaysremain
imperfecttosomeextent.Ontheotherhand,iftheattackistime-invariantthenitcanbeestimatedwitharbitrary
reliabilitybyusingsufficientlylargetrainingdata.Inthispaper,weproposeachannelmodelthathasarandom
multiplicativeandanadditivecomponent.Thiscanbethoughtofasadata-hidingchannelwithafadingcomponent
(analogoustothefadingincommunicationtheory).Wenotethatmostoftheattackmodelsstudiedsofarinthe
literaturearesubsetsoftheproposedone.Theprobabilitydistributionsoftherandomcomponentsofthedata-hiding
ToappearinProc.ofSPIEVol.4314,SecurityandWatermarkingofMultimediaContentsII,2001.
++G_d
G_rW
NZ
Figure1.Mathematicalmodelfortheimperfectlyknowndata-hidingchannel.
channelneednotbeperfectlyknownatthereceiver.However,itispossibletoestimatethedeterministiccomponent
oftherandomchannel.Thisgivesuspartialknowledgeaboutthechannel.Ourgoalistocomputeboundsonthe
data-hidingcapacityandstudytheeffectoftheimperfectknowledgeofthechannelonthedatahidingcapacity.
Wegivebothlowerandupperboundsforthecapacity.Theoreticalresultsarefurtherexplainedusingnumerical
results.Wenotethatouranalysisisvalidevenforobliviouswatermarkingschemeswherestatisticalestimatorsare
usedtoestimatethehostsignalorsomeofitsparameters.Thetime-varyingattackmodelcanalsobeinterpretedas
time-varyingchannelcharacteristicswhenthehostsignaltogetherwiththehiddendataistransmittedoverrandomly
fluctuatingreal-lifetransmissionchannels.Forexample,considerthecasewhereasenderusesanimagetohidea
messageintendedforaparticularreceiver.Whenthisimageissenttothereceiveritmayundergocompressionand
alsoexperienceadditivenoiseeffects.Thereceivermaynotknowthecompressionratio(or,eventhecompression
algorithm)andthemeanandvarianceoftheadditivenoise.Ifthereceiverdoesnothaveaccesstotheoriginalimage
(whichisusuallythecase)itmayhavetoestimatesomeoftheseunknownparameterstoextractthehiddendata.
Atypicalexampleisanobliviouswatermarkingscheme.Theparameterestimationprocessresultsinvariouskinds
oferrorsandeffectsthathavesignificanteffectsonthedatahidingcapacityperchanneluse.
Theorganizationofthepaperisasfollows.TheproposedproblemisdefinedmathematicallyinSection2.
Theoreticalresultsarederivedinthissection.TheseareexplainedfurthervianumericalanalysisinSection3.The
conclusionsofthisstudyaregiveninSection4.
2.PROBLEMDEFINITION
Weattempttocomputeand/orboundthedata-hidingcapacitywhentheknowledgeaboutthedata-hidingchannel
(orattack)isimperfectandremainstobeimperfect.Figure1showstheproposedmathematicalmodelforthe
proposedproblem.Inthefigure,therandomvariableWstandsforthehiddensignal(ordata),GdandGrarethe
deterministicandtherandomcomponentsofthetime-varyingattack,i.e.,therandomgain,Gisdecomposedas
G=Gd+GrandNistheadditivenoisecomponent.Formally,thereceivedhiddensignalcanbewrittenas
Z=GW+N(1)
=GdW+GrW+N(2)
where,therandomvariablesG,W,andN∈areassumedtobestatisticallyindependentofeachother.The
time-varyingnatureoftheactiveadversaryischaracterizedbythecondition,σ2G=Variance(Gr)>0.Thismeans
thatthereisalwayssomeuncertaintyintheestimationoftheattack.Italsoquantifiestheerrorinestimatingthe
randomgain,G,atthereceiverinordertoundotheeffectoftheattack.Further,wemakethefollowingreasonable
assumptions:
•E(W2)≤σ2W
•N∼Gaussian(0,σ2N)
•E(G)=Gd,E(Gr)=0,andE(G2r)=σ2G
