Watermarking 2D-Vector Data for Geographical Information Systems

Transcription

1 Waterarn D-ector Data for Georaphcal Inforaton Systes Mchael ot and Chrstoph Bsch Unversty of Technoloy Darstadt, D-68 Darstadt, Gerany Franhofer Insttte for Copter Graphcs, D-68 Darstadt, Gerany BSTRCT Ths paper deals wth the sse of waterarn D-vector data whch are sed n Georaphcal Inforaton Systes (GIS) The waterar s ebedded n the tolerance rane of the coordnates, where one bt of the waterarn nforaton s represented by one PN-seqence, whose eleents consst of the two vales tolerance and tolerance To robstly ebed one bt of the waterarn nforaton the lenth of the PN-seqence has to be ch reater than the sqare of the a coordnate vale leadn to non-acceptable seqence lenths de to hh coordnate vales To acheve a PN-seqence lenth that s stable to the sze of the data doan we do not consder the whole coordnate vale bt only those decal dt postons of the coordnate vale, where chanes are snfcant bt not volatn the tolerance reqreents De to ths restrcton on a saller rane of vales, overflow and nderflow has to be consdered drn the ebeddn process Wthn the retreval process we frst etract ths fracton of the coordnate vale before correlatn t wth the PN-seqence The proposed ethod s robst aanst attacers chann the coordnates wthn the tolerance rane Keywords: Georaphcal Inforaton Systes (GIS ), Waterarn and PN-seqence eal: vchael@nest-darstadtde eal: bsch@dfhde INTRODUCTION Georaphcal Inforaton Systes (GIS) are sed for acqston, anaeent, processn and presentaton of data whch bear a spatal relaton These data descrbe obects, whch are characterzed n the frst place by ther eoraphcal data (coordnates) and secondly by ther topoloy (cobnaton of ponts to seantc obects e ndvdal peces of real estate or hose) Frtherore each obect ay be assocated wth certan descrptve attrbtes, e propertes of the obects ndependent of ther eoraphcal poston (e nae, nber of the townsfol etc) GISdata represent a hh ateral vale de to hh efforts and costs n ther acqston and antenance of the data, whch s ether perfored throh anal land srvey or throh analyss of aeral photoraphy De to the hh vale and wde dstrbton of sch data nsde e car navaton systes or va the nternet there ests an nterest of the copyrht holders to protect these data aanst nlcensed copyn Wdely sed are coordnates whch are specfed n the Gass-Krüer syste, whch constttes an ndeed erdan strpe syste The strpes are centered arond an erdans, whch lay at ltple of derees The strpes have an epanson of abot 00 to the left and rht fro the an erdan The Gass-Krüer coordnates of a pont are specfed n eter and consst of the so-called R-ale and H-ale The R-ale of a pont specfes the perpendclar dstance fro the assocated an erdan n west-east drecton Snce the an erdan s eqvalent to 500,000, the R-ales for a pont n each strpe vary between 00,000 and 600,000 The H-ale specfes the poston of a pont n north-soth drecton and s eqvalent to the dstance fro the eqator The R- and H-ales for D data sets, whch bld the basc ateral for or consderatons There est of corse the choce of other coordnate systes le the Unversal Transversal Mercator proecton (UTM), whch s the coonly sed proect ethod n the Unted States The reander of ths paper abstracts fro a specfc proecton ethod, bt asses the data to be represented n a rectanlar coordnate syste wth ven easreent precson of a snle pont The precson of a coordnate s stronly dependent on the acqston ethod of the data Nevertheless data vendors arantee a certan precson whch ay be nterpreted as tolerance reon arond the pont the spatal reon n whch odfcaton of the pont coordnates throh or waterarn alorth s acceptable We reard two nds of D data sets wth dfferent

2 tolerance reons The frst nd represents obects le real estates and have tolerances n the d-c rane The other rop nclde obects that descrbe land sae aps whch have a precson n the rane of eters We se the tolerance rane of the coordnates to ebed the waterar, where one bt of the waterarn nforaton s represented by one psedo-nose (PN)-seqence The detecton process s a sple correlaton between the PNseqence and the waterared data To robstly ebed resp detect one bt of the waterarn nforaton n a waterar seent of the data the lenth of the PN-seqence has to be ch reater than the sqare of the a coordnate vale De to the last stateent the ebeddn of a bt-waterar for R-ales between 00,000 and 600,000 wth a tolerance of e ± wold lead to non-acceptable seqence lenths resp to non-acceptable error rates for stable seqence lenths To crcvent ths restrcton we propose the follown ethod PROPOSD MTHOD be the raw GIS-data wth a tolerance of whch shold be waterared and let Let (,,, ) (,,, ) be a secret PN-seqence, nown eclsvely to the ebedder and the decoder, where { ±} (,,) tolerance vale s abbrevated wth The waterared vales are denoted as a ( a, a,, a ) consder attacs by addn the an vales We collect the nto the vector (,,, ) a coordnate vale of t as a( ) for We and denote the The proposed ethod s vald for an attacs satsfyn The nderlyn asspton s that for arbtrary an attacs > the data set wold snfcantly loose ts vale (e rectanlar obects sch as hoses wold not appear n rectanlar shape after the attac) The waterared vales after a an attac are called a ( a, a,, a ) Wth s we denote the fncton whch yeld the s-snfcant decal dt postons of a vale, e for the GIS-data 567,8 the -snfcant decal dt postons vale s Possble vales for the case s are the nbers 0,,, 99 For the follown eplanatons we concentrate on beddn Procedre: The follown flow-chart shows the ebeddn procedre for the case of two snfcant decal dt postons tractn the -snfcant decal dt postons of Y > 0 N Y N Y N a a a 99 a F Flow-chart for the ebeddn procedre The alorth ensres that attacs of ± do not lead to an overflow or nderflow (OF/UF), whch wold nflence the net snfcant decal dt (e the -snfcant poston vale 8 n or eaple) (Chann the vale 99 nto the vale 9 ves the procedre for one snfcant decal dt poston)

3 ttacn Procedre: s attacs we consder addtve dstrbances of the waterared data a The vale after an attac s a, so that t can be wrtten as a a Retrevn Procedre: For retrevn the waterar we se the correlaton between the PN-seqence ( ) ebeddn) and a 50, that s a 50 To facltate the follown calclatons of a, whch represents the dfference between (sed for ntrodce an addtonal ter and a 99 and respectvely Wth ths dfference ter whch ensres that no OF/UF occrs we can decopose a a as a and wrte for the rando varable 50 For a better nderstandn of the follown calclatons le the epectaton and the varance of the net table ves an overvew of the relatonshp between dfferent vales for the sed qanttes le, and The pper part of the table treat the case and the lower part s for a a X, we a a X- F Relatonshp between dfferent vales for the qanttes, and The rando varable s the s of other rando varables Therefore we ae the asspton that posses an N µ, σ µ Gassan dstrbton, that s ( ) The net steps wll be the evalaton of the epectaton vale { } and the varance { } valaton of { } : Wth { } 0 σ, thereby assn ndependence of the epectaton of, and

4 { } { } { } { } 50 redces to { } { } Usn { } ( ) ( ) the epectaton becoes { } ( ) ( ) 00 The specal case leads to { } valaton of { } : For the calclaton of the varance { } we ae se of the forla { } { } { } ssn ndependence of, and and wth { } 0 the frst epectaton { } redces to { } { } { } ( ) { } { } { } { } and the sqare of the epectaton of s { } { } { } So the varance coes to { } { } { } { } { } { } { } ) ( It s portant that the ters wth a - dependency are cancelled t frst we evalate the epectaton vales of the ters whch are ndependent fro the an: { } 5 8 and { } 5 9 { } ( ) ( ) ( ) ( ) ( ) [ ] The specal case leads to { } { } ( ) ( ) 00, see the evalaton of the epectaton { } The specal case leads to { } ( ) 0000

5 [ ( ) ] { } ( ) ( ) ( ) The specal case leads to { } ( 6 ) 600 Net we consder dfferent attac cases and evalate the varance: No attac: { } 0 and { } 0 leads to { } (,, ): wth { } and { } we et 99 { } (,, ): wth { } 97 and { } 960 we et { } ± ( at rando) (,, ): wth { } 0 97 and { } { } yeldn to We have now copleted the evalaton of the two paraeters (epectaton and varance) of the Gassan dstrbton for dfferent cases of attacs and are ready to specfy the We consder the error we ae when we ebed a waterar and we can not detect t That s, we need to calclate the error probablty P err erf µ σ The theoretcal crves of the err P for data lenths of 5 and 50 are shown n the net two fres for tolerances between and 9 Of corse the decreases f we choose loner seqence lenths

6 0 0 perran00: datalenth 5 no attac tolerance n eter F rror rate P err for a seqence lenth of 5 (no attac) 0 0 perran00attac: datalenth 50 attac tolerance attac -tolerance attac tolerance(at rando) tolerance n eter F rror rate P err for a seqence lenth of 50 (attac case -) Loon at F and F t stres that the starts to ncrease after reachn a n vale Ths s de to the nonlneartes drn the ebeddn procedre F shows that a rando attac n order of antde of the tolerance s bonded by the two crves for attacs tolerance (constant) and tolerance (constant) respectvely

7 SIMULTION RSULTS To verfy the theoretcal crves ven n F and F the net two daras show soe slaton reslts 0 0 an00attacrep: datalenth no attac Theoretcal Slaton tolerance n eter F5 rror rate for P err a seqence lenth of 5 (no attac) 0 0 an00attacrep: datalenth 50 attac *tolerance (at rando) Theoretcal Slaton tolerance n eter F6 rror rate for a seqence lenth of 50 (attac case )

8 The net dara shows the slaton reslts for the transforaton of easreents wth tolerances fro to 9 nto easreent vales wth optal tolerance resp nal 0 - an00attactransf:datalenth 50 (0000 repettons) attac *tolerance (at rando) tolerance n eter F7 rror rate after transforn the easreents nto the optal pont wth nal CONCLUSION ND FURTHR WORK We have shown that a odfed correlaton ethod s stable for ebeddn and retrevn waterars n GIS-data sn the tolerance rane of the data Unle the lnear case where the contnosly decreases wth the tolerance resp an here the s nal for a tolerance vale lower than the aal tolerance To acheve ths nal for all tolerances we transfor the nto ths optal pont The proposed ethod s robst aanst attacs wthn the tolerance rane and t can be eneralzed to all nd of easred data havn tolerances ttacs reater than the tolerance on snle data can case overflows whch ae worse the detecton procedre We wll nvestate how any of ths attacs the ethod can handle nother nd of attac s the reovn or addn of data We wll analyse f t s stll possble to detect the waterar after sch an attac RFRNCS Y Yacob, Iproved Boneh-Shaw Content Fnerprntn, D Naccache(d): CT-RS 00, LNCS 00, pp 78-9, 00 Papols, Probablty, Rando arables, and Stochastc Processes, McGraw-Hll Internatonal dtons, 99