Watermarking Integer Linear Programming Solutions

Transcription

1 Watermarkig Iteger Liear Programmig Solutios Seapah Megeria Computer Sciece Departmet Uiversity of Califoria Los Ageles Mileko Driić Computer Sciece Departmet Uiversity of Califoria Los Ageles Miodrag Potkojak Computer Sciece Departmet Uiversity of Califoria Los Ageles ABSTRACT Liear programmig (LP) i its may forms has prove to be a idispesable tool for expressig ad solvig optimizatio problems i umerous domais. We propose the first set of geeric watermarkig techiques for iteger-lp (ILP). The proof of authorship by watermarkig is achieved by itroducig additioal costraits to limit the solutio space ad ca be used as effective meas of itellectual property protectio (IPP) ad autheticatio. We classify ad aalyze the types of costraits i the ILP watermarkig domai ad show how ILP formulatios provide more degrees of freedom for embeddig sigatures tha other existig approaches. To demostrate the effectiveess of the proposed ILP watermarkig techiques, the geeric discussio is further cocretized usig two examples, amely Satisfiability ad Schedulig. Categories ad Subject Descriptors K.6.5 [Maagemet of Computig ad Iformatio Systems]: Security ad Protectio -- Digital Watermarkig Geeral Terms Algorithms, Ecoomics, Theory, Legal Aspects. Keywords Digital Watermarkig, Itellectual Property Protectio. INTRODUCTION Recetly, the advet of modularized hardware ad software libraries ad core-based system-o-chip (SOC) desig methodologies fueled a flurry of research focused o itellectual property (IP) protectio (IPP). Oe of the techiques that has bee show to be extremely effective for IPP is watermarkig (WM) due to its potetial simplicity, strog authorship proofs, flexibility, ad low overhead. Watermarkig techiques utilize the fact that geerally, tasks used i IP desig rely o computatioally itractable optimizatio problems havig umerous solutios of similar quality. By itroducig additioal costraits correspodig to the author s sigature, they limit the solutio space ad hece make a covicig authorship proof if a particular solutio ca be show to belog to the costraied solutio space. Numerous Watermarkig methods have bee proposed for differet stages of the desig process, icludig system sythesis, logic sythesis, techology mappig, ad physical desig [Kah98, Ho95, Meg00]. Liear programmig is a efficiet method for miimizig a liear objective fuctio uder liear iequality costraits. Iteger liear programmig (ILP) is a special form of liear programmig with Permissio to make digital or hard copies of all or part of this work for persoal or classroom use is grated without fee provided that copies are ot made or distributed for profit or commercial advatage ad that copies bear this otice ad the full citatio o the first page. To copy otherwise, or republish, to post o servers or to redistribute to lists, requires prior specific permissio ad/or a fee. DAC 00, Jue 0-4, 00, New Orleas, Louisiaa, USA.. Copyright 00 ACM /0/0006 $5.00. the additioal restrictio that variables are itegers. While ay liear programmig istace ca be solved i polyomial time, solvig ILP is NP-hard. Still, umerous effective techiques ad packages have bee developed that solve fairly large ILP istaces i reasoable ru times. The popularity of ILP as a backboe to may optimizatio algorithms has prompted the eed for effective ILP watermarkig techiques. I this paper we classify ad discuss a geeric set of techiques that ca be used to watermark the solutios of ILP formulatios. I additio to the geeric-level discussio, we illustrate the methods usig two specific problems, amely Satisfiability (SAT) ad Schedulig. I additio to their wide rage of applicatios, these two problems are well suited to complemet our discussios sice SAT is decisio orieted while Schedulig is a optimizatio problem. Although problem-specific watermarkig techiques have already bee proposed for both SAT ad Schedulig, we show the effectiveess ad power of ILP watermarkig techiques ad discuss why they ca be superior... A ILP Watermarkig Example To motivate the ILP watermarkig discussio, we use a very simple SAT example. We watermark the solutio of the simple istace by embeddig sigature costraits i its correspodig ILP formulatio i two ways. Cosider the five-variable formula: F = (x + x + x 3 ) (x 4 + x 5 ) There are a total of possible satisfyig truth assigmets out of the 3 total assigmets for F. We assume that ay of the valid solutios are equally likely to occur so the probability that a specific solutio is selected as the fial aswer is /3. The ILP formulatio correspodig to the SAT istace represeted by F ca be stated as: Miimize: Where: x + x + x 3 x 4 + x 5 For the sake of brevity, we igore the complemeted variables ad assume that all variables are defied as 0- itegers. A watermark is essetially a set of characteristics imposed i the fial ILP solutio by limitig the solutio space. The followig formulatio demostrates oe such costrait usig the ILP objective fuctio: Maximize: x - x - x 3 + x 4 - x 5 Where: x + x + x 3 x 4 + x 5 This ew ILP formulatio still represets the origial SAT istace correspodig to F. The ew objective fuctio sigature here is obtaied usig the pseudo-radom bit strig 000 (see 5.). The ew solutio space has bee reduced to out of the origial valid assigmets, a idicatio of the WM stregth. Note that for this example, this is the strogest possible watermarkig solutio. Aother method for imposig watermarkig costraits i ILP formulatios is the use of augmeted variables. Cosider the additio of a ew iteger variable x a i the origial formulatio:

2 Maximize: x a Where: x + x + x 3 - x a x 4 + x 5 - x a x a 0 It is easy to verify that the ew formulatio has 3 solutios of equal quality. Such use of augmeted variables i order to impose ew ad complex relatioships amog urelated costraits is oe of the mai advatages of usig ILP WM. Previously proposed WM methods do ot provide the flexibility ad power of ILP techiques to impose complex relatioships amog groups of costraits ad variables. Geerally, practical solutio istaces are much larger ad provide a rich eviromet for embeddig log sigatures by simultaeously applyig several differet WM techiques... Paper Orgaizatio The remaider of this paper is orgaized as follows: Sectio cotais the related work. Sectio 3 cotais the prelimiary discussio icludig defiitios, otatios, ad assumptios. Sectio 4 presets the geeric ILP watermarkig techiques ad costrait classificatios. Sectios 5 ad 6 preset the details of ILP watermarkig specific to SAT ad Schedulig with supportig experimetal results ad aalysis, followed by the coclusio.. RELATED WORK Imperfectios of the huma visual ad auditory system have bee used as the basis for hidig data i still images, audio, ad video [Bor96]. However, watermarkig IP where o errors are allowed, such as desigs ad programs, require differet techiques. Protocols for watermarkig active IP have bee developed at the physical level [Kah98], techology mappig ad template matchig [Meg00], ad behavioral sythesis [Ho95]. Qu et al. address the questios of the quality of the solutio subspace ad the difficulty of fidig a solutio from such subspace i [Qu00]. Boolea Satisfiability ad Schedulig have umerous applicatios i CAD ad other domais. May desig problems such as test patter geeratio, circuit delay computatio, logic optimizatio, ad combiatioal equivalece checkig ca be formulated as SAT problems [Mar00]. A example of a heuristics for solvig SAT problem is preseted i [Bay96]. Summery of importat classical schedulig theory results for real-time computig ca be foud i [Sta95] ad [Pau89]. A example of a ILP-based formulatio for Schedulig is preseted i [Geb9]. 3. PRELIMINARIES 3.. SAT Let U={u, u,, u m } be a set of Boolea variables. A clause over U is a set of literals that represets their disjuctio. A clause is satisfied by a truth assigmet if ad oly if at least oe of its members is true uder that assigmet. The Satisfiability problem is hece defied as [Gar79]: Problem: SATISFIABILITY (SAT) Istace: A set U of variables; A collectio C of clauses over U. Questio: Is there a satisfyig truth assigmet of C? Usually SAT problems are stated i the from of a formula. Sectio. cotais a example of a SAT formula ad its correspodig ILP formulatio. I our otatio, x i idicates the logical complemet of the variable x i. For simplicity, we use the same ames to represet the Boolea SAT variables ad the ILP variables. We assume that all SAT ILP variables are defied as 0- itegers uless otherwise specified. 3.. Schedulig We use the sychroous data flow (SDF) as our primary computatioal model [Lee87]. SDF is a special case of geeral data flow, i which the umber of data samples produced or cosumed by each ode is specified a priori. The sytax of a targeted computatio is defied as a hierarchical cotrol-data flow graph (CDFG) [Rab9]. The CDFG represets the computatio as a flow graph, with odes, data edges, ad cotrol edges. The sematics uderlyig the sytax of the CDFG format is that of the SDF flow model. Operatio schedulig is the process of partitioig the set of operatios i a CDFG ito groups such that the operatios i the same group are executed cocurretly i the same cotrol step. The task of operatio schedulig is to determie the amout of resources ecessary for executio of operatios i a CDFG. The two geeral approaches for schedulig are heuristic-based methods [Pau89] ad ILP [Geb9]. I this paper, we restrict our attetio to ILP for time-costraied schedulig, where for a give umber of cotrol steps, the set of arithmetic ad logical operatios i a CDFG is partitioed such that each partitio correspods to a sigle cotrol step. The goal of schedulig here is to miimize the umber of fuctioal uits ecessary for the executio of all operatios i the CDFG. The problem ca formally be stated as: Problem: Time-Costraied SCHEDULING of a CDFG. Istace : A CDFG C, a time costrait t, parameter. Questio: Does there exist a Schedulig of C ito t cotrol steps, such that the umber of operatios from C i each cotrol step is less tha ad all precedece costraits i C are satisfied? The problem of Time-Costraied Schedulig of a CDFG is computatioally itractable as it is essetially the same problem as precedece costraied schedulig, a well kow NP-complete problem [Gar79]. To preset the ILP formulatio of the timecostraied schedulig problem [Geb9], we defie variables x ij as follows: x = if operatio j is scheduled i cotrol step i. i, j 0 Otherwise. i, j m, where is the umber of the give clock cycles ad m is the umber of operatios i the CDFG. The coditio that each operatio has to be scheduled i exactly oe cotrol step is expressed through the equatio: x i, j = () j This coditio is usually replaced with a tighter coditio takig the earliest ad the latest possible times whe a operatio ca be scheduled ito cosideratio. l x i, j = k () j where k = ASAP(x i,j ) i, ad l = ALAP(x i,j ) i. ASAP(x i,j ) ad ALAP(x i,j ) refer to as soo as possible ad as late as possible heuristic algorithms for schedulig. The data flow i the CDFG is restricted with precedece costraits, formulated as: (3) ( ( i + )) xi j + ( ( i + )) xi, j, where j is the operatio i the CDFG whose output is used as a iput by the operatio j. We also defie variables x i as: m (4) i x i = x i, j j=

3 The variable x i represets the actual umber of fuctioal uits for each cotrol step i. The problem of miimizig the umber of fuctioal uits ecessary for executio of the CDFG uder the above costraits ca ow be formulated as: Miimize: y Where: x i < y i This defiitio ca trivially be exteded to the allocatio problem, separatig the coditios for umber of fuctioal uits to differet types of uits. 4. GENERAL ILP WATERMARKING The global flow for ILP watermarkig is show i Figure. The mai strategy is to add a umber of additioal costraits ad try to satisfy as may of them as possible. The watermarkig costrait selectio is made accordig to a pseudoradom umber geerated i correspodece with a sigature. The costrait ecodig procedure yields sufficietly radomized costraits to hider the detectio ad forgig of sigatures. Figure shows the ecodig process: SHA is a oe-way hash fuctio, RSA is a public key ecryptio system, ad RC4 is a stream cipher (SHA ad RSA are used i the PGP software package). RC4 outputs a cryptographically strog pseudoradom bit-stream that the costrait ecodig step uses to geerate the WM costraits. Verificatio of a sigature ivolves both testig for sigature presece ad matchig the sigature to the text file of the alleged ower. Demostratig that the sigature is i the solutio is doe by showig a uusual presece of eough WM costraits. A sigature ca be show to correspod to the text file ad the ower s public key usig PGP. The types of methods used to embed watermarks i ILP formulatios have three mai classificatios: (i) augmeted variable defiitio, (ii) objective fuctio, ad (iii) ILP costraits. I the followig subsectios, we provide a overview of each of these methods. 4.. Variable Defiitios The variables used i ILP formulatio of optimizatio ad decisio problems fall ito two categories: (i) atural ad (ii) augmeted. Natural variables are variables that directly represet aspects of the origial problem. Augmeted variables o the other had, have o apparet relatioship to the problem ad ofte do ot have ay physical meaig. The ability to defie ad use augmeted variables i ILP formulatios is a powerful method of imposig additioal costraits i order to improve watermarkig stregth, which is ofte ot possible with other methods. Augmeted variables eable the impositio of complex relatioships amog the atural variables ad costraits. These ew depedecies are key factors that make the ILP based watermarkig methods more powerful tha previously proposed algorithmspecific techiques. Sectio 5 ad 6 provide several examples to demostrate how augmeted variables ca be used to impose specific watermarkig relatioships amog idepedet costraits. 4.. Objective Fuctio Geerally, each ILP formulatio is composed of a objective fuctio F 0 ad a set of costraits. Ay umber of additioal objective fuctios ca be used for sigature ecodig. The geeral objective fuctio F WM used i WM ILP has the followig structure: F WM = α 0 F 0 + α F + α F + α 3 F 3 + where F i (i>0) deote augmeted watermarkig objective fuctios ad coefficiets α i are real valued costats. Sigature (biary data) Figure. Global Flow For ILP Watermarkig PGP Text File SHA Hash RSA Bitstream Problem Istace ILP Formulatio Costraits Watermarkig Modified Costraits ILP Solver Watermarked Solutio Private Key Costrait Ecodig Seed File Costraits RC4 Figure. ILP Watermarkig - Costrait Ecodig Note that i optimizatio itesive applicatios, α 0 >>α i for i>0 sice the quality of the fial solutio must ot be allowed to degrade. The augmeted objective fuctios F i ca be ay arbitrary liear fuctio of the atural ad augmeted ILP variables. Problem specific attributes ca guide the costructio of such fuctios i order to improve WM stregth ad preserve solutio quality. Certai decisio problems such as SAT, do ot have ay iheret objective fuctios i their correspodig ILP formulatio ad hece provide the most flexibility for embeddig such sigatures ILP Costraits Modifyig the origial ILP costraits is the third ad most broad method of embeddig WM costraits. To esure solutio correctess, almost all such techiques maipulate the origial set of ILP costraits such that the ew resultig costraits are more restrictive. As the case with ILP variables, ILP costraits ca be either atural or augmeted. Natural costraits arise from the problem specificatio ad care must be take to esure that they are ot violated as a result of the watermarkig process. Augmeted costraits do ot geerally have ay obvious relatio to the problem at had ad ca provide great freedom i embeddig sigatures. Furthermore, each type of costrait ca be classified as positive or egative. A positive costrait specifies the value or rage of values that each variable should be assiged to, while a egative costrait specifies udesirable assigmets. 5. ILP WATERMARKING SAT 5.. SAT WM: Variable Defiitios Normally, there is little freedom i redefiig the atural set of variables i SAT. However, augmeted variables allow the impositio of complex relatioships amog the existig ad/or augmeted costraits. Cosider the followig costraits: x + x + x 5 + x 9 + x 3 x 3 + x 7 + x + x 7 Defie augmeted iteger variable x a ad modify costraits:

4 x + x + x 5 + x 9 + x - x a = 3 x 3 + x 7 + x + x 7 - x a = x a 0 I this simple example, the ew variable x a defies a ew relatioship betwee the two origial costraits. So, although previously each costrait could be satisfied idepedetly, here, there is a ew uderlyig depedecy that relates seemigly urelated costraits. This otio ca further be used to iclude multiple sets of costraits ad multiple augmeted variables. However, care must be take to esure that the additioal costraits do ot cause a satisfiable formula to become usatisfiable ad vice versa. 5.. SAT WM: Objective Fuctios As stated earlier, SAT is purely a decisio problem ad hece does ot pose ay iheret objective fuctios. This additioal degree of freedom provides powerful meas of ecodig SAT watermarkig costraits. For istace, cosider a SAT formulatio with 0 variables ad a 0-bit pseudo-radom bit sequece (sigature) S: S = 0000 Defie a ew SAT ILP objective fuctio F by mappig each variables x i to a bit s i S. We add the variable x i if the correspodig bit b i = ad subtract x i if b i =0: F = x - x + x 3 + x 4 - x 5 - x 6 + x 7 x 8 + x 9 + x 0 Maximizig F i the ILP i effect is a attempt to preserve the sigature S i the form of variable assigmets. Here, if all variables are assiged accordig to the bits i S, the the WM stregth is maximized. However, such a assigmet may ot be a valid solutio. The scheme ca be ehaced usig the first- ad secodorder heuristic objective fuctios preseted i [Qu99]. The fuctios essetially calculate the likelihood of particular assigmets for each variable i a valid solutio. This likelihood iformatio ca be used to improve the mappig process of the sigature strig to the WM objective fuctio, thus icreasig the WM stregth. Specifically, the heuristics ca be used to decide if each bit b i S should be mapped to the variable x i or its complemet x i (+ or i the objective fuctio) i order to icrease the likelihood that assigig variables accordig to S would result i a valid solutio SAT WM: ILP Costraits Let us ow cosider ILP costrait maipulatio techiques for watermarkig. Each atural costraits i the ILP correspods to a clause i the SAT formula. Cosider the costrait: x + x + x 5 + x 9 + x We ca reduce the solutio space by removig variables from such costraits. This has the same effect as deletig literals from correspodig clauses that is discussed i [Qu99]. For example, we ca replace the costrait above with the ew costrait assumig that the ew costrait does ot make the formula usatisfiable: x + x 5 + x 9 We ca further limit the solutio space by addig ew augmeted ILP costraits. I additio to addig ew costraits similar to the oes above (see addig clauses i [Qu99]) we may opt to impose special characteristics o groups of variables such as: x 3 + x 4 + x 9 + x + x 7 3 x + x 5 + x 6 + x + x 0 4 x + x + x 5 + x 9 Augmeted costraits ca be combied with augmeted variables to produce eve more sigature embeddig optios SAT Experimetal Results We compared the performace of the ILP WM approach to the optimizatio itesive SAT-solver results preseted i [Qu99]. We use the same set of DIMACS bechmark istaces. Table lists the DIMACS bechmarks ad the umber of variables ad clauses for each istace. The Opt. Bits colum shows the umber of watermarked bits as reported i [Qu99]. For each istace, we embedded a comparable size radom sigature i each istace ad used LP_SOLVE, a freely available (GNU) mixed iteger liear program solver. I all cases, the ILP solver produced the watermarked results with the ru-times reported i miutes i colum six, o a 333 MHz Su Ultra 5 Workstatio with 8 MB RAM. It is expected that larger istaces ca be watermarked equally effectively usig more sophisticated commercial ILP solvers. Table. SAT WM: ILP vs. Optimizatio Itesive Techique Istace Vars Clauses Opt. Bits ILP Bits ILP Ru time ii8a.cf ii8a.cf ii8a3.cf ii8a4.cf ii8b.cf ii8b.cf ii8c.cf ILP WATERMARKING SCHEDULING Formulatig schedulig as a ILP problem opes ew dimesios of imposig watermarkig costraits that are impossible or difficult to achieve usig other methods. I this sectio we preset a few examples to demostrate how geeric ILP watermarkig ca apply to Schedulig. 6.. Schedulig WM: Variable Defiitios Defie augmeted variables x i,j that satisfy the equatio l = i k i j x, = ad impose a umber of precedece costraits expressed usig equatios of type (3). The ew variables are ot added to equatios for variable x i as they are ot affectig the quality of the fial solutio. Here, we exploit a degree of freedoms specific to ILP formulatios of schedulig that does ot have a physical iterpretatio. Defie a augmeted variable x i correspodig to the additio of a ew cotrol step. The direct cosequece is the quality of the fial solutio. As show i Table, this techique does ot reduce the umber of solutios ad it should be used i cojuctio with other techiques. However, the feasible solutios are altered as a cosequece of the costraits ad which ca serve as a proof of authorship. The oly exceptio is whe this techique eables the reductio of umber of fuctioal uits ecessary for executio of operatios i a CDFG. Defie a augmeted variable z i, z i = xi j + xi j + K+ xig j + x g i + xi + K+ x,,, ih uder the costrait that a i z i b i,, where a i ad b i are specified positive itegers. Note that this type of costrait does ot have ay real iterpretatios i the Schedulig domai, yet provides aother degree of freedom for watermarkig solutios. May more costraits of this ature with varyig structures ca be imposed.

5 6.. Schedulig WM: Objective Fuctios Itroduce correlatio costraits for variables x i : x + x + + x = x + x + K+ x i i K ig ig+ ig+ i g+ h This techique directly affects the objective fuctio. The itroduced correlatio does ot have ay direct associatio to the schedulig problem. Maximize α, the weighted total umber of satisfied costraits, w jci = α j C, where C is the set of costraits. g Various watermarkig costraits itroduced i this sectio ca be applied aloe or i cojuctio to oe or more of the other techiques. The ILP solver may be uable to solve the problem uder all posed costraits. Here, we itroduce aother objective fuctio which will allow the ILP solver to satisfy as may WM costraits as possible, while still obtaiig a good quality solutio Schedulig WM: ILP Costraits Fix variable x i,j to : x i, j =, ad x g, j = 0 for g i. For solutio feasibility, the rage of i is i [ASAP, ALAP]. This techique correspods to fixig a operatio from CDFG to a particular cotrol step. Fix variable x i,j to 0: xi = 0, = 0,,, = 0,, j xi, j K xi h j k i i l. This costrait i effect forbids a operatio from beig scheduled i oe or more cotrol steps ad is thus a egative costrait. Eforce miimum distace betwee two variables with existig precedece costrait: ( ( i + )) x i, j dmi + ( ( i + )) x i, j Miimum distace d mi, where d mi is a positive iteger, represets the miimum umber of clock cycles that j is scheduled before j. Eforcig maximum distace betwee two variables with existig precedece costrait: The costraits i this case have essetially the same form as the previous case with the exceptio that the positive iteger variable d max is used istead of d mi ad the iequality is istead of. Variable d max represets the maximum umber of clock cycles that j is scheduled before j. Eforcig miimum distace betwee two variables without existig precedece costrait: ( ( i + )) x i, j dmi + ( ( i + )) x i, j Variables that do ot have explicit precedece costraits but have implicit precedece costraits as a result of trasitivity of precedig relatios are costraied by this techique i such a way that the implicit order is preserved. Eforcig maximum distace betwee two variables without existig precedece costrait: Implicit precedece costrait restricts this techique to cases where d max is greater tha the critical path betwee j ad j Impacts of WM o Solutio Space Ecodig the sigature ito costraits is the fial step i embeddig the sigature i the ILP formulatio. This step relies o the selectio of WM techiques ad their associated parameters. We use the first block of bits of a sigature bit-stream for techiqueselectio ad the rest for b c parameter selectio. For each of the preseted a C h d e f g techiques, we attempt to C embed the WM such that i C j C 3 C 0 after satisfyig all costraits, there is still a C 4 C 6 C 5 o-zero probability that a valid solutio exists. For C 8 C 7 example, if a variable x i,j is C 9 beig fixed to, the applied procedure must x y z choose i i the rage Figure 3. A CDFG Example [ASAP, ALAP]. Similarly, if the critical path i the CDFG betwee two variables is more tha d max while eforcig a maximum distace costrait, the the solutio to the problem becomes ifeasible. Due to the itractability of Schedulig, the proposed procedure caot guaratee that the solutio will exist. Thus, such decisios almost always rely o problem specific heuristics. Oe attempt at such a heuristic is the α maximizatio techique preseted i Sectio 6., that relaxes the costraits ad eables easier solutio search for the ILP solver. The weights for each costrait are iversely proportioal to the ratio of their resultig average solutio space reductio. Cosider the CDFG show i Figure 3, where variable x i,j correspods to operatio C j ad variable x i correspods to the cotrol step i. Suppose that the required umber of cotrol steps for the executio of this CDFG is 8. Here, the miimum umber of fuctioal uits required for executio is 3. Table lists the correspodig ASAP ad ALAP assigmet for each operatio. Table. ASAP/ALAP Schedulig for the CDFG i Figure 3. OP. C C C 3 C 4 C 5 C 6 C 7 C 8 C 9 C 0 C ASAP ALAP There are 44,75 valid operatio-to-cotrol-step (schedulig) assigmets of the CDFG i Figure 3. Let us cosider the 36-bit sequece 39D6B4C86 hex as our sigature. If we oly use the fix variable x i,j to techique the our ecodig will produce: x 7,8 =; x i,8 = 0, i 7 x 6, =; x i, = 0, i 6 x,5 =; x i,5 = 0, i x 7,9 =; x i,9 = 0, i 7 We use the first 5 bits of the sequece for selectio of the particular variable x i,j. Idex j is obtaied by the formula j = ( code) mod +, where code refers to the decimal umber correspodig to the first 5 bits of the sequece. The reaso for usig 5 bits istead of 4, the miimal umber, is to make the selectio as fair as possible [Qu00]. Usage of oly 4 bits would make the likelihood of selectig the first 6 variables twice as large as the rest. Idex i is selected similarly, usig 4 bits, ad ormalizig the umber to the rage of feasible solutios (i the [ASAP, ALAP] rage). If the sequece decodig results i the selectio of a variable that is already scheduled, that selectio is discarded. After applyig the first costrait, the possible umber of solutios is reduced to,55, after the secod costrait the umber of solutios drops to,793, ad after the ext two costraits, to 9 ad 7 respectively. If we select the fix variable x i,j to 0 techique from Sectio 6.3, usig the same ecodig as i the previous case (x 7,8 =0; x 6, = 0; x,5 = 0; ad x 7,9 = 0), the umber of possible solutios is reduced at

6 a much slower rate. After applyig the first costrait, the umber of solutios drops to 3,00, after the secod costrait, to 8,437, ad after the last two costraits to 8,5 ad 4,37 respectively. Clearly, differet methods of covertig the sigature to WM costraits are possible. We ca expect that the fix variable x i,j to reduces the solutio space more tha the fix variable x i,j to 0. Due to the ature of Schedulig ILP formulatios, the positive costrait of settig a variable to forces several more to 0. The solutio space reductio caused by other techiques ca be see i Table 4. The umber of possible solutios ad miimal umber of fuctioal uits eeded for executio of all operatios of the CDFG i Figure 3 is show i Table 3, as the allowed maximum umber of cotrol cycles is chaged. For some techiques, the legth of the sigature i our example is ot sufficiet to show eough correlatio betwee the techique ad the chage i solutio space size. The impact of the remaiig techiques highly depeds o the selected variables ad the actual eforced costraits. Table 3. Impact of Icreasig Cycles o Number of Solutios Total umber of cycles available F. Uits Solutios 55 4,75 44,75 57,506,435,7 6,944 Table 4. Impact of Watermarkig o Solutio Space Techique WM Sequece [bits] Defie ew variable x i,j Fix variable x i,j = Fix variable x i,j = Mi distace betwee variables with existig precedece. 5 Max distace betwee variables with existig precedece. 6 Mi distace betwee variables without existig precedece. 7 Max distace betwee variables without existig precedece. Table 5. Schedulig WM Experimetal Results Applicatio 4K bits sigature 6K bits sigature # of Perf. Perf. Name P Ops c P Overhead c Overhead G % % Epic % % PEGWIT % % PGP % 0-78.% GSM % 0-5.3% JPEG.c % % MPEG.d % N/A N/A 6.5. Schedulig Experimetal Results To quatify the performace of the ILP-based watermarkig approach for Schedulig, we compared the performace to the localized WM methodology preseted i [Kir99]. Table 5 lists the istaces used ad the umber of operatios i each CDFG. The ext set of results are for embeddig 4K bit- ad 6K-bit radom sigatures. The P c colum shows the calculated likelihood of coicidece as defied i [Kir99]. It estimates the ratio of the umber of high quality solutios i the costraied watermarked space with respect to the o-costraied solutio space, which ca be used as a approximatio for WM stregth. Note that the exact calculatio of this ratio is itractable due to the expoetially-icreasig size of the solutio space. The performace overhead colum lists the ru-time overhead i solvig the watermarked ILP formulatio vs. the origial formulatio. I most cases, the ILP-based method was able to produce the specified WM solutios fairly rapidly. However, the limitatio of our ILP solver (LP_SOLVE) became apparet o larger istaces such as MPEG.d. 7. CONCLUSION We preseted a set of geeric watermarkig techiques for ILP solutios. We classified the geeral types of costraits i the ILP watermarkig domai ad discussed how each ca be used to embed a sigature. We also provided specific examples ad detailed discussio from the SAT ad Schedulig domais ad showed several tradeoffs ad problem specific attributes i each case. Experimetal results idicate that ILP watermarkig methods have tremedous potetials for embeddig strog watermarks with little overhead. 8. REFERENCES [Bay96] R.J. Bayardo ad R. Schrag. Usig CSP Look-Back Techiques To Solve Exceptioally Hard SAT Istaces. Priciples ad Practice of Costrait Programmig, pp.46-60, 996. [Bor96] A.G. Bors ad I. Pitas. Image Watermarkig Usig DCT Domai Costraits. Iteratioal Coferece o Image Processig, Vol.3, pp.3-44, 996. [Geb9] C.H. Gebotys, ad M.I. Elmasry. 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