Gossip Codes for Fingerprinting: Construction, Erasure Analysis and Pirate Tracing

Transcription

1 Journal of Unversal Computer Scence, vol. 11, no. 1 (005), submtted: 15/7/04, accepted: 6/1/04, appeared: 8/1/05 J.UCS Gossp Codes for Fngerprntng: Constructon, Erasure Analyss and Prate Tracng Rav S. Veerubhotla (Insttute for Development and Research n Bankng Technology, Hyderabad, Inda vrsankar@drbt.ac.n) Ashutosh Saxena (Insttute for Development and Research n Bankng Technology, Hyderabad, Inda asaxena@drbt.ac.n) V.P. Gulat (Insttute for Development and Research n Bankng Technology, Hyderabad, Inda vpgulat@drbt.ac.n) A.K. Pujar (Department of Computer & Informaton Scences, Unversty of Hyderabad, Inda akpcs@uohyd.ernet.n) Abstract: Ths work presents two new constructon technques for q-ary Gossp codes from t- desgns and Traceablty schemes. These Gossp codes acheve the shortest code length specfed n terms of code parameters and can wthstand erasures n dgtal fngerprntng applcatons. Ths work presents the constructon of embedded Gossp codes for extendng an exstng Gossp code nto a bgger code. It dscusses the constructon of concatenated codes and realsaton of erasure model through concatenated codes. Keywords: codng and nformaton theory, multmeda nformaton system, dgtal lbrares Categores: H.1.1, H.5.1, H Introducton Protectng dgtal content from llegal copyng and dstrbutng s one of the key ssues worryng owners and dstrbutors n the dgtal world. Copyrghts are an accepted means of legally encouragng the creaton of works and ther rghtful explotaton. Recent developments n dgtal nformaton processng and dstrbuton have had a tremendous mpact on the creators and ther copyrghts. On the one hand, these advancements open new ways of creatng and explotng dgtal works, whle on the other, the same advancements also open up novel ways of crcumventng copyrghts. These technques can be used aganst the nterests of copyrght holders by creatng less number of copes and dstrbutng them effectvely. Therefore, new dgtal rghts protecton technques are requred, whch allow the creators and other rght-holders to enforce ther legal rghts and nterests. The overall goal of these technques s to ensure suffcent ncentves for creatng works and makng them avalable n the dgtal form.

2 Veerubhotla R.S., Saxena A., Gulat V.P., Pujar A.K.: Gossp Codes Fngerprntng dgtal objects help n checkng copyrght volatons n the dgtal world. The scope of dgtal fngerprntng also ncludes pay per vew, pay per use and other broadcast applcatons. Dgtal fngerprntng refers to the act of embeddng a unque dentfer n a dgtal object, n such a way that t s dffcult for others to fnd and destroy the dentfer. The fngerprnts are typcally embedded nto the content usng watermarkng technques, whch are desgned to be robust to a varety of attacks. Ths markng makes every user s copy unque, whle stll beng close to the orgnal. It allows the dstrbutor to detect any unauthorsed copy and trace t back to ts orgn. Snce a marked object can be traced back to the owner, users wll be deterred from releasng an unauthorsed copy. A cost-effectve attack aganst such dgtal fngerprnts s colluson, where several dfferently marked copes of the same content are combned to dsrupt the underlyng fngerprnts. A coalton of users may detect the marks by comparng multple fngerprnted objects and modfy or erase them. In ths process, prates may further try to frame nnocent users. To prevent ths, Boneh and Shaw [Boneh and Shaw 98] ntroduced frameproof codes n whch the prates cannot frame nnocent users outsde ther coalton. Moreover, t s necessary for the dstrbutor to fnd at least one user nvolved n creatng a prate copy. Identfable Parent Property (IPP) codes, ntroduced by Hollman et al. [Hollman, van Lnt, Lnnartz and Tolhu-zen 98] exactly for ths purpose, can dentfy at least one parent of the prate word. Non-bnary (alphabet sze q > ) IPP codes can handle a large colluson sze c ( > ) [Barg, Cohen, Encheva, Kabatansky and Zémor 01] unlke bnary IPP codes. In [Trappe, Wu, Wang and Lu 03], the authors nvestgate the problem of desgnng fngerprnts that can wthstand colluson attacks and allow the dentfcaton of colluders usng Ant-Colluson codes (ACC), whch were orgnally proposed n [Trappe, Wu, Wang and Lu 0]. Ther work addresses the colluson problem consderng the addtve embeddng and then study the effect that an averagng colluson has upon orthogonal modulaton. Ther fngerprntng scheme s based upon code modulaton and the code constructon requres only O( M ) orthogonal sgnals n order to accommodate M users, for a gven number of colluders. The work presents four dfferent detecton strateges that can be used wth ACC for dentfyng a suspect set of colluders. Traceable schemes are cryptographc systems that provde protecton aganst llegal copyng and redstrbuton of dgtal data. A closely related concept s traceablty systems ntroduced by Chor et al [Chor, Fat, Naor 94] used n the context of broadcast encrypton schemes. Broadcast encrypton systems [Fat, Naor 93], [Stnson, We 99] allow targetng of an encrypted message to a prvleged group of recevers. Each recever has a decoder wth a set of keys that allows hm to decrypt the encrypted messages f he s a member of the target group. When a group of colluders (up to c ) construct a prate decoder to decrypt the content, broadcast encrypton systems can dentfy at least one of the colluders who contrbuted to the prate decoder. In the tracng mechansm proposed n [Stnson and We 98], the merchant fnds the numbers of common keys between the prate decoder and all the authorsed decoders. The merchant then chooses the decoder wth the maxmum number of common keys wth the prate decoder, whch exposes a culprt. These schemes are known as Traceablty schemes and are also called key fngerprntng schemes, whch allow tracng of one of the colluders. In a Traceablty Scheme [Chor,

3 14 Veerubhotla R.S., Saxena A., Gulat V.P., Pujar A.K.: Gossp Codes... Fat, Naor 94], [Stnson and We 98], [Kurosawa, Desmedt 98], [Kurosawa, Desmedt 98],[Garay, Staddon, Wool 99], each authorsed user has a decoder wth a set of k keys, from a base key set X of sze v that unquely determnes the owner and allows hm to decrypt the encrypted broadcast. Ths work present the constructon methods for non-bnary IPP codes, namely, c- Gossp codes [Lndkvst 01] from t -desgns, Traceablty schemes and vce versa. The constructons attan the mnmum possble code length specfed for Gossp codes, n terms of alphabet sze q, number of codewords M and colluson sze c. We descrbe the constructon of embedded Gossp codes and also generalse Gossp codes. We dscuss the prate tracng mechansm for these codes n presence of erasures. We also present a constructon method and an analyss of Concatenated codes. The organzaton of the paper s as follows: Secton presents the prelmnares, namely Traceablty schemes, t- desgns, fngerprntng codes and related work. In secton 3, we descrbe the constructon of c- Gossp codes wth shortest possble length from c -Traceablty schemes, t -desgns and vce versa. Ths secton also presents the constructon of embedded Gossp codes and generalsed Gossp codes. The secton 4 models erasures n Gossp codes, secton 5 analyses Gossp codes and concatenated codes followed by the conclusons n Secton 6. Prelmnares Ths secton presents the background and related work on fngerprntng. Defnton.1 Consder a code Γ of length N on an alphabet Q wth Q = q whose symbols are denoted by {0,..., q 1}. Then N Γ Q and we wll call t an ( N, n, q) -code f Γ= n. The elements of Γ are called codewords. Each codeword s gven by x = ( x1,..., x N ), where x Q, 1 N. For any subset of codewords C Γ, we defne the set of descendents of C, denoted D( C ) by N D( C) = { x Q : x { a : a C}, 1 N}. (see [Stnson and We 98]) The set D( C) conssts of N -tuples that could be produced by a coalton holdng the codewords of the set C..1 Gossp codes Gossp codes are fngerprntng codes ntroduced n [Lndkvst 01] whch can provde protecton aganst llegal copyng of dgtal objects. They are also c -IPP codes, whch can dentfy at least one user nvolved n creatng an llegal copy. We frst explan the constructon of Gossp codes orgnally presented n [Lndkvst, Löfvenberg, and M. Svanström 0]. Let B( M, q) be the 0/1- matrx consstng of l columns and M rows such that each column s created by placng q 1ones and M q + 1 zeros. The parameters q and M are so chosen satsfyng q 3 and M q + 1. The codeword matrx G( M, q) s

4 Veerubhotla R.S., Saxena A., Gulat V.P., Pujar A.K.: Gossp Codes constructed from B( M, q) by replacng the q 1 ones n each column, wth the q 1 dfferent non-zero symbols of Q and retanng the zeros unaltered. The code matrx G( M, q) s called a Gossp code and each code column s known as a gossp column. In a gossp column, the occurrence of non-zero alphabet symbols (among themselves) s mmateral as long as they are dstnct. We denote G( M, q) as c- Gossp( l, M, q ) code where c s colluson sze, M s the number of code words, q s the alphabet sze and l s the length of the code. Wth the above constructon method a -Gossp (7, 7, 4) code constructed s as follows. Example.1.1 -Gossp (7, 7, 4) code Defnton.1.1 A gossp column s a code column of a q-ary code (Gossp code), wth all non-zero symbols of the alphabet Q appearng exactly once, and the symbol zero n the remanng postons. (see [Lndkvst 01]).1.1 Tracng usng gossp columns Consder the 6 th column of the Gossp code presented n e.g In ths gossp column the non-zero symbols are dstnct and appear at postons 3 rd, 5 th and 6 th. Thus ths gossp column s responsble for tracng users {3 rd or 5 th or 6 th } (based on 6 th poston n the prate word) f they are nvolved as trators. The colluson groups whose members can be traced (accused) by a gossp column are known as accusaton groups of that column. Thus ths gossp column accuse the colluson groups {3, 5}, {5, 6} and {3, 6} of colluson sze c =. The remanng colluson groups are accusaton groups for some other columns. The colluson groups {3, }(for =1,, 4 or 7) certanly can create a prate word wth a one at that column's poston. But the tracng of these groups s left over to other gossp columns. Thus a gossp column contrbutes to tracng trators nvolved n creatng a prate word. Lemma.1.1 A Gossp code s the code wth a gossp column for each colluson group to trace. (see [Lndkvst 01]) The tracng method s based on the fact that every llegal fngerprnt contans at least one non-zero symbol. If a partcular colluson group s traceable by a gossp column, then all ts subgroups are also traceable by the same gossp column snce they wll have a subset of possble alphabet symbols. If there exsts a gossp column for each c -group, t s possble to trace at least one member of all prate groups whose

5 16 Veerubhotla R.S., Saxena A., Gulat V.P., Pujar A.K.: Gossp Codes... sze s less than or equal to c. So, the Gossp code wll always fnd a member of the prate group and thus t s a c-ipp code. Defnton.1. The set of non-zero postons of a gossp column s termed as column key correspondng to that column. In e.g..1.1 the column key for the 6 th column s {3, 5, 6}. Column keys of Gossp code dentfy the set of prate groups that each gossp column s capable of q 1 accusng. From a gossp column of a Gossp code, where c < q < M, exactly c colluson groups are traceable. If M s the total number of codewords n a Gossp M code, then the total number of possble prate groups of sze c s gven by c. For q 1 c q 1, each gossp column at the most can accuse groups snce there are c ( q 1) non-zero symbols, whch are dstnct. Hence, n a Gossp code the total M q 1 number of gossp columns l c c for c < q < M. Ths gves a lower bound on length of the Gossp code. If the Gossp code length attans the equalty condton t has the shortest possble length. Hereafter we denote ths shortest length by l. The structure of Gossp code s such that every non-zero symbol n the prate word wll lead to one culprt. (see Lemma 4.1.1) Snce Gossp codes are q ary IPP codes and come up wth a determnstc tracng algorthm unlke q ary c-secure code wth error probablty ε [Boneh and Shaw 98], they have a sgnfcant role to play n fngerprntng applcatons.. Traceablty Schemes A c -Traceablty scheme c- TS( k, b, v ) [Stnson,We 99], [Stnson and We 98] s a broadcast encrypton scheme, where k s the number of keys provded to each user, b s the total number of users, v s the total number of base keys and c s the colluson sze of prates that the scheme can wthstand. The Trusted Authorty generates a set T of v base keys and assgns k keys chosen from T to each user. th The user s personal key or prvate key s denoted by P () that unquely determnes the owner and allows hm to decrypt the encrypted broadcast. The broadcast message conssts of an enablng block E and a cpher block Y. The cpher block s the encrypton of the actual plantext data m usng a secret key a. That s, Y = ξa ( m), where ξ () s the symmetrc encrypton functon for some cryptosystem. The enablng block conssts the shares of a, whch are encrypted usng some or all of the v keys n the base set T. The decrypton of enablng block wll allow the recovery of the secret key a. Every authorzed user should be able to recover a usng hs personal key and then decrypt the cpher block usng a to obtan the plantext data.e., m = ζ ( Y ), where ζ () s the decrypton functon for the a

6 Veerubhotla R.S., Saxena A., Gulat V.P., Pujar A.K.: Gossp Codes cryptosystem. A colluson of users may conspre and gve an unauthorzed user a prate decoder F. Ths prate decoder wll consst of a subset of base keys such that F P(), where W s the coalton of trators. Once a prate decoder s found, the W broadcaster can trace those who have partcpated n producng the prate decoder. Trator detecton wll be done by computng F PU ( ) for all users U. If F PU ( ) F PV ( ) for all users V U, then U s defned to be exposed user. Defnton..1 An encrypton scheme c- TS( k, b, v ) s called a c -Traceablty scheme, whenever a prate decoder produced by a coalton C and C c, the exposed user U s a member of the coalton. Suppose X s a set and B s a famly of k -subsets of X where each k -subset s called a block. A Traceablty scheme c- TS( k, b, v) can be consdered as a set system ( X, B) where each block corresponds to a decoder wth k keys from the key set X wth the followng property. Theorem..1 There exsts a c- TS( k, b, v) f and only f there exsts a set system ( X, B) such that X = v, B = b and P = k for every P B, wth the property that for every choce d c blocks B 1,..., Bd B and for any k -subset d F B, there does not exst a block P B { B1,..., B d } such that F B > F P for 1 j d. ([see Stnson and We 98]) Defnton.. A t-( v, k, λ) desgn [Colbourn, Dntz (96)] s a set system ( X, B ), where X = v, B = k for every B B, and every t - subset of X occurs n exactly λ blocks n B..3 Related work and known bounds The constructon of frameproof codes from Traceablty schemes s presented n [Stnson and We 98]. In [Safav-Nan and Wang 01] lower bounds on maxmum number of codewords for frameproof codes and Traceablty schemes are presented. Gafn et al [Gafn, Staddon, Yn 99] presented a method for addng any desred level of broadcastng capablty to any Traceablty scheme and vce versa. Wth regard to secure codes Boneh et al [Boneh and Shaw 98] presented an explct constructon for colluson secure codes. Ther code has a length of ( 3 log ( )) 1 O c ε and attans securty aganst coaltons of sze c wth ε error. Pekert et al [Pekert, Shelat, Smth 03] demonstrated (on lower bound of code length) that no ( ) secure code can have a length of log 1 ( ) j= 1 o c ε. Guth and Ptzmann [Guth, Pftzmann 03] constructed bnary c -secure codes wth ε error under weak markng assumpton, whch assumes that adversary can create only a certan percentage of erasures n place j

7 18 Veerubhotla R.S., Saxena A., Gulat V.P., Pujar A.K.: Gossp Codes... of embedded marks n prate fngerprnts and these erasures are (approxmately) randomly dstrbuted. In [Trappe, Wu, Wang and Lu 0] a new class of codes, called Ant-Colluson codes (ACC) are proposed whch have the property that the composton of any subset of c or fewer code-vectors s unque. Usng ths property, t s possble to dentfy groups of c or fewer colluders. Ther work presents a constructon of bnary-valued ACC under the logcal AND operaton that uses the theory of combnatoral desgns ( t - (, v k,1) desgns). For erasures n non-bnary codes Safav-Nan et al [Safav-Nan, Wang 03] gave a constructon of q -ary c -secure code that can recover the deleted marks of a shortened fngerprnt. In [Safav-Nan, Wang 03] the codeword s repeated adequate number of tmes so that at least one copy of the embedded codeword can be recovered and thereby ncreasng erasure tolerance. Gossp code s ntroduced as IPP code n [Lndkvst 01] for colluson secure fngerprntng applcatons. The Gossp code wth c = q 1 has been studed n [Lndkvst, Löfvenberg, and M. Svanström 0]. The condtons for a Gossp code to become Traceablty (TA) code are presented n [Lndkvst, Löfvenberg, and M. Svanström 0]. For c = q 1 the code G( M, q) s a constant weght code whose M 1 weght s equal to q. For q = 3 the use of ths code as an error correctng code was analysed n [Svanström 99]. Fernández and Sorano [Fernández, Sorano 0] presented concatenated fngerprntng codes wth effcent dentfcaton. In ths case the nner code use effcent decodng algorthms (Chase algorthms) that correct beyond error correcton bound of the code. 3 Constructon We present the methods to construct Gossp codes that attan shortest possble length from Traceablty schemes and t -desgns. A program DISCRETA [Betten, Laue, Wassermann 97] to compute t -desgns s avalable n publc doman. In Gossp codes that acheve the bound, the gossp columns accuse dstnct groups and thus every colluson group s an accusaton group for some gossp column or the other. Lemma 3.1 A c- Gossp( l, M, q) code s shortest f and only f each gossp column q 1 accuses dstnct colluson groups ( c -groups). (see Lndkvst 01]) c The condton for gossp code to acheve ts equalty bound of code length s that each gossp column should accuse dstnct groups. In order to have shortest length each q 1 gossp column should accuse maxmum possble groups that are dstnct. c

8 Veerubhotla R.S., Saxena A., Gulat V.P., Pujar A.K.: Gossp Codes Gossp codes from Partton method and t-desgns Gossp codes that acheve the bound consttute a partton on Ŝ the set of all user groups (colluson groups of sze c ). Ths means there exsts an equvalence relaton R, whch parttons Ŝ nto equvalence classes. Let C and C j be two colluson sets of sze c. We say C RC j ( C s related to C j ) f they are accusatons groups for the same gossp column. Let X be a set of M users. Let Ŝ be the set of all subsets of X of sze c. Let C and C j be two members of Ŝ. Consder a t - (, vkλ, ) desgn wth t = c, v = M, k = q 1 and λ = 1. We defne a relaton R as follows. C R C f and only f j C and C j are subsets of the same block n the t-( v, k, λ) desgn. It s transparent that the relaton R s reflexve, symmetrc and transtve and hence an equvalence relaton that parttons Ŝ nto equvalence classes. The constructon of c- Gossp( l, M, q) code from partton method s as follows. Let C1, C,..., Cr be the exhaustve members of the th equvalence class. We now compute the set r U = 1 C, whch gves the non-zero postons (column key) for the th gossp column. Repeatng the same process for all equvalence classes, one can arrve at all of the column keys. From these column keys the ndces of non-zero postons for each gossp column are known, and hence a c- Gossp( l, M, q) code can be constructed. Theorem A c- Gossp( l, M, q) code exsts f and only f c-( M, q 1, 1) desgn M q 1 exsts where l = c c. Proof. Consder a t-( v, k, λ) desgn wth t = c, v = M, k = q 1 and λ = 1. Then there exsts a set system ( X, B ) where X = v, B = k for every B B, and every c -subset of X occurs n exactly one block n B. Let B be the th column key of the Gossp code. Then the accusaton groups for the th gossp column are dentcal to the c -subsets of B. Snce every c -subset of X occurs n exactly one block, the accusaton groups of all gossp columns are dstnct. It s known that number of M q 1 blocks n c-( M, q 1, 1) desgn s. Thus the t -desgn c c c -( M, q 1, 1) completely defnes c- Gossp( l, M, q ) code. The converse of the theorem s straghtforward.

9 130 Veerubhotla R.S., Saxena A., Gulat V.P., Pujar A.K.: Gossp Codes... Lemma presents the condton for exstence of -Gossp codes for alphabet szes 4, 5 and 6. Lemma 3.1. presents the exstence of a Gossp code wth colluson sze 3. These lemmas are based on Theorem Lemma For 3 k 5, a - Gossp( l, M, k + 1) code exsts f and only f M 1 or k mod( k k). Proof. We have shown n Theorem that c- Gossp( l, M, q) code exsts f and only f c-( M, q 1, 1) desgn exsts. Thus - Gossp( l, M, k + 1) code exts f and only f -( M, k, 1) desgn exsts. It s known that f 3 k 5, -( M, k, 1) desgn exsts f and only f M 1 or k mod( k k) (see Chapter I. n [Colbourn, Dntz (96)]). Lemma 3.1. There exsts 3- Gossp (9 8, 8, 11) code. Proof. It s known from [Colbourn, Dntz (96)] that 3-( p + 1, p+ 1, 1) desgn exst when p s prme power. Let p = 9 then t follows that 3-(8, 10, 1) desgn exsts. Usng Theorem construct c -Gossp code from 3-(8, 10, 1) desgn. We have v = 8, c = 3, q = k+1 = 11 v q = ; = c 6 c 6 l = 9 8 It follows 3- Gossp (9 8, 8, 11) code exsts. M It may be noted that for c = q 1, c- Gossp, M, q code exsts for all values c of M. For a chosen M, q and c the exstence a shortest Gossp code s accordng to Theorem However for any chosen values of M, q and c, there exsts a Gossp code such that c < q < M, whch may not acheve the shortest length. The followng are two known results [Colbourn, Dntz (96)] about t-desgns, whch are related to ths work. Theorem 3.1. If ( X, B ) s a t-( v, k, λ ) desgn and S s any s- element subset of X, wth 0 s < t, then the number of blocks contanng S s v s k s λs = { P B: S P} = λ. t s t s. Theorem If ( X, B ) s a t-( v, k, λ ) desgn and S s any s- element subset of X, wth 0 s t, then the number of blocks does not contan any pont of S s v s v t λs = { P B: P S = φ} = λ. k k t.

10 Veerubhotla R.S., Saxena A., Gulat V.P., Pujar A.K.: Gossp Codes In Theorem we present some of the propertes of shortest Gossp codes. Theorem For a shortest c- Gossp( l, M, q) code M q 1 Length of the code l = n( M, q) = c c M 1 q Weght of the code w( M, q ) = λ 1 = c 1 c 1 M M c Dstance of the code dmq (, ) = l q 1 q 1 c q 1 Proof. From a gossp column of a Gossp code, where c < q < M, exactly c colluson groups are traceable. If M s the total number of codewords n a Gossp M code, then the total number of possble prate groups of sze c s gven by c. For q 1 c q 1, each gossp column at the most can accuse groups snce there are c ( q 1) non-zero symbols, whch are dstnct. In a Gossp code the total number of M q 1 gossp columns l c c for c < q < M. Ths gves a lower bound on length of the Gossp code. Hence the shortest Gossp code length attans the equalty condton. We have shown that (Theorem 3.1.1) a shortest c- Gossp( l, M, q) code exsts f and only f t-( vk,,1) desgn exsts wth t = cv, = M, k= q 1exsts. Lookng at the t - desgn s blocks we can tell whch group s prate words wll have 0 at a gven poston, and whch groups wll not. Let W be the colluson group nvolved n pracy. We shortly wrte W = {, j} f w and w are members of W. Consder W = 1, thus from Theorem 3.1. the number of blocks that contan W are λ 1 where λ 1 s computed wth t = cv, = M, and k = q 1 from Theorem Thus λ 1 s the number postons not equal to 0 n any codeword of a Gossp code. Dstance between two codewords s the number of postons n whch they dffer. If the dstance between any two codewords s constant, then dstance of the code wll be equal to that. Consder W =. If W X B (mples W B = φ ) and W t then the colluson group W cannot detect the th poston snce the th poston s zero n all the codewords of W. From Theorem the number of blocks that do not contan W are λ. Ths means the number of undetectable postons (zeros) for W s λ. Thus the number of undetectable postons for any colluston of sze s λ. Thus the dstance between any two code words s l λ where λ s as defned

11 13 Veerubhotla R.S., Saxena A., Gulat V.P., Pujar A.K.: Gossp Codes... above wth t = cv, = M, k= q 1. So the dstance of the code becomes M M c l q 1 q 1 c. 3. Gossp codes from Traceablty Schemes In ths secton, we present the constructon of shortest length Gossp codes from Traceablty Schemes c- TS( k, b, v ) and t-( v, k, 1) desgns. The requred condton for constructon of Gossp codes from a c -Traceablty scheme s that the number of v k prvate keys n the c -Traceablty Scheme should be b =. Stnson et al c c [Stnson and We 98] proved the exstence nfnte class of Traceablty Schemes used n our case. Theorem 3..1 presents the constructon of Gossp codes from Traceablty Schemes. Lemma 3..1 presents a condton on c -Traceablty schemes to be used n constructon of Gossp codes. Lemma 3..1 Let ( X, B) denote the set system correspondng to c- TS( k, b, v ) wth B s representng the prvate keys of c- TS( k, b, v ) and cgosspb - (, v, k+ 1) denote the Gossp code constructed from c- TS( k, b, v ). If B B j c for j, then the accusaton groups of the gossp columns are not dstnct. Proof. Assume B B j c. Then there exst at least c elements say { x1,..., x c } whch are common n both B and B j. Snce the c- group { x1,..., x c } s formed by both B and B j, the c -groups formed by B s are not dstnct. Further, the set { x1,..., x c } wll be accusaton set for both th and j th gossp columns. Hence the accusaton sets correspondng to gossp columns n the Gossp code are not dstnct. v k Theorem 3..1 If c- TS k,, v c c exsts. v k exsts then cgossp -, v, k+ 1 c c code Proof. For a c- TS( k, b, v ), the total number of c -groups that can be formed from v users s v. Then the maxmum number of c- groups that can be formed by each c prvate key contanng k elements s k. If the maxmum number of c -traceable c keys s equal to v k, then c -groups formed by each prvate key should be c c dstnct. It follows that B B j < c for j where B s correspond to

12 Veerubhotla R.S., Saxena A., Gulat V.P., Pujar A.K.: Gossp Codes ( X, B) representaton of a c- TS( k, b, v ). Let each B be equal to one column key n the Gossp code. Then the c -groups created by k elements of each B.e., k groups wll be the accusaton groups for the th gossp column. Snce c B B j < c, from Lemma 3..1 each column n Gossp code accuses k c dstnct groups. If we consder v codewords n the Gossp code, equal to the number of base v k keys n c- TS( k, b, v ), then the code s a cgossp -, v, k+ 1 code, and each c c column contans k + 1dstnct elements ( 0 to k ). Snce we have Gossp code acheves the mnmum code length. v k l = c c, the The followng theorem presents the converse of Theorem 3..1,.e., the constructon of Traceablty Schemes from Gossp codes. Theorem 3.. If M q 1 w-ts q 1,, M c c M q 1 c- Gossp, M, q code exsts then c c exsts where w= ( q ) ( c 1). Proof. Let ( X, B) be the set representaton of the Gossp code. Then we have X = M, and B = q 1 for every B B, where each block B correspond to one M q 1 column key. The number of blocks n B s equal to. Snce the Gossp c c code acheves the mnmum code length the accusaton sets of each gossp column are dstnct. Ths mples that ths ( X, B ) system represent a t-( v, k, λ ) desgn, where t = c, v = M, k = q 1, and λ = 1. M q 1 w-ts q 1,, M c c ( 1) ( 1) Ths mples there exsts correspondng (see Theorem 3. n [Stnson and We 98]) where w= k c. Lemma 3.. There exsts a Gossp v( v ) v 1 or 5 mod 0. - ( 1 0, v, 6) code for all Proof. Stnson and We (see Theorem 3.4 n [Stnson and We 98] showed that there exsts TS(5, v( v 1) 0, v) whenever v 1 or 5 mod 0. Observe that

13 134 Veerubhotla R.S., Saxena A., Gulat V.P., Pujar A.K.: Gossp Codes... v 5 = v( v 1) 0. From Theorem 3..1 t follows that Gossp( v( v 1) 0, v, 6) code exsts. 3.3 Embedded Gossp codes In many cases, the number of users n a scheme wll ncrease after the system s set up. Intally, the data suppler wll construct a scheme that wll accommodate a fxed number of users say M. If the number of users exceeds M, we need to extend the scheme that s compatble wth the exstng scheme. Such scenaros are possble n Traceablty Schemes apart from dgtal fngerprntng applcatons. In Traceablty Schemes, t s not advsable to change the keys already ssued, when the users n the system grow. Embedded Traceablty schemes extend the scheme compatble to the exstng Traceablty Scheme. Embedded Gossp codes (see Defnton 3.3.1) can be used to construct these Embedded Traceablty schemes apart from fngerprntng codes. The constructon of embedded Gossp codes s as follows. Defnton Let Γ be a c- Gossp( l, M, q) code and Γ be a c Gossp( l, M, q) code, where M < M and l < l. Suppose that for every codeword x Γthere exsts a codeword x Γ such that the frst l symbols are the same as that of x. Then we say Γ s embedded nto Γ. It s smpler to understand the embeddng n terms of set systems. Let ( X, B) and ( X, B ) be two set systems. ( X, B) s sad to be embedded nto ( X, B ) f X X and B B. Suppose ( X, B ) correspond to t-( v, k, λ ) and ( X, B ) to t-( v, k, λ) desgn. Then t-( v, k, λ ) s embedded nto t-( v, k, λ) f and only f ( X, B) s embedded nto ( X, B ). Lemma A c- Gossp( l, M, q) code can be embedded nto c- Gossp( l, M, q) code f and only f the respectve c-( M, q 1, 1) desgn s embedded n c-( M, q 1, 1). Proof. The proof follows from Theorem snce a c- Gossp( l, M, q) code s dentcal to c-( M, q 1, 1) desgn n set system representaton. As an example - Gossp (7, 7, 4) code can be embedded nto - Gossp(35, 15, 4) code. 4 Erasures n Gossp codes We consder erasures n Gossp codes that acheve mnmum possble code length. In our dscussons, erasure means a non-alphabet symbol chosen by the prates n a detected poston of the embedded fngerprnt. The prates can fnd only those alphabet symbols that match wth any one of ther copes at each detected poston.

14 Veerubhotla R.S., Saxena A., Gulat V.P., Pujar A.K.: Gossp Codes We assume that ths markng assumpton [Boneh and Shaw 98] holds good n our model. In general, the embeddng mechansm, whch encodes codewords nto dgtal objects, wll determne the type of alphabet symbols or erasures that are possble at each detected poston for creatng an llegal copy. Our model assumes that prates can create one of the alphabet symbols matchng any one of ther copes, or an erasure n the place of a detected mark. When the prates choose a symbol that matches wth a symbol n the alphabet, t s not be possble to dfferentate between the symbols chosen by the prates and the vald symbols embedded under usual fngerprntng methods. Ths may lead to errors n tracng. For ths reason, the embeddng algorthm should be so chosen that t hdes the actual alphabet from beng detected by the prates (through some encodng mechansm). We now dscuss the erasures n Gossp codes. Defnton 4.1 A Fngerprntng System wth Optonal Erasures (FSOPE) s a system where the prate controllng a group of trators may optonally choose to create erasures at the detected marks,.e., the prate may create an alphabet symbol that he can fnd or a non-alphabet symbol. FSOPE nclude the followng cases: No Erasures Selectve Erasures Only Erasures 4.1 Gossp codes wth no erasures In ths case, the prates are not allowed to create erasures, however, at each poston they are free to choose the alphabet symbols they fnd n ther copes. The tracng algorthm dentfes a culprt from the extracted codeword. The tracng mechansm s obvous when the coalton sze s equal to one. Ths s because the prate s codeword has to match wth one of the vald codewords of the Gossp code. We consder e.g...1,.e., -Gossp (7, 7, 4) code to llustrate tracng n Gossp codes wth no erasures Tracng wth no erasures 1 Let the prate set be W = { w, w } where w s the th codeword n e.g Let the prate word found n the llegal copy be x = {, 0, 0, 0, 0, 0, 0}. Clearly 1 x Dw (, w), but x also belongs to j descendent set of { w, w }. 3 j D( w, w ) where D( w, w ) s the Consder the non-zero value n the prate word x of the above example. Its poston s frst. Observe that the second codeword contans n the frst poston. We accuse the nd user as culprt because the poston of non-zero alphabet symbol and the value n the prate word match wth that of the second user s codeword. It may also be noted that the nd user s the member of both the coaltons that are capable of creatng the prate word. These codes can be constructed from Theorem and Theorem Please see the Appendx for detals.

15 136 Veerubhotla R.S., Saxena A., Gulat V.P., Pujar A.K.: Gossp Codes... Lemma Every prate word created from a Gossp code contanng at least one non-zero alphabet symbol can reveal one member of the colluson. Proof. The constructon of the Gossp code makes t clear that every non-zero symbol n the prate word can reveal one member of the colluson. Let the th poston n the prate word be a vald alphabet symbol say 1, then there exsts exactly one codeword say j th codeword, whch contans 1 n the th poston. It s clear that wthout the nvolvement of j th user, t s mpossble to create the prate word x and so the algorthm accuses j th user. The general constructon of a Gossp code that can trace all actve colluders (n no erasures case) who contrbuted to prate copy s as follows: Choose the number of codewords equal to the number of alphabet symbols.e., M = q. Let the colluson sze c be equal to q 1 and the number of gossp columns be l = q. Construct q gossp columns, where each column contans the alphabet symbols {0,..., q 1} only once. It s known from [Lndkvst 01] that when c = q 1, there exsts a Gossp code such that each column accuses one of the colluson sets. If M = q then c- Gossp(1, M, M ) code s shortest code snce all symbols are dstnct. But for concatenated codes we consder nner code wth l = q = M. Lemma A (q-1)-gossp(q, q, q) code has a determnstc tracng algorthm and the tracng algorthm accuses the actve trators. Proof. In a ( q-1)- Gossp( q, q, q) code every symbol n the alphabet appears only once n each column. So the alphabet symbol 0 also contrbutes to tracng and each q gossp column can trace prate groups. Ths leads to tracng of all the actve users c who contrbuted n creatng the prate word by usng ther legtmate copes. Example Gossp (5, 5, 5) code Gossp codes wth selectve erasures In ths case users are free to create erasures or the alphabet symbols they found n ther copes at each detected poston of the colluson set. When a prate copy s created by a colluded set the undetected postons of the colluson reman the same n the copy.

16 Veerubhotla R.S., Saxena A., Gulat V.P., Pujar A.K.: Gossp Codes Tracng wth selectve erasures Lemma holds good for all Gossp codes even n presence of erasures, thus any prate word contanng one non-zero symbol can reveal one trator. In a c- Gossp( l, M, q) code f the prates create only erasures at all detected postons then the tracng s same as n only erasures case. Lemma 4..1 The ( q-1)- Gossp( q, q, q ) code can trace all the prate words except ( eeeee,,,,... l tmes) to reveal at least one culprt. Proof. In selectve erasures case of a ( q-1)- Gossp( q, q, q ) code every alphabet symbol (ncludng 0) can dentfy a culprt. But f all the symbols are erasures nobody can be accused. In e.g...1 t can be seen that the prate word ( e, eeee,,, ) contans all erasures and so the tracng wll not yeld any prate. It wll be the same as choosng only erasures n detected postons, snce all the postons are detected for a colluson of sze c =. 4.3 Gossp codes wth only erasures An embedded poston s detected when ts value s found dfferent n two copes durng comparson. In only erasures case, the prates are allowed to create only erasures n the detected postons and cannot choose any vald alphabet symbol. In only erasures case (of any shortest Gossp code) t s always true that each prate group can create only one prate word. Let c- Gossp( l, M, q) code be a code wth c = q 1 and M =.( q 1). Then the number of zeros n each gossp column s equal to the number of prates c,.e., c = M ( q 1). Ths wll guarantee a unque coordnate (n the prate word) for each prate group, whch has only zeros. Thus the prate words created by dfferent prate groups are all dstnct. Here s an example. Example Gossp(6, 4, 3) code Here, the number of prate sets of sze are equal to =6. The exhaustve lst of the prate words (equal to number of prate groups here) created by all the colluson sets are as follows, where e denotes an erasure at all detected postons and 0 means the postons are undetected.

17 138 Veerubhotla R.S., Saxena A., Gulat V.P., Pujar A.K.: Gossp Codes... S.No Prate Sets Descendent sets of Prates 1 {1, } (e, e, e, e, e, 0) {1, 3} (e, e, e, 0, e, e) 3 {1, 4} (e, e, 0, e, e, e) 4 {, 3} (e, e, e, e, 0, e) 5 {, 4} (e, 0, e, e, e, e) 6 {3, 4} (0, e, e, e, e, e) Tracng wth only erasures Table 1: Colluson sets vs. prate fngerprnts We observe that each colluson set creates a unque fngerprnt n only erasures case,.e., the number of prate fngerprnts s equal to number of colluson sets. Moreover all the prate words are dstnct. Snce there s one-to-one matchng between a partcular colluson set and the prate fngerprnt, t s possble to fnd the colluson M group as a whole. Ths tracng method has O complexty where c s colluson c sze. An alternatve method s to fnd all the zeros n the prate word, and accuse everybody who has zeros n all these postons. Ths works even f the actual coalton sze s less than c, and runs n tme O( M l) where M s the number of users and l M s the length of the code. Ths s sgnfcantly more effcent than O, whch s the c runnng tme of the earler algorthm General case for only erasures Let ( X, B ) denote the equvalent set system of a t -desgn that corresponds to a c- Gossp( l, M, q) code. We shortly wrte W = {, j} f w and w are members of the colluson group W. Lookng at the t -desgn s blocks we can tell whch group s prate words wll have 0 at a gven poston, and whch groups wll have erasures. For example the block B gves the lst of prate groups, whch wll not be able to detect the th poston. If W X B (mples W B = φ ) and W t then the colluson group W cannot detect the th poston. W wll create an erasure (snce the mark s detected) n j th poston f W Bj φ. Let x be the prate word created by a colluson group W and x be the th poston n the prate word. The prate word created by W can be completely specfed by the blocks of the t -desgn. x j s zero (undetected poston) f W Bj = φ and x j s e (detected) f W Bj φ

18 Veerubhotla R.S., Saxena A., Gulat V.P., Pujar A.K.: Gossp Codes From the above dscusson and Theorem 3.1. can be seen that the number of non-zero symbols n each codeword of a c- Gossp( l, M, q) code s equal to λ 1. Thus each codeword contan l λ1 zeros. Lemma If c- Gossp( l, M, q) code s a Gossp code wth mnmum code length, then the number of undetectable postons (number of zeros n only erasures case) for a colluson W of sze d c n the prate word s d 1 d l ( 1) λ where λ s defned as n Theorem 3.1. above, wth = 1 λ = 1, v = M and t = c. Proof. Let W be the colluson set of sze d c. The number of blocks n the t-( v, k, λ) desgn dsjont from W,.e., total number of blocks excludng the blocks that contan any member of W, s equal to the number of undetectable postons for the colluson. The proof follows from Theorem and the above dscusson. For example consder -Gossp (7, 7, 4) code presented n e.g...1. The t - desgn correspondng to ths Gossp code s -(7, 3, 1) desgn, whch s gven by X = {1,, 3, 4, 5, 6, 7} B1 = {1,, 3}, B = {1, 4, 5}, B3 = {1, 6, 7}, B4 = {, 4, 6}, B5 = {, 5, 7}, B6 = {3, 5, 6}, B7 = {3, 4, 7} Let W be a colluson set, and let W = {1, }. Number of blocks that do not contan any member of W = Total number of blocks Number of blocks that contan 1 Number of blocks that contan + Number of blocks that contan {1, } = l. λ1 + 1 =7-( 3)+1= Ths mples n only erasures case for -Gossp (7, 7, 4) code each prate word contan two zeros. In shortest Gossp code, f c < M ( q 1) then the number of zeros n a column s more than c. Ths ensures that there s at least one undetected poston for the prate words. It s known that ( p + p+ 1, p+ 1, 1) desgn exsts when p s prme power. When a Gossp code s constructed from -( p + p+ 1, p+ 1, 1) desgn, the number of zeros n each Gossp column = M ( q 1) = p + p ( p + 1) = p > c. Also the number of c -groups for whch, a p gven poston n the prate code word s undetectable s. Thus for the code n c 4 e.g..1.1, the number of zeros n each column s 4, and = 6 prate words contan 0 n a gven poston.

19 140 Veerubhotla R.S., Saxena A., Gulat V.P., Pujar A.K.: Gossp Codes... Consder the Gossp code as gven n e.g..1.1, where M = 7, q = 4 and c =. 7 Here, the number of prate sets of sze are equal to =1. The exhaustve lst of the prate words (equal to number of prate groups here) created by all the colluson sets are as follows. S.No Prate Sets Descendent sets of Prates 1 {1, } (e, e, e, e, e, 0, 0) {1, 3} (e, e, e, 0, 0, e, e) 3 {1, 4} (e, e, e, e, 0, 0, e) 4 {1, 5} (e, e, e, 0, e, e, 0) 5 {1, 6} (e, e, e, e, 0, e, 0) 6 {1, 7} (e, e, e, 0, e, 0, e) 7 {, 3} (e, 0, 0, e, e, e, e) 8 {, 4} (e, e, 0, e, e, 0, e) 9 {, 5} (e, e, 0, e, e, e, 0) 10 {, 6} (e, 0, e, e, e, e, 0) 11 {, 7} (e, 0, e, e, e, 0, e) 1 {3, 4} (e, e, 0, e, 0, e, e) 13 {3, 5} (e, e, 0, 0, e, e, e) 14 {3, 6} (e, 0, e, e, 0, e, e) 15 {3, 7} (e, 0, e, 0, e, e, e) 16 {4, 5} (0, e, 0, e, e, e, e) 17 {4, 6} (0, e, e, e, 0, e, e) 18 {4, 7} (0, e, e, e, e, 0, e) 19 {5, 6} (0, e, e, e, e, e, 0) 0 {5, 7} (0, e, e, 0, e, e, e) 1 {6, 7} (0, 0, e, e, e, e, e) Table : Colluson sets vs. prate fngerprnts (-Gossp(7, 7, 4) code) Theorem If c- Gossp( l, M, q) code s a Gossp code wth shortest length such that c < M ( q 1), then no two colluson groups can create the same prate word n fngerprntng wth only erasures. Proof. In only erasures case (of any Gossp code) t s always true that each prate group can create only one prate word. If c < M ( q 1) n a Gossp code wth mnmum possble length the pattern of undetectable zeros s dfferent for each group. Thus the prate words are unque. If two prate words can create same prate word then each gossp column cannot accuse q 1 dstnct groups, whch s a c contradcton. Examples for these codes are Gossp codes constructed from -( p + p+ 1, p+ 1, 1) and 3-( p + 1, p+ 1, 1) desgns where p s prme power and

20 Veerubhotla R.S., Saxena A., Gulat V.P., Pujar A.K.: Gossp Codes greater than. Specfc examples are 3- Gossp(30, 10, 5), and - Gossp(1, 1, 6) codes apart from - Gossp(7, 7, 4) code. 5 Concatenated Codes A Concatenated code conssts of an nner code and an outer code, where the symbols n the outer code are the codewords of the nner code. Thus, every codeword n the Concatenated code s the concatenated collecton of nner codewords. Further, the number of codewords n nner code s equal to the number of alphabet symbols n the outer code. We denote the codewords of the nner code as {0,..., q 1} for convenence. We replace the symbols of outer code wth nner codewords, for constructng the concatenated code. When a frameproof code s used as nner code, the prate members cannot create alphabet symbols not found n ther copes n the detected postons of the concatenated code. Thus, whenever a mark s detected durng comparson n the concatenated code, the prates need to create an alphabet symbol they fnd n ther copes, or a non-alphabet symbol n the detected postons. Thus ths concatenated code presents a sensble way of mplementng fngerprntng schemes under the prate model assumng markng assumpton [Boneh and Shaw 98] for non-bnary codes. Consder a frameproof code wth M = 4, q = and c =. Let 0 denote the frst nner codeword, 1 denote the second, and so on. Example 5.1 -FP(3, 4, ) code If we use the Gossp code n e.g...1 as outer code for the Concatenated code, the Concatenated code wll be: Example 5. -Gossp(7, 7, 4) concatenated code w w : : : : w

21 14 Veerubhotla R.S., Saxena A., Gulat V.P., Pujar A.K.: Gossp Codes... The colluson sze c s two for the concatenated code n e.g. 5.. Now, we dscuss the tracng for the Concatenated code defned above. 1 Consder the colluson set W, consstng of w and w. The descendent set s DW ( ) = {( a1,..., a5, 0, 0) / a1 {1, }, a {1, 0} for 4} snce the last two postons are undetected. The choces n the other postons are based on the choces avalable n the nner code (.e., code gven n e.g 5.1.) Let a prate word created by 1 the colluson set W = { w, w } be x = { }. Ths can be re-wrtten as {, e, e, 1, e, e, 0, 0}. Ths s selectve erasures case of smple Gossp code. Further the prate word contans a nonzero symbol. In ths case the tracng algorthm runs n tme O( M l) and wll trace w. 5.1 Concatenated Gossp codes wth Selectve erasures In ths constructon, the outer code s also a Gossp code (that has a tracng algorthm) and the nner code has a determnstc decodng algorthm, so the Concatenated code also traces back the members of the prate group. Dealng wth erasures n nner Gossp codes would automatcally mean dealng wth erasures n Concatenated codes. Consder a Gossp code wth M = q = 4, and c = 3. It would result n the followng code, where 0 denotes the frst nner codeword, 1 denote the second, and so on. Example Gossp(4, 4, 4) code Ths code can trace all ts prates f the colluson sze s less than or equal to three (refer Lemma 4.1.). If we use the Gossp code n e.g..1.1 as outer code for the Concatenated code, the Concatenated code wll be: Example 5.1. Gossp(7, 7, 4) concatenated code w w : : : : w

22 Veerubhotla R.S., Saxena A., Gulat V.P., Pujar A.K.: Gossp Codes The colluson sze c s two for the concatenated code. Now, we dscuss the optonal erasures case for the Concatenated code defned above. In ths case, the prates are allowed to create an alphabet symbol they see n the detected postons Tracng 1 Consder the colluson set W, consstng of w and w. The descendent set s gven by DW ( ) = {( a1,..., a5, 0, 0) / a1 {1, }, a {1, 0} for 4} snce the last two postons are undetected. The choces n the other postons are based on the choces avalable n the nner code (.e., code gven n e.g ) Let the prate word created by the colluson set W be x = { }. Ths can be re-wrtten as { e,1, e,0, e,0,0}. Ths s same as smple Gossp code wth selectve erasures. Ths 1 wll reveal a prate namely w. Suppose the prateword s x = { }. Ths can be rewrtten as { e, e, e, 0, e, 0, 0}. Ths does not contan any non-zero symbol of the outer code. But here the nner codewords that are exposed by applyng tracng algorthm on nner prate word { } are{1, }. Smlarly, all other nner prate words, reveal {0, 1} except the last two prate words, whch reveal 0. Wth ths nformaton, we 1 can completely dentfy the set of prates. Ths traces the prate set to be { w, w }. Lemma 5.1: From an nner c- Gossp( l, M, q ) ) code and an outer c- Gossp( l, M, q ) code s a concatenated IPP code can be constructed provded M = q. To construct a generc concatenated code, the condton that must be fulflled s that the number of nner codewords s equal to the number of alphabet symbols of the outer code Further, ths concatenated code s an IPP code, snce the nner and outer codes are IPP codes. Such a concatenated code can be constructed by consderng e.g as nner code and e.g..1.1 as outer code. The no erasures and only erasures cases n these concatenated codes are straghtforward and omtted here as the tracng n the nner codewords s smlar to Gossp codes for no erasures and only erasures. 5. Performance of Gossp codes One advantage of Gossp codes s that they can accuse a trator determnstcally. Nevertheless all the prevous constructons were probablstc n some way or other. In general, the focus n all probablstc code constructons s to keep the length of each codeword short. Short length s mportant to desgn effcent and effectve embeddng technques of a codeword nto an object. Some of the prevous probablstc codes managed to acheve polylog sze n the number of the users. But the length of the code tends to be very large (to be nterpreted as proportonal to the number of users) f determnstc full tracng solutons are sought. Thus Gossp codes n general, have much greater length than probablstc codes. Our constructons for Gossp codes acheve the mnmum possble code length specfed for Gossp codes.

23 144 Veerubhotla R.S., Saxena A., Gulat V.P., Pujar A.K.: Gossp Codes... The addtonal advantage of Gossp codes comes from the fact that they can wthstand erasures and tracng s possble n presence of erasures. For mproved results concatenated codes wth outer code as Gossp code and nner codes wth probablstc tracng can be used. 5.3 Embeddng fngerprnts We present smple watermarkng technque, whch are based on wavelet transforms to embed (hde) the codewords of a Gossp code nto mages. The dstrbutor may choose any of the embeddng method from the avalable watermarkng technques based on the requrement and type of the meda Wavelet watermarkng The Dscrete Wavelet Transform (DWT) separates an mage nto a lower resoluton approxmaton (LL) as well as horzontal (HL), vertcal (LH) and dagonal (HH) detal components. The process can be repeated to compute multple scale wavelet decomposton smlar to the two-scale wavelet transform as shown below. One of the advantages of the usng wavelet transform s that t s beleved to more accurately model the Human Vsual System (HVS) as compared to Fast Fourer Transform (FFT) and Dscrete Cosne Transform (DCT). Ths allows us to use hgher energy watermarks n regons that the HVS s known to be less senstve to; such as the hgh-resoluton detal bands LH, HL, and HH. Fgure 1: Wavelet Decomposton 5.3. Algorthm We use a straghtforward technque for embeddng the Gossp codeword (fgure ) as a watermark sequence n the detal bands accordng to the equaton shown below: V + α. V. wm, a, b HL, LH IV = ab, V ab, LLHH,

24 Veerubhotla R.S., Saxena A., Gulat V.P., Pujar A.K.: Gossp Codes where V denotes the coeffcent of the transformed mage, wm one bt of the watermark to be embedded, and a scalng factor. In order to detect the watermark, we generate the same watermark sequence and determne ts correlaton wth the two transformed detal bands. If the correlaton exceeds some threshold T ˆ, the watermark s detected. Thus, n ths method watermark detecton does not requre the orgnal mage. Ths method can be easly extended to embed multple watermarks nto the mage. The robustness evaluaton (and tracng) was lmted to testng aganst JPEG compresson and the addton of random nose. (a) (b) (c) Fgure : a) Orgnal Image b) Watermarked Image c) Embedded Watermark d) Recovered Watermark 6 Concluson Gossp codes that acheve the shortest code length can be constructed from class t - desgns and c -Traceablty Schemes. In ths work, two methods of constructon for shortest Gossp codes are presented. The converse part.e., constructon of Traceablty Schemes and t -desgns from Gossp codes s also feasble. These results are relable to determne whether a shortest Gossp code exsts or not, for the chosen code parameters M, q and c. Gossp codes, as IPP codes, can dentfy one parent of the prate copy durng tracng. They allow determnstc tracng of prates and also come up wth a tracng algorthm, whereas earler works on fngerprntng are usually probablstc n some way or the other. The constructon of embedded Gossp code s also possble and t can be used for extendng an exstng Gossp code to a bgger code. These (embedded) codes and Traceablty schemes can expand a broadcast applcaton such that t s compatble to the exstng scheme. When a frameproof code s used as nner code, the prate members cannot create alphabet symbols not found n ther copes n the detected postons of the concatenated code. Thus, durng comparson, whenever a mark s detected n the concatenated code, the prates need to create an alphabet symbol they fnd n ther copes or a non-alphabet symbol n the detected postons. Thus, ths concatenated code presents a sensble way for mplementng fngerprntng schemes under prate model (FSOPE), assumng markng (d)