Wide-sense fingerprinting codes and honeycomb arrays

Transcription

1 Wide-sense fingepining codes and honeycomb aays Anasasia Panoui Technical Repo RHUL MA Febuay01 Depamen of Mahemaics Royal Holloway, Univesiy of London Egham, Suey TW0 0EX, England hp://

2 Wide-Sense Fingepining Codes and Honeycomb Aays Anasasia Panoui Thesis submied o he Univesiy of London fo he degee of Doco of Philosophy 01

3 Wide-Sense Fingepining Codes and Honeycomb Aays Depamen of Mahemaics Royal Holloway, Univesiy of London

4 To my paens, Maia and Thanasis

5 Declaaion of Auhoship I, Anasasia Panoui, heeby declae ha his hesis and he wok pesened in i is eniely my own. Whee I have consuled he wok of ohes, his is always clealy saed. Signed: (Anasasia Panoui) Dae: 3

6 Acknowledgmens I would like o hank my supeviso Pof. Simon Blackbun fo his guidance which led o he composiion of his hesis. Moeove, his useful and shap commens on my eseach led o he impovemen and fuhe developmen of my mahemaical educaion. The pleasan envionmen of ou eseach discussions and his posiive view, helped me no only ovecome any poblems and disappoinmens ha so ofen occu in eseach, bu also o be pleased and poud of he good eseach esuls, as small o big as hey migh be. I wish o hank my fiends fom Geece, who despie he gea disance, neve ceased o encouage and suppo me. Many hanks o my fiends and fellow Ph.D. eseaches who made my say in Egham vey enjoyable and he envionmen in McCea building a gea place o wok. Special hanks o Liz Quaglia fo he fiendship and suppo and o Gaven James Wason, who was always willing o offe advice and help. I am gaeful o Jean Paul Degabiele, fo in sessful imes he helped me view maes in a diffeen and moe elaxed pespecive. I would also like o expess my gaiude fo he College Reseach Scholaship ha was offeed o me and made he hee yeas of my Ph.D. sudies possible. Finally, I would like o hank my paens and my bohe fo hei consan suppo and faih in me houghou he yeas of my sudies. 4

7 Summay The hesis is divided ino wo independen pas. The fis pa examines he main ypes of fingepining codes unde fou descendan models, while he second invesigaes he combinaoial objec called a honeycomb aay. Digial fingepining is a echnique ha is used o poec inellecual ighs by pevening illegal edisibuion of digial daa (films, music, sofwae, ec.). This echnique is faciliaed by he collecion of codes called fingepining codes. The hesis focuses on he following fou fingepining codes: aceabiliy, IPP, secue famepoof and famepoof. These codes ae sudied unde fou models, namely naow-sense, expanded naow-sense, wide-sense and expanded wide-sense. These models efe o he abiliy of malicious uses (aios) o poduce he fingepin in he illegal copy. In paicula, following an idea of Boneh and Shaw, i is shown ha hee only exis ivial wide-sense aceabiliy and IPP codes. In he mae of widesense famepoof codes, enhancing he elaion beween hese codes and Spene families fis inoduced by Sinson and Wei, we impove hei uppe bound on he size of his ype of fingepining codes. The las wo esuls ae oiginal and we egad he lae o be he main oiginal conibuion of his pa of he hesis. A honeycomb aay of adius is a se of n = +1dos placed on he hexagonal gid in such a way ha he disance of evey do fom a fixed cell, he cene, is a mos. I is also equied ha in each column and in each diagonal only one do occus and ha he veco diffeences beween all pais of dos ae disinc. In he hesis i is poved ha honeycomb aays 5

8 can only be consuced using Cosas aays, which ae configuaions of dos in he squae gid simila o honeycomb aays. Using he exising Cosas aay daabase, all honeycomb aays wih apple 14 ae deemined, and wo new aays of adius 7 ae pesened. 6

9 Peface This wok is a composiion of ideas and esuls fom wo independen aeas. Thus, i is appopiae o divide he hesis ino wo pas. The fis is called Fingepining Codes, and examines he inepeaion of fingepins ino he digial wold as a mehod of poecing inellecual popey ighs. The second pa ansfes Cosas aays o he hexagonal gid and sudies he popeies and he behaviou of he esuling combinaoial objec, called Honeycomb Aays. 7

10 Conens I Fingepining Codes 14 1 Inoducion Poecion of Inellecual Popey Ouline Se Theoy 18.1 Spene Theoy Inesecing Families The Fingepining Poblem Digial Fingepining and Applicaions The Descendan Se Fingepining Codes Famepoof Codes Secue Famepoof Codes IPP Codes Taceabiliy Codes Relaed Wok Famepoof Codes Secue Famepoof Codes Idenifying-Paen-Popey Codes Taceabiliy Codes Beyond he Main Types of Fingepining Codes

11 4.5.1 Secue "-Eo Codes Relaions Beween Fingepining Codes The Naow-Sense Model Wide-Sense and Expanded Naow/Wide-Sense Models on Taceabiliy and IPP Codes Famepoof and Secue Famepoof Codes Secue Famepoof Codes Famepoof Codes Unifying he Relaions Beween Fingepining Codes Wide-Sense -Famepoof Codes Popeies of -wfp codes Small Lengh Case Abiay Lengh Case wfp Codes of Lengh II Honeycomb Aays Honeycomb Aays Fom Rooks o Semi-Queens Cosas Aays Honeycomb Aays Consucion of Honeycomb Aays Compuaional Resuls Concluding Remaks Appendices 136 A Seach of Honeycomb Aays in C 136 Bibliogaphy 141 9

12 Lis of Figues 5.1 Relaions of fingepining codes unde he naow-sense model Relaions of aceabiliy and IPP codes unde he expanded naowsense, wide-sense and expanded wide-sense model Relaions of naow-sense famepoof and secue famepoof codes wih aceabiliy and IPP codes unde all fou models of descendan se Relaions of naow-sense famepoof and secue famepoof codes and aceabiliy and IPP codes unde all fou models of descendan se A Cosas aay A honeycomb aay The hexagonal gid Hexagonal egion Golomb and Taylo consucion of honeycomb aays fom Cosas aays The egion S i (m) The iangula boad and how is coveed by he egion S i (n) The ansfomaion of he hexagonal laice ino he squae laice The analogy beween he neighbous in he hexagonal and squae aay The black hexagonal sphee of ode is ansfomed ino he incomplee squae S 0 of ode n 0 =

13 7.11 Illusaion of he inducive sep. The exaced shapes show he neighbou elaion of he maked cell, as he hexagonal is ansfomed ino he squae laice The use of slope fo he deeminaion of whehe o no a diagonal in he hexagonal sphee conains wo dos The wo membes of he fis equivalence class, whee he second honeycomb aay is he veical eflecion of he fis The wo membes of he second equivalence class, whee again he second honeycomb aay is he veical eflecion of he fis The 3 3 Cosas aay and he coesponding A 1 class of he honeycomb aays The 7 7 Cosas aays and he coesponding A 3 class of he honeycomb aays The 7 7 Cosas aay and he coesponding B 3 class of he honeycomb aays The 9 9 Cosas aay and he coesponding A 4 class of he honeycomb aays The 9 9 Cosas aay and he coesponding B 4 class of he honeycomb aays The Cosas aay and he coesponding A 7 class of he honeycomb aays The Cosas aay and he coesponding B 7 class of he honeycomb aays The Cosas aay and he coesponding C 7 class of he honeycomb aays The 1 1 Cosas aay and he coesponding A 10 class of he honeycomb aays The 7 7 Cosas aay and he coesponding A 13 class of he honeycomb aays

14 7.5 The Cosas aay ha poduces he A he honeycomb aay The A class of he honeycomb aays geneaed fom he Cosas aay The symmey of he honeycomb aays o Cosas aays The hexagonal symmeies wih espec o he lines defined by he hee diecions of he hexagonal gid

15 Lis of Tables 3.1 The se of q` keys ha ae used o encyp keys {s 1,...,s`} The encypions in he enabling block, whee ` denoes he numbe of segmens and q he numbe of maks Uppe bounds on he size m of a -wfp code of odd lengh ` The Spene family X 0 and he coesponding codewods The Spene family X 0 and he coesponding enay code The numbe of n n Cosas aays found by exhausive seach Enumeaion esuls

16 Pa I Fingepining Codes 14

17 Chape 1 Inoducion 1.1 Poecion of Inellecual Popey Digial infomaion, whehe soed in a CD, DVD o eaching he ecipien hough he Inene, is disibued in a coninually inceasing manne. Howeve aacive is he ease of accessing and manipulaing digial daa, such as movies, music, sofwae o documens, he abuse of i is anamoun o a ciminal ac. The poecion of inellecual popey is he afemah of he invenion of pining, which allowed he effoless disibuion of lage numbes of copies of documens and books. Today, due o he easy and diec ways of exchanging digial daa, he necessiy of ensuing ha he ighs of he ceao/owne ae inac is even geae. As a consequence, he eseach on developing mehods ha poec hese ighs is paiculaly impoan. The infingemen of inellecual popey ighs, o in ohe wods he illegal edisibuion of daa, is summaised in he wod piacy. Digial fingepining, and in paicula fingepining codes, povide a means of limiing piacy. Simila o human fingepins which ae unique fo each peson, he pupose of his ype of code is o make idenical copies of objecs unique. 1. Ouline The aim of his secion is o descibe in bief he conens of each chape of he fis pa of he hesis, which invesigaes fingepining codes. As he mahemaical backgound of his ype of codes is based on se heoy, he 15

18 peliminay secion ha compises Chape inoduces he necessay definiions and noions of se heoy. In paicula, he saemens and poofs of main esuls in he aea of Spene heoy and he heoy of inesecing ses ae pesened. As he name fingepining codes indicaes, coding heoy also plays a ole in his opic. Howeve, a elaed peliminay secion on coding heoy is omied, as only he basic noions ae equied and ae assumed o be familia o he eade. A deailed pesenaion of fingepining codes can be found in Chape 3. The envionmen wihin which fingepining codes ae used consiss of he following chaaces: he disibuo, he egiseed uses and he aios. The disibuo is he owne of he digial daa, who is esponsible fo is disibuion o he egiseed uses, who ae he ecipiens of he daa. The chaaceisaion aios, efes o he subse of he se of egiseed uses who ac maliciously and illegally edisibue he daa. Chape 3 descibes wo main applicaions of fingepining codes ha ae based on diffeen seings. The fis, inoduced by Cho, Fia and Nao [18], assumes ha he digial daa eaches he ecipiens hough boadcas ansmission, while in he second, which can be found in he suvey pape by Blackbun [8], he disibuion is caied ou via he Inene o in a CD/DVD foma. The same chape also efes o he making assumpion, accoding o which he aios geneae fingepins fo illegal copies of he daa. Moeove, fou main ypes of fingepining codes ae descibed, namely famepoof, secue famepoof, IPP and aceabiliy codes, which coespond o diffeen secuiy noions, viewed fom he disibuo s pespecive. On he ohe hand, aking ino accoun he aios capabiliies in geneaing he illegal fingepin, he following fou advesay models ae defined: naow-sense, expanded naow-sense, wide-sense and expanded wide-sense descendan model. In ode o povide good codes ha capue boh he secuiy noion fom he disibuo s poin of view and he advesay powe, he fou main ypes of fingepining codes ae defined unde he fou afoemenioned ad- 16

19 vesay models, esuling in oal in sixeen diffeen ypes of fingepining codes. Lasly, Chape 3 includes wo new esuls. The fis (Poposiion ), shows ha expanded wide-sense IPP codes and oally secue codes ae he same objec. The second esul (Poposiions 3.3. and 3.3.5), efes o wo naual ways of defining Hamming disance in he conex of fingepining codes and poves ha codes which use hese wo ypes of disance ae equivalen. The nex chape, Chape 4, is devoed o pesening elaed and pevious wok on fingepining codes. Fo each ype of code, known consucions ae descibed and known bounds on he size of he codes ae given. Chape 5 invesigaes he elaions amongs he sixeen diffeen ypes of fingepining codes. Finally, Chape 6 is he chape ha pesens he main oiginal conibuion of his pa of he hesis. I focuses on he sudy of a paicula ype of fingepining codes, namely wide-sense famepoof codes. Using he connecion beween his ype of codes wih Spene and inesecing families, we obain an impovemen of he known uppe bound on he size of such codes (Theoem 6.3.8). Addiionally, he chape includes oiginal esuls on he size of wide-sense famepoof codes of small lengh. 17

20 Chape Se Theoy The pupose of his chape is o selecively pesen known esuls fom se heoy, ha will be used in he subsequen chapes. These esuls elae o Spene heoy and he heoy of inesecing ses..1 Spene Theoy In 198, Spene [46] sudies he family of ses wih he popey ha no membe of he family belongs eniely o anohe. An impoan esul of his sudy is an uppe bound on he size of he maximal such family, and moeove, he ype of ses ha he family mus conain, in ode o achieve his bound. Fo he emainde of he hesis, he noaion n-se (accodingly n-subse), denoes a se of size n. Definiion.1.1 (Spene family). Le n be a posiive inege and F be a family of ses ove he gound se {1,...,n}. The family F is called Spene o has he Spene popey, if fo all A, B F he ses A and B ae incompaable, ha is A 6 B. Theoem.1. (Theoem 1.1.1, [6], [46]). Le n be a posiive inege and F be a Spene family. Then 18

21 (a) 8 >< F apple >: n n, if n is even n n 1, if n is odd. (.1.1) (b) Equaliy holds if and only if 8 {X {1,...,n} : X = n }, if n is even >< F = {X {1,...,n} : X = n 1} o >: {X {1,...,n} : X = n+1 }, if n is even. Poof. I is easy o see ha in boh he even and he odd case, he family F pesened in he second pa of he claim is Spene and achieves he bound fom pa (a). Theefoe, i suffices o pove ha hee does no exis a lage Spene family. We fis begin wih he necessay noaion. Le F be a maximal Spene family, and define l =min{s : 9F F s.. F = s}, u =max{s : 9F F s.. F = s} o denoe he size of he smalles and he lages ses in F, especively. Moeove, le G = {X F : X = l}, H = {Y {1,...,n} : Y = l +1and 9X G s.. X Y }, F 0 =(F G) [ H. We nex pove ha F 0 is Spene. Assume fo a conadicion, ha i is no. Clealy, F G is a Spene family as a subse of F, and H also saisfies he Spene popey, because i conains disinc ses of he same size. Thus, hee exis Y H and Z F G, such ha Y Z. By he definiion of H, 19

22 hee also exiss X G F, such ha X Y, which implies ha X Z, a conadicion, since boh X and Z belong o F, and F is Spene family. Le l apple n 1. Also, le N be he numbe of pais (X, Y ) such ha X G, Y H and X Y. If X is fixed, hen by adding one of he elemens of {1,...,n} X o X, we obain exacly n l diffeen ses Y ha conain X. This leads o N = G (n On he ohe hand, if we fix a se Y he numbe of l-ses ha ae conained in Y is Y l = l +1, bu since no all of hem belong o G, we have l). N apple H (l +1). (.1.) Combining he above esuls on N wih l apple n 1, yields H G n l l +1 n n 1 n 1 +1 =1, whee equaliy is aained if l = n 1. The fac ha F is Spene means ha F \ H = ;, hus F 0 = F G + H F, and he equaliy implies l = n 1. Since we have chosen F o be he Spene family of he maximum size, fom he analysis above i is clea ha he case l apple n 1 leads o conadicion, because i implies he exisence of a Spene family of size geae han F. n+1 If u, hen we also each a conadicion. The poof of his case n 1 follows he same agumens as in he case whee `. The only diffe- ence is ha hee he family F 0 is fomed by eplacing all u-ses S in F by he (u 1)-ses R, such ha R S. n 1 Hence, we can assume ha l and u apple n+1. When n is even, his implies ha he maximal Spene family has size a mos his way he claim. n, poving in 0

23 Le n be odd. If l = u, hen F apple n n 1 = n n+1. Hence, assume ha l = n 1 and u = n+1. Fom he above, we have ha n F apple F 0 apple n+1, and since F is maximal i mus hold F 0 = F, which means ha in (.1.) we have N = H (l +1). This can occu only when all he l-subses of a se Y H belong o G, fo all Y H. Examine he ses Y H and Z F G, such ha Y \ Z is he maximum. Since F conains only ses of size l = n 1 and l +1= n+1, hen Y = Z = l +1. Clealy Y 6= Z and Y \ Z being he maximum implies ha hee exiss y Y Z and z Z Y. As peviously menioned G conains all l-subses of any Y H, which implies ha Y {y} is a membe of G. Hence, Y 0 =(Y {y}) [ {z} is a membe of H. Howeve, Y 0 \Z = Y \Z +1, which conadics he fac ha he inesecion beween Z and Y is maximal. Hence, in he case whee n is odd, he maximal Spene family has size n n 1 and is aained by aking all ses of size n 1 o n+1. The nex poposiion, which can be found in he book Spene Theoy by Engel [6], povides uppe bounds on he size of a Spene family ha depends on he sizes of he ses ha i conains. The lemma ha pecedes he poposiion pesens a necessay esul fo he poof of he poposiion. Lemma.1.3 (Coollay.3., [6]). Le F be a family of k-ses ove a se of size n 3 and define he families H and D as follows: H = {Y {1,...,n} : Y = k +1and 9F F s.. F Y }, D = {D {1,...,n} : D = k 1 and 9F F s.. D F }. (a) If k n +1, hen D F n. (b) If k apple n 1, hen H F n. 1

24 Poposiion.1.4 (Coollay.3.3, [6]). Le F = {F 1,F,...,F } be a Spene family ove he se {1,...,n} and l := min{s : 9i {1,...,} s.. F i = s}, u := max{s : 9i {1,...,} s.. F i = s}. If l apple n apple u, hen 8 >< apple >: n b nc n b nc (u l) n, if n is even, (u l 1) n, if n is odd. Poof. The cases n =1, ae ivial, so le n 3. We pove he claim using inducion on he diffeence u l and when n is even. The odd case can be poved analogously. If u l =0, hen we obain he bound (.1.1) fom he pevious heoem, Theoem.1.. Hence, u l 1. Fo he base case u l =1and since l apple n apple u, eihe l = n 1 and u = n, o l = n and u = n +1. Due o he symmey of hese cases, we pove he claim when l = n 1 and u = n. Simila o he poof of Theoem.1., we define he following: G = {X F : X = l}, H = {Y {1,...,n} : Y = l +1and 9X G s.. X Y }, F 0 =(F G) [ H. Since F consiss of l-ses and (l +1)-ses, by eplacing G wih H he esuling family F 0 conains only (l +1)-ses. Clealy, he ses in H ae diffeen fom he ses in F G, because ohewise he Spene popey of F would be violaed. This implies ha F 0 is also Spene. Since F \ H = ;, applying Lemma.1.3 on he family G, we obain he following F 0 = F G + H F + n ) F apple n n n,

25 which poves he claim fo he base case of he inducion. Fo he inducive sep we assume ha whee k = u u l = k +1, hen Define G and F 0 F apple n n k n, l and we pove ha when F is a Spene family fo which F apple n n (k +1) n. = (F G) [ H as peviously, bu his ime he ses in F 0 have diffeen size. We nex show ha F 0 is Spene. Assume ha i is no. Clealy, F G and H ae boh Spene families, as a subse of F he fome and as a family of disinc ses of he same size, he lae. Hence, hee mus exis a se Y H such ha Y Z, fo some Z F G. By definiion of H, hee also exiss X G, such ha X Y Z, which is a conadicion o F being a Spene family. Having emoved G fom F, leads o he esuling family F 0 consising of ses of size a leas l 0 =(l +1). This means ha he diffeence beween he ses of he lages and smalles size in F 0 is u l 0 = u l 1=k. By he inducive sep, n F 0 n k n. (.1.3) Applying once again Lemma.1.3 on G, we have ha H G n. Combining his esul wih he bound (.1.3) and he fac F 0 \ H = ;, we obain F 0 n n = F G + H F ) F apple n (k +1) n, which concludes he poof.. Inesecing Families This secion pesens impoan esuls on inesecing families, an objec ha has capued he inees of mahemaicians fo many yeas. 3

26 Definiion..1. Le F be a family of ses ove a gound se E. Then, F is called -inesecing if fo evey pai of ses A, B F we have A \ B. The fis esul we pesen, is he Edős-Ko-Rado Theoem [7], ha povides an uppe bound on he size of inesecing Spene families ha conain ses of specific size. Apa fom he esul iself, he Edős-Ko-Rado heoem plays a significan ole in exemal se heoy, as i inoduces hough is poof he mehod of shifing. The poof ha is pesened hee can be found in he pape by Fankl and Gaham [8], who have fomalised he shifing mehod, and pesen he oiginal poof of Edős, Ko and Rado using his fomalisaion. Fo convenience, when is no specified -inesecing families will be called inesecing. Below, we give he definiion of he (i, j)-shif, followed by some popeies ha ae needed fo he poof of he Edős-Ko-Rado heoem. The poof of hese popeies is omied, as hey can be easily deived fom he definiion of he (i, j)-shif. Definiion.. (The (i, j)-shif, [8]). Le F be a family of ses ove {1,...,n}. Fo 1 apple i<japple n, define S ij (F) ={S ij (F ):F F}, whee 8 < F 0 =(F {j}) [ {i}, if j F, i / F and F 0 / F, S ij (F )= : F, ohewise. Poposiion..3 (Poposiion., [8]). If F is a family of ses ove {1,...,n}, hen (a) S ij (F ) = F, (b) S ij (F) = F, (c) if F is inesecing, hen so is S ij (F). 4

27 Now we can sae and pove he Edős-Ko-Rado hoeem. Theoem..4 (Theoem 1, [8]). If F = {F 1,...,F } is an inesecing Spene family ove he se {1,...,n}, such ha fo all i =1,..., we have F i = s wih 1 apple s apple n, hen apple n 1. s 1 Poof. The claim is poved by inducion on n. We disinguish wo cases: Case 1: n =s Le F be a s-se ove {1,...,n} and F = {1,...,n} F be is complemen. Then, all s-ses ove {1,...,n} can be paiioned ino 1 of complemenay ses. Clealy, if F F hen F / F. Hence, F apple 1 s n 1 =, s s 1 which poves he claim. Case : n>s s s pais Define F 0 = F and fo i =1,...,n 1 le F i = S in (F i 1 ). Accoding o Poposiion..3(b) and (c), families F and F n and since F is inesecing, F n G and H as follows: G = {F F n 1 : n/ F }, H = {F {n} : n F F n 1 } 1 have he same size 1 is inesecing oo. Define he families We have ha F = G + H. By definiion, G is an inesecing family ove he se {1,...,n 1}, which by inducion leads o (n 1) 1 n G apple =. s 1 s 1 We nex pove ha H is also inesecing. Assume fo a conadicion, ha hee exis ses H, H 0 H such ha H\H 0 = ;. Since he size of he 5

28 ses in H is s 1, we have ha H [H 0 =(s 1) <n 1, which implies ha hee exiss i {1,...,n F = H [ {n} belongs o F n 1}, such ha i/ H [ H 0. By definiion, 1. Moeove, n F, which implies ha hough he shifing pocess, none of he membes of F wee eplaced by n. In ohe wods, fo all i {1,...,n 1}we have S in (F )=F, which means ha (F {n})[{i} = H [{i} F i 1 and hence H [{i} F n 1. By assumpion, H \ H 0 = ;. Thus, (H [ {i}) \ (H 0 [ {n}) =;, a conadicion o he fac ha F n 1 is inesecing. As boh G and H ae inesecing families ove {1,...,n consising of s-ses and H of (s 1)-ses, we have n n F = G + H apple + = s 1 s which is he desied bound. n 1, s 1 1}, wih G The Edős-Ko-Rado heoem efes o inesecing Spene families ha consis of ses of ceain size. Resuls egading he geneal case, whee he inesecing Spene family conains ses of abiay size, wee povided by Milne in [37]. The poof of Milne s esul is based on he following heoem by Kaona, which will no be poved hee. Theoem..5 (Theoem, [31]). Le F be a family of `-ses ove {1,...,n}, ha is k-inesecing. Le 1 apple g apple `, 1 apple k apple `, g + k ` and B = {B : B = g and B F, fo some F F}. Then, B ` g ` ` k F, k whee sic inequaliy holds in he following wo cases: 6

29 (a) g = ` (b) F consiss of all `-ses ove he se E {1,...,n} of size E =` k. Theoem..6 (Theoem 1, [37]). If F = {F 1,...,F } is a k-inesecing Spene family ove he se {1,...,n}, hen n apple b n+k+1. c Poof. Le = b n+k+1 c. Clealy, if all ses in F have size exacly, hen he claim is ivially ue. We conside wo cases, depending on he sizes of he ses ha compise he family F. Case 1: Fo all i {1,...,}, F i apple and hee exiss some i, such ha F i <. In ohe wods, if ` is he size of he smalles se in F, hen we assume ha fo 1 apple s apple n ` = F 1 =...= F s < F s+1 apple...apple F apple. Fo he ses F 1,...,F s, le C 1,...,C q be he disinc (` +1)-ses, such ha fo some i {1,...,s} and j {1,...,q} we have F i C j. Then, since fo evey i, j {1,...,} F i \ F j k, i also holds ha C i \ F j k, fo all i {1,...,q} and j {s +1,...,}, which means ha he family {C 1,...,C q,f s+1,...,f } is k-inesecing. Fuhemoe, if hee exis i {1,...,q} and j {s +1,...,} such ha C i F j, hen F i F j, which violaes he Spene popey of F. Hence, he ses C 1,...,C q,f s+1,...,f fom a k-inesecing Spene family of size q + s. Fo a se A, le A = {1,...,n} A denoe he complemen of A. Then, C i = n (` +1)and hee exiss i {1,...,q}, such ha C i F j, fo j {1,...,s}. Also, fo evey i, j {1,...,s} F i \ F j = F i [ F j = n ( F i + F j F i \ F j ) n + k `. 7

30 Noice, ha by assumpion ` apple 1 and hence n+k ` 1. Applying Kaona s heoem (Theoem..5) on {F 1,...,F s }, we have (n `) (n + k `) n k q n ` 1 s = (n `) (n + k `) n ` n ` 1 s s, (..1) n k n ` whee he las inequaliy holds because n + k ` 1. If ` = 1, hen fo all {1,...,q} we have C i =, which leads o {C 1,...,C q,f s+1,...,f } being a k-inesecing Spene family of -ses and size q + s. Using (..1) we obain he desied bound fo. If ` < 1, hen he claim is poved using inducion on `. Case : Thee exiss some i {1,...,}, such ha F i >. In his case, he family F consiss of ses wih diffeen sizes, hence we can assume he following: F 1 apple...apple F s <apple F s+1 apple...apple F p < F p+1 =...= F = `0, whee 1 apple s apple p<. Le D 1,...,D q be he disinc -ses, fo which hee exiss i {1,...,s} such ha F i D j, fo some j {1,...,q}. Then, following he same agumen as in he pevious case we obain q s. (..) Le B 1,...,B u be he disinc (l 0 1)-ses, such ha hee exiss an i {p +1,...,} fo which B j F i, fo all j {1,...,u}. Nex, fo j {1,...,u} and i {p+1,...,} we coun in wo ways he numbe N of pais (B j,f i ) such ha B j F i. Recall, ha B j = l 0 1 and F i = l 0. Fo fixed j, hee exis exacly n (l 0 1) ses F i, such ha B j F i, and since we have u ses B j we ge N = u(n l 0 +1). (..3) 8

31 On he ohe hand, if we fix an i, he numbe of (l 0 l 0, bu as no all of hem belong o {B 1,...,B u }, we have N apple ( p)l 0, which combined wih (..3) yields As l 0 >, we have ha 1)-subses of F i is u(n l 0 +1) ( p)l 0. (..4) `0 >n+ k +1>n+1, (..5) since F is inesecing and hus k>0. Inequaliy (..5) implies ha `0 n `0 +1 > 0 and so (..4) gives u> p. (..6) I is easy o check ha he ses D 1,...,D q,f s+1,...,f p,b 1,...,B u fom a k-inesecing Spene family. If l 0 = +1, hen his family consiss of ses of size l 0 1 and l 0, and has size q +(p s)+u. By Case 1, he size of he family is q +(p s)+u apple n `0 1 = n. Combining inequaliies (..1) and (..6) we have n = q +(p s)+u apple and he claim is poved. If l 0 >+1, hen he desied bound is obained by applying an inducion agumen on l 0. 9

32 A diffeen appoach on he sudy of inesecing Spene families, was inoduced by a conjecue made by Pudy and poven ue by Schonheim [44]. Insead of looking a inesecing Spene families, he conjecue consides Spene families wih he popey ha he union of any wo disinc pais of ses, does no cove he gound se. Definiion..7. Le F be a family ove a gound se E. Then F is called non -coveing if fo evey pai of ses A, B F we have A [ B 6= E. Theoem..8 ([44]). Le {1,...,n} be a se of even size. If F = {F 1,F,...,F } is a non -coveing Spene family ove {1,...,n}, hen n apple. n 1 Poof. Le B 1,...,B s be ses in F ha have size geae han n. Accoding o a esul in [0] by De Buijn, Tengbegen and Kuijswijk, he se of all subses of {1,...,n} can be decomposed ino paiwise disjoin symmeic chains. Since F is Spene, B 1,...,B s belong o diffeen chains. Based on he esul of De Buijn e al., fo i =1,...,she se B i can be eplaced by he n -se C i fom he chain conaining B i. Then, since C i B i, he condiion B i [ B j 6= {1,...,n} implies ha C i [ C j 6= {1,...,n}. Fuhemoe, if F is a se in F of size less han n, hen i also holds ha C i [ F 6= {1,...,n}, since C i [ F < n = n. Hence, his eplacemen does no desoy he popey of he iniial family, ha he union of any wo ses do no cove he gound se {1,...,n}. Le C i = {1,...,n} C i be he complemen of C i, fo all i =1,...,s. Then C i is also a n -se, and since F canno conain complemenay ses, we have s apple n n n, which is smalle han o equal o whee F = {B 1,...,B s }. n n. This poves he claim in he case 1 30

33 Conside again he decomposiion ino paiwise disjoin chains and le D 1,...,D be ses of size less han n. Then, since F is Spene, D 1,...,D belong o diffeen chains. Thee ae of size less han n, hence apple n n 1 disjoin chains ha conain ses n n 1, and he claim is poved when F = {D 1,...,D }. We now conside he geneal case. If X is a ( n 1)-subse of a n -se which belongs o F, hen since F is Spene, no se fom he chain conaining X, belongs o F. Clealy, he n -ses C 1,...,C s ae paiwise inesecing, since ae membes of F. Applying Theoem..5, we ge ha he numbe of ses X is a leas n 1 n 1 1 n n s = n 1 n 1 s = s. n 1 n In oal, we have ha F = s + apple s + which complees he poof. n n 1 s = n n 1, 31

34 Chape 3 The Fingepining Poblem This chape inoduces he noion of fingepining and he moivaion behind he eseach of fingepining codes. In paicula, wo applicaions of fingepining ae pesened, followed by he definiion of he fou advesay models ha descibe he capabiliies of he aios: naow-sense, expanded naow-sense, wide-sense and expanded wide-sense model. Nex, fou main ypes of fingepining codes ae descibed, namely famepoof, secue famepoof, IPP and aceabiliy codes, which coespond o fou diffeen secuiy noions. The combinaion of hese secuiy noions wih he advesay models gives ise o he definiion of sixeen ypes of fingepining codes. Fuhemoe, an oiginal esul egading expanded wide-sense IPP codes (Poposiion ) poves ha his ype of codes is equivalen o he ype of fingepining codes called oally secue codes. The chape concludes wih a new esul on aceabiliy codes, involving wo naual ways of defining he Hamming disance. Poposiions 3.3. and show ha aceabiliy codes unde he wo diffeen ypes of disance ae in fac equivalen. 3.1 Digial Fingepining and Applicaions The ease of access o he vas collecion of digial daa ha is povided by he Inene, as well as by ohe means of exchanging daa, equies he use of mehods ha peven he illegal disibuion of he daa, known as piacy. Jus as human fingepins make each one of us unique and consiue a way of idenificaion, digial fingepins give he popey of disincness 3

35 amongs he copies of he digial daa. The idea of fingepining is no new. As menioned in [15] by Boneh and Shaw, hundeds of yeas ago mahemaicians used his echnique in he logaihm ables. In ode o make each copy unique, hey aleed he leas significan digi of andomly chosen values of log x, so ha each copy had a diffeen se of log x values aleed. In his way, once an illegal copy of he ables was found, i was possible o idenify he aioous owne. Digial fingepining is a mehod of pesonalising digial daa, such as music, films, documens, sofwae, in ode o eliminae illegal edisibuion (piacy), by acing he malicious uses (aios). Waemaking echnologies povide anohe way of making digial daa and in some cases ae egaded o be he same as digial fingepining. Howeve, in he pesen conen we conside waemaking o be a mehod of indicaing he owne/ceao of he objec, wheeas fingepining seves he pupose of deecing he malicious uses. Fo a bee undesanding of he noion of digial fingepining, wo main applicaions ae pesened hee, aken fom he suvey pape [8] by Blackbun. The fis was inoduced in 1994, by Cho, Fia and Nao in [18]. Accoding o hei model, he daa o be disibued eaches he egiseed uses hough boadcas ansmission. This implies ha he daa can also be eceived by unegiseed uses, since hee is no way of conolling boadcas signals. To avoid his siuaion fom occuing, he disibuo applies cypogaphic echniques and insead of ansmiing he clea daa, he ansmis is encyped fom. Fo he encypion, i is necessay ha he disibuo ceaes a base se of keys S and a se Q of q maks. Then, he divides he daa ino blocks and each block m, ino ` segmens. Nex, he andomly chooses wo ses of keys fom S, namely a se {s 1,...,s`} of ` keys and a se of q` keys, as pesened in Table 3.1. The fis se of keys, is used by he disibuo in consucing he session key s = s 1... s`, while he second se, in encyping he keys s 1,...,s` (Table 3.) in he following way: evey 33

36 k 1,1 k 1, k 1,` k,1 k, k,` k q,1 k q, k q,` Table 3.1: The se of q` keys ha ae used o encyp keys {s 1,...,s`}. E k1,1 (s 1 ) E k1, (s ) E k1,`(s`) E k,1 (s 1 ) E k, (s ) E k,`(s`) E kq,1 (s 1 ) E kq, (s ) E kq,`(s`) Table 3.: The encypions in he enabling block, whee ` denoes he numbe of segmens and q he numbe of maks. key s j is encyped unde keys k i,j, fo all i =1,...,q. Finally, he disibuo fis ansmis he encypions of he keys s 1,...,s` (enabling block), and nex he encypion of he daa block m unde he session key s (ciphe block). A he eceives end, i is clea ha only he uses who possess he keys used fo he encypion, ae able o decyp he ansmied daa. Hence, in ode fo he egiseed uses o obain he clea daa, he disibuo povides each one of hem wih a smacad, which he uses use as inpu o a decode box. Each smacad conains a diffeen se of ` keys, ha allow he uses o decyp he encyped message. Specifically, he se consiss of one key fom each column of he able of keys k i,j (Table 3.1). In his way, all auhoised uses ae able o decyp he eceived E i,j (s j ), since he encypion was caied ou using keys k i,j, fo all i =1,...,q and j =1,...,`. Finally, he decypion of he ciphe block E s (m) is now possible, by consucing he key s = s 1... s`. Wih egad o he acions of he aios, we conside wo cases. The 34

37 fis case involves only one aio, wheeas he second a coaliion, which is fomed by a mos auhoised uses. The case of a sole aio is easily addessed, as his smacad uniquely idenifies him. On he ohe hand, he case of a coaliion is moe complex. The aios ae aiming o help an unauhoised use (piae) ceae an illegal smacad ha will decyp he ansmied encyped daa. In ode o achieve successful decypion, he aios mus give away he keys fom hei smacads. Since he smacads uniquely idenify he uses, o avoid being capued, he aios load he piae smacad wih a combinaion of hei keys. This shuffle of keys fom diffeen uses esul in beaking he connecion beween hem and he new ceaed smacad. Anohe applicaion of fingepining, which can also be found in he suvey pape by Blackbun [8], is he case whee he digial daa is disibued hough he Inene o in a CD/DVD fom. In his scenaio, he uses do no eceive he same copies of he daa, bu copies which ae maked diffeenly. Le us conside he case whee he ceao/owne of he daa, disibues copies of a film wien on DVD, o he egiseed uses. The disibuo, associaes each use wih a codewod chosen fom a code C of lengh `. Nex, he maks he copy wih his codewod and sends he maked DVD o he use. We call he collecion of maks in each copy, a fingepin. In his way, each use coesponds o a diffeen copy of he film, which uniquely idenifies him. This implies ha if only one use acs aioously and edisibues his copy, hen he illegal DVD ivially indicaes him as a aio. In he case whee a coaliion is fomed, he aios can hide hei ideniy by combining he diffeen fingepins ha ae embedded in hei copies. By he consucion, he maks ae impecepible and hence he only way o deec hei pesence is o examine he diffeences amongs he copies of he coaliion. This means ha in he piae copy, he aios have he feedom o modify he posiions whee hei copies diffe. In paicula, hey can apply 35

38 one of he following modificaions in hese deecable posiions: (a) use he values of he coesponding posiions fom hei copies, (b) use he values of he coesponding posiions fom hei copies, delee he value o un i uneadable, (c) use abiay values ha comply wih he alphabe ha is used, (d) use abiay values ha comply wih he alphabe ha is used, delee he value o un i uneadable. Fo he pevious applicaion i is clea ha he only opion fo consucing he illegal fingepin is (a), as he fingepin plays he ole of he cypogaphic key. Moeove, even if he fingepin does no have key popeies, mehod (a) could be applied by he aios in he case whee i is impossible o deec he posiions of he maks. Thus, he coaliion can only combine hei fingepins, wihou being able o modify hem. In he case whee he maks ae paially deeced, he aios could apply he model (b) and emove he fingepins fom he known posiions. In opposiion o he fis applicaion, in he example wih he DVD disibuion, i is no essenial fo he aios o use he values fom hei fingepined copies. If he maks ae visible, bu fo some eason hei exacion is eihe no feasible o i would cause qualiy degade of he daa, hen in ode o hide hei ideniy he membes of he coaliion would modify he fingepins by changing hei value (model (c)). Finally, when he fingepins ae visible and (paially) emovable, model (d) povides he bes saegy, as i ceaes a piae copy whose fingepin is as unconneced o he coaliion s fingepins as possible. 3. The Descendan Se Boh applicaions menioned above, indicae ha he aios capabiliy is esiced. This esicion is descibed by he Making Assumpion: 36

39 he membes of he coaliion can only ale hose coodinaes of he fingepin in which a leas wo of hei fingepins diffe, as saed in [4]. To summaise, he fingepining poblem focuses on he consucion of a code C wih he popey ha he disibuo is always able o idenify a leas one membe of he coaliion, which has size a mos. Le Q denoe fo he emainde of he fis pa of he hesis, an alphabe of size q. Also, le U = {u 1,...,u } denoe he se of aios and D = {y 1,...,y } he fingepins ha coespond o each u i, fo all i =1,...,. The se of he illegal fingepins is called he descendan se, as all hese fingepins deive fom he fingepins of he coaliion. Howeve, in he lieaue he descendan se could be found unde he name of envelope [4] o feasible se [50, 15]. Thee ae fou diffeen advesay models o ceae he descendan se ha coespond o he fou opions (a)-(d) above: he naow-sense, he wide-sense and hei expanded vesions. The fis, he naow-sense descendan se, denoed by desc(d), is defined as he se of all x Q` which ae geneaed using lees only fom he codewods in D: desc(d) ={x Q` : x i {y 1 i,...,y i}}. The wide-sense descendan se, denoed wdesc(d), allows he aios o subsiue he maks ha hey ae able o deec by any mak of he alphabe Q: x wdesc(d) if and only if x = x 1...x`, whee 8 < x i = yi 1, if yi 1 = yi =...= yi : x i Q, ohewise. Each one of he defined descendan ses can be exended by inoducing he symbol?, which epesens deleion o an uneadable mak. The expanded naow-sense and expanded wide-sense descendan ses, denoed desc (D) and wdesc (D) accodingly, ae defined as follows: 37

40 desc (D): x desc (D) if and only if x = x 1...x`, whee 8 < x i = yi 1, if yi 1 = yi =...= yi : x i {yi 1,...,yi} [ {?}, ohewise. wdesc (D): x wdesc (D) if and only if x = x 1...x`, whee 8 < x i = yi 1, if yi 1 = yi =...= yi : x i Q [ {?}, ohewise. The example ha follows descibes in a concee way he definiion of he descendan ses. Example Le D = {1001, 01, 001} be a subse of a code C ove Q = {0, 1, }. Then desc(d) ={1001, 101, 001, 01}, desc (D) ={1001, 101, 10?1, 001, 01, 0?1,?001,?01,?0?1}, wdesc(d) ={0001, 0011, 001, 1001, 1011, 101, 001, 011, 01}, wdesc (D) ={0001, 0011, 001, 00?1, 1001, 1011, 101, 10?1, 001, 011, 01, 0?1,?001,?011,?01,?0?1}. 3.3 Fingepining Codes This secion pesens he fou main ypes of fingepining codes, namely famepoof, secue famepoof, idenifying paen popey (IPP) and aceabiliy codes Famepoof Codes As menioned in boh applicaions, if hee is only one auhoised use who acs aioously and edisibues his copy, hen he illegal copy will idenify him, since i beas his fingepin. Howeve, i is possible ha he aio peends o be innocen and asses ha he has been famed. Theefoe, 38

41 i is necessay fo he disibuo o be able o idenify he ue aio and peven he capue of innocen uses. In 1995, Boneh and Shaw [15] inoduced a new noion of fingepining, which possesses exacly his popey. Insead of concenaing on acing a leas one aio, he aim is o peven he membes of he coaliion fom faming a membe ha does no belong in he coaliion. A code C ha ensues ha an innocen auhoised use is no famed by he aios is called a famepoof code. Definiion A code C ove he alphabe Q is called -famepoof o naow-sense -famepoof, denoed by -FP, if fo evey subse D of C wih D apple, we have ha desc(d) \ C = D. Definiion A code C ove he alphabe Q is called expanded naowsense -famepoof, denoed by -FP, if fo evey subse D of C wih D apple, we have ha desc (D) \ C = D. Definiion A code C ove he alphabe Q is called wide-sense -famepoof, denoed by -wfp, if fo evey subse D of C wih D apple, we have ha wdesc(d) \ C = D. Definiion A code C ove he alphabe Q is called expanded wide-sense -famepoof, denoed by -wfp, if fo evey subse D of C wih D apple, we have ha wdesc (D) \ C = D. Noice, ha he absence of any chaaceisaion elaed o he ype of descendan se, implies ha he naow-sense model is used Secue Famepoof Codes The second ype of fingepining code is called a secue famepoof code and was fis inoduced in [49] by Sinson, van Tung and Wei. Moe pecisely, hey defined secue famepoof codes unde he wide-sense descendan model. The siuaion ha iggeed he idea of his ype of codes, was he discouaging esul of Boneh and Shaw [15] poving he non exisence 39

42 of deeminisic wide-sense fingepining codes, ha can idenify a leas one aio. On he ohe hand, famepoof codes do no povide any fom of aceabiliy. Hence, secue famepoof codes wee defined in ode o senghen he family of wide-sense fingepining codes. Le x be an illegal fingepin. Since x could have been poduced by moe han one coaliion, ideally we would like o idenify as a aio, he use whose fingepin belongs o he inesecion of all possible coaliions ha could ceae x. As his is an impossible siuaion (accoding o he esul of Boneh and Shaw), Sinson, van Tung and Wei equied he following popey: fo evey pai of disjoin coaliions D 1 and D, hei descendan ses ae also disjoin. Similaly o he famepoof codes, secue famepoof codes ae defined diffeenly, depending each ime on he descendan se model. Definiion A code C ove he alphabe Q is called -secue famepoof, denoed by -SFP, if fo all disinc subses D, D 0 of C such ha D apple and D 0 apple, we have ha if desc(d) \ desc(d 0 ) 6= ;, hen D \ D 0 6= ;. Definiion A code C ove he alphabe Q is called expanded naowsense -secue famepoof, denoed by -SFP, if fo all disinc subses D, D 0 of C such ha D apple and D 0 apple, we have ha if desc (D) \ desc (D 0 ) 6= ;, hen D \ D 0 6= ;. Definiion A code C ove he alphabe Q is called wide-sense -secue famepoof, denoed by -wsfp, if fo all disinc subses D, D 0 of C such ha D apple and D 0 apple, we have ha if wdesc(d)\wdesc(d 0 ) 6= ;, hen D\D 0 6= ;. Definiion A code C ove he alphabe Q is called expanded wide-sense -secue famepoof, denoed by -wsfp, if fo all disinc subses D, D 0 of C such ha D apple and D 0 apple, we have ha if wdesc (D) \ wdesc (D 0 ) 6= ;, hen D \ D 0 6= ;. 40

43 3.3.3 IPP Codes Codes wih he idenifying paen popey wee fis inoduced by Hollmann, van Lin, Linnaz and Tolhuizen [30] (he case of wo piaes, =) and by Saddon, Sinson and Wei [47] fo any se of aios wih a mos membes. In conas o he peviously defined ypes of fingepining codes, IPP codes possess a song aceabiliy popey, as hey ensue he deecion of a leas one membe of he coaliion. This is achieved by idenifying as a aio he use whose fingepin belongs o he inesecion of fingepins of all he coaliions of ceain size, ha could geneae he illegal fingepin. Fo he definiion of his ype of codes, i is necessay o define fis he se of poenial paens of a wod and he descendan se of a code. Definiion Le C be a code of lengh ` ove he alphabe Q. Fo x Q` define P,C (x) ={D C : D apple and x desc(d)} o be he se of all possible subses of codewods ha x descended fom. The se P,C (x) is called he poenial paen se of x. In he case whee he wide-sense, expanded naow-sense and expanded wide-sense descendan is being used, he coesponding poenial paen ses ae denoed by P,C w ( ), P,C ( ) and Pw,,C ( ). Apa fom he descendan se of a subse of a code, we can also define he descendan se of a code as follows: desc (C) = desc (C) = wdesc (C) = wdesc (C) = [ D C, D apple [ D C, D apple [ D C, D apple [ D C, D apple desc(d) desc (D) wdesc(d) wdesc (D) (naow-sense model), (expanded naow-sense model), (wide-sense model), (expanded wide-sense model). 41

44 As expeced, each of he fou ypes of he descendan se leads o diffeen definiions of IPP codes. Definiion A code C ove he alphabe Q has he -idenifiable paen popey, if fo all x desc (C) we have ha We denoe his code by -IPP. \ DP,C (x) D 6= ;. Definiion A code C ove he alphabe Q has he expanded naowsense -idenifiable paen popey, if fo all x desc (C) we have ha We denoe his code by -IPP. \ DP,C (x) D 6= ;. Definiion A code C ove he alphabe Q has he wide-sense -idenifiable paen popey, if fo all x wdesc (C) we have ha We denoe his code by -wipp. \ DP w,c (x) D 6= ;. Definiion A code C ove he alphabe Q has he expanded wide-sense -idenifiable paen popey, if fo all x wdesc (C) we have ha We denoe his code by -wipp. \ DP w,,c (x) D 6= ;. Befoe inoducing he fouh fingepining code, we pesen a new caegoy of codes, called oally secue codes ha wee fis defined by Boneh and Shaw in [15]. The eason hese codes ae examined hee, is because hey ae equivalen o IPP codes, as we will sholy pove, and hence he esuls of oally secue codes can be applied o IPP codes and vice vesa. Like IPP 4

45 codes, oally secue codes also possess he popey of idenifying a aio and in ode o achieve his, he pesence of a acing algoihm is equied. This algoihm is hough of as a funcion A : Q`! {1,...,}, which on inpu he illegal fingepin x Q` oupus a membe of he coaliion. In he oiginal pape [15], oally secue codes wee defined ove {0, 1} bu hee ae genealised ove he alphabe Q. Definiion (Definiion 4.1, [15]). Le Q be an alphabe of size q. A code C is oally -wsecue code of lengh `, if hee exiss a acing algoihm A : Q`! {1,...,} saisfying he following condiion: if a coaliion D of a mos uses geneaes a wod x wdesc (D), hen A(x) D. Nex, we pove ha hese codes ae equivalen o -wipp codes. Poposiion A code C is -wipp if and only if C is a oally -wsecue code. Poof. Fis, assume ha C is a -wipp code of lengh `, ove he alphabe Q. Le D 0 C wih D 0 apple and x wdesc (D 0 ). In ode o pove ha C is oally -wsecue, we need o show ha hee exiss an algoihm A : Q`! {1,...,}, such ha A(x) D 0. Since C is -wipp, hen fo evey x wdesc (C) \ DP w,,c (x) D 6= ;. As D 0 P w,,c (x), we have ha given he elemen x, hee exiss a y T DP w,,c (x) D, which implies ha y D 0. In ohe wods, hee exiss an algoihm A : Q`! {1,...,}, such ha A(x) =y D 0, which means ha C is oally -wsecue. Fo he evese diecion, assume ha C is a oally -wsecue code ove Q and le x 0 be an elemen of wdesc (C). Since C is oally -wsecue, hee exiss an algoihm A : Q`! {1,...,} such ha on inpu a x Q` oupus a membe of he coaliion ha poduced x. Le y 0 = A(x 0 ) and assume fo 43

46 a conadicion ha he inesecion of all poenial ses of paens of x 0 is empy, ha is \ DP w,,c (x 0) D = ;. This implies ha hee exiss a se D P w,,c (x 0), such ha y 0 / D, which means ha he popey of oally -wsecue code C failed fo he se D. A conadicion Taceabiliy Codes Taceabiliy codes wee he fis ype of digial fingepining o be inoduced and wee defined by Cho, Fia and Nao [18] in ode o peven illegal edisibuion of digial daa. As hey guaanee he idenificaion of a aio once he illegal fingepin is found, aceabiliy codes ae a subse of he family of IPP codes. Howeve, hei impoan feaue is he algoihm hey povide in ode o accomplish he idenificaion of he aio. This algoihm is deeminisic and is based on he examinaion of he Hamming disance beween codewods and wods of he descendan se. Fis ae defined he naow-sense and wide sense aceabiliy codes. Definiion A code C Q` is a -aceabiliy code, denoed -TA, if fo evey D C wih D apple and fo evey x desc(d), hee exiss a leas one y D such ha d(x, y) <d(x, z) 8z C D, whee d(, ) is he Hamming disance. Definiion A code C Q` is a wide-sense -aceabiliy code, denoed -wta, if fo evey D C wih D apple and fo evey x wdesc(d), hee exiss a leas one y D such ha d(x, y) <d(x, z) 8z C D, whee d(, ) is he Hamming disance. 44

47 Befoe he definiion of he expanded naow-sense and expanded widesense aceabiliy codes, we inoduce he following definiions of he disance beween a codewod and a wod ha belongs o he descendan se. This is a necessay definiion, as in he expanded cases a wod fom he descendan se migh conain he? symbol, which sands fo he deleion of he value on ha posiion o an uneadable mak. The exisence of? gives ise o wo diffeen ways of defining he disance. The fis, denoed by d 1 (, ), is he known Hamming disance, wheeas he second, d (, ) eas he? as being he same as he lee ha is compaed o, and hus he disance is zeo. Definiion Fo evey a Q and b Q [ {?} he disance d 1 (, ) beween a and b is defined as 8 1, if a 6= b >< d 1 (a, b) = 1, if b =? >: 0, if a = b. Definiion Fo evey a Q and b Q [ {?} he disance d (, ) beween a and b is defined as: 8 1, if a 6= b and b Q >< d (a, b) = 0, if b =? >: 0, if a = b. I is easy o noice he elaion beween hese wo definiions of disance: fo all a Q and b Q [ {?} we have 8 < d 1 (a, b), if b 6=? d (a, b) = : d 1 (a, b) 1, if b =?. (3.3.1) The above definiions can be easily genealised o he disance of wods of lengh geae han 1. Le n be an inege, and a = a 1...a n Q n, b = b 1...b n (Q [ {?}) n be wods of lengh n. Then, nx d 1 (a, b) = d 1 (a i,b i ) i=1 45