Efficient Dynamic Traitor Tracing

1
Efficient Dynamic Traitor Tracing

• R91922005 ???
• R91922006 ???

2
Reference

• Efficient Dynamic Traitor Tracing O. Berkman,
  M. Parnas, and J. Sgall SODA 2000
• Dynamic Traitor Tracing A. Fiat and T. Tassa
  CRYPTO99

3
Outline

• Introduction
• Dynamic Traitor Tracing
• Previous Works by Fiat and Tassa
• Preliminaries
• Proposed Algorithms
• The clique algorithm
• The optimal algorithm
• Many algorithms as building-blocks

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Introduction

• Dynamic Traitor Tracing
• Number of traitors is unknown

5
Dynamic Traitor Tracing

• Use watermarking techniques
• Two important concerns
• Number of versions/colors
• Limited bandwidth
• Number of rounds

6
Previous Works by Fiat

• Any deterministic algorithm must use at least p1
  colors to locate a traitor
• Use 2p1 colors in O(p log n) rounds
• Use p1 colors in O(3pp log n) rounds

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Proposed in This Paper

• Use p1 colors in T(p2p log n)
• Optimal solution using only p1 colors
• Use pc1 colors in O(p2/cp log n)
• 1 ? c ? p
• Use pc1 colors in O(p logc n)
• 2 ? c
• A tight lower bound for pc colors
• O(p2/cp logc1 n)

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Preliminaries

• Notation
• U the set of users
• n the number of users, Un
• T the set of traitors, T is in U
• p the number of traitors, T p

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Lemma 2.1

• Modify mp-1 to m is trivial
• How to modify m to mp-1?
• A locates a traitor in m rounds
• B locates all traitors in mp-1 rounds

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From m to mp-1

• At the mth round of algorithm A, all traitor must
  have a unique color

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Representation

• Represent the current state of the algorithm by
  an undirected graph G(V,E)
• Each vertex represents a subset of users
• Each user belongs to exactly one vertex
• An edge (X,Y) means that X?Y contains a traitor
• A special vertex I

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Basic Algorithm

• Start with IU, VI, Eempty, t0
• t is the lower bound of traitors
• Find a vertex X that contains a traitor
• Use 2t1 colors in O(p log n) rounds
• Use t1 colors in exponential rounds
• If XI, split I into two vertices, and connect
  them. Set Iempty, tt1
• Else set II?Y where (X,Y) is in E. Split X into
  two vertices and connect them

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Basic Algorithm Example
14
(t,k)-Graph
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Some Facts of (t,k)-Graph

• A clique Qi of ti vertices contains at least ti-1
  traitors
• The number of vertices of a (t,k)-graph is at
  most 2t1

X3
X4
X1
I
X5
X2
A valid (3,2)-graph
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The Clique Algorithm

• Vertices are partitioned into two zones
• Z1 is partitioned into blocks
• Each block is induced by the vertices of two
  cliques Qi and Qj
• Z2 contains I and possibly a clique Qi

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Blocks in Z1

• A block does not contain a clique of size titj-1
• There are four distinct vertices X1,X2 in Qi, and
  Y1,Y2 in Qj, such that (X1,Y1), (X2, Y2) are not
  in E

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Z2

• I and the clique Qi do not form a clique of size
  ti1
• There exists a vertex X in Qi, such that (X,I) is
  not in E

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The Algorithm

• Two phases
• Phase 1 Distributing the colors
• Allocate ti-1 colors to each clique Qi, and one
  color to I
• Use only t1 colors
• Phase 2 Reorganizing after pirates answer
• Start with a (0,0)-graph with IU

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Phase 1-1

• For blocks in Z1
• Total available number of colors titj-2
• Use two colors for (X1,Y1),(X2,Y2)
• Remains titj-4 colors
• Use one color for each of the remaining titj-4
  vertices

X1
Y1
tj-2
ti-2
X2
Y2
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Phase 1-2

• For blocks in Z2
• Total available number of colors ti
• Use one color for (X,I)
• Remain ti-1 colors
• Use one color for each of the remaining ti-1
  vertices

ti-1
X
I
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Phase 2-1
Broadcasted color is?
New clique QX1,X2 kk1
No
given to a pair X,Y
Phase 2-2
QiQi\X where Qi contains X
Qi1?
given to only one vertex X
Split X into X1,X2 Connect them
Add the vertex to I QiX1,X2
Yes
No
New clique QX1,X2 Iempty,tt1,kk1
XI?
Yes
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Phase 2-2
given to a pair X,Y
Add edge (X,Y)
Connect between?
New clique QQiUQj Add X to I where X is not in
Q kk-1
pair of clique, and form a titj-1 clique
I and a vertex of Qi, and form a ti1 clique
Add I to Qi Set Iempty, tt1
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Phase 2-3

• When create a new clique Q
• If Z2 contains contains only I
• Place Q in Z2
• Otherwise, Z2 contains also a clique
• Pair these two cliques and create in Z1 a new
  block containing them

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Correctness and Efficiency
26
Lemma 3.1 Proof
27
Lemma 3.1 Proof Illustration
I splits
O(p log n) splits
O(p) times
I splits
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Theorem 3.1 and Proof
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Theorem 3.1 Proof
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Theorem 3.1 Proof
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4 An Optimal Algorithm
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Data Structures and Invariants

• The Graph
• G(V,E)
• The vertices are the subsets of the user set U,
  and form a partition of U.
• If (X,Y)??E, then X?Y contains a traitor.
• Zones and blocks
• G????????????Z1, Z2, Z3, Z4?
• Z1, Z2???,????blocks Bi
• bi ?block Bi????traintor??
• ???block(?zone)????,??????!

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??Zone???

• Z1
• ?????blocks Bi
• ??block Bi?traitor???3?? bi ? 7
• Z2
• ?????blocks Bi
• ??block Bi?vertices???
• bi2(?traitor??2)
• ???vertices??singletons!
• ??block Bi??????cliques(???????)?
• ??Bi????????bi1?clique

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??Zone???

• Z3
• ?????
• ??
• ???clique(?????2),
• ????vertices??singletons!
• Z4
• ?? (t,k)-graph,t ? 2
• Special vertex I(???Innocent users)
• ??????K

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Number of Traitors

• ?zz1z2z3z4
• ziZone Zi????traitors??
• ??T?????traitors??????T,??T ? z?
• Implies the algorithm may use (T1) colors.
• T????????zgtT?,?Tz?

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Marks

• ?Z2?Z3???,??singletons?
• ???????marked
• ?marked?????
• ?marked??????,???traitor!
• ??,
• ???traitors?????T1
• ?T???,???marks??????

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??q-good block

• q-good block????(q1)?usersA1, , Aq, and
  J,??,
• ??set Ai????traitor??
• ??block????q?traitors?

A1
A2
Aj
Aq
J
Aj



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???Z1?blocks?????

• ???????,????????Algorithm (I)(V)?
• (I)(IV)??????(V)?building block?
• ???????,??????
• ?????a subset of 2 users that contains a traitor
• ??????,??????????(?????????,????????)
• ????
• ???????????,???????singleton?

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???????????
40

• ???????????q-good block?
• ??????set Ai?????Ai,0?Ai,1

A1
A2
Aj
Aq
J
Aj



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(1)????coloring
A1
A2
Aj
Aq
J
Aj
A1,0
A1,1
A2,0
A2,1
Aj,aj
Aj,1-aj
Aq,0
Aq,1
Aj,aj
Aj,1-aj
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(1)????coloring
A1
A2
Aj
Aq
J
Aj
A1,0
A1,1
A2,0
A2,1
Aj,aj
Aj,1-aj
Aq,0
Aq,1
Aj,aj
Aj,1-aj
???1(q-2)1q???
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• (2)
• ??pirate???Unique???set Ai,a????
• ????Ai,1-a??J
• Ai,a????traitor,???????Ai,1-a?????
• Set Ai Ai,a,?Ai????Ai,0?Ai,1,??Step (1)
• ????? rounds?,?? ??A???,pirate???outp
  ut???J????????,??block?????q?traitors?
• ????block????q?traitor,??
• A1Aq????traitor,??Ai???????,??????traitor?
• J??traitor
• ?? ??????,????????????J????????,??????tra
  itor,?????? ??A????,???output???J???????
• ??,??block???q?traitors?

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• (3)
• ?????????,??????renumbering?,pirate????????J ?
  A1,1 ? ? Aq,1?????
• ??pirate?????? Aj,0 ? Aj,0 for all j?? j
• Case1??Aj?Aj?singletons
• ?? Aj,0?Aj,0??????traitor??????????!

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• ?(3)
• Case2????Aj???????singleton set
• ???????????traitor?subset Aj,1
• ?Aj,1??J,??AjAj,0,??Step(1)
• (proof)
• A1,0 , , Aq,0??q-cliques(?q-1?traitors)
• ?J?A1,0?A2,1??Aq,1????
• ?A1,1?A2,0?????
• ???????,?????????traitor?subset Aj,1?

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• Algorithm(III) (4.2)
• ???input?clique on 5 vertices,?????4?traitors??4.2
  ????,????There is a 4-good block with Jempty
  set, one of the set being A1A2B, and the other
  3 sets being the remaining vertices.???????C?D?E?
  ???traitor(s),?A1?A2?B???traitor(s)?????5-vertex
  clique???,?????????traitor,??????
• ???4?traitors
• ??traitor???A?B????????traitor,??(A1?A2?B)??set??
  ????traitor???Set1(A1?A2?B), Set2C, Set3D,
  Set4E???4-good block?
• ??traitor???C(?D?E)???????A1?A2?????????traitor,?
  ???traitor??A2, ??A2C??edge????????(4.2)????,??A1A
  2B???????????,??edge????,?????
• ?4???traitors?4-good block

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Step (1)
?3?traitors
II (q3)
?4?traitors
k cliques with k3 vertices k?1,3
k1
k3
Step (2)
3 clique with 6 vertices ????2,2,2?? ????clique???
???traitor, ?????clique???sets, ????3-good block
1 clique with 4 vertices ? A 4-clique
IV
3-good block
3-good block
I (q3)
?4?traitors
Step (2)
48
Step (2)
?4?traitors
II (q4)
k cliques with k4 vertices k?1,3
?5?traitors
Step (2.1)
Step (2.2)
k1
k3
Step (3)
1 clique with 5 vertices ? A 5-clique

• 3 clique with 7 vertices
• ???(??)?1,1,2?traitor(s)
• ???block????disjoint subsets
• clique with at least 1 traitor,
• the other 2 cliques (with at least 3 traitors)

III
2 subsets with 1 and 3 traitors
4-good block
??
I (q4)
??
?5?traitors
Step (3)
49
Step (3)
?5?traitors
II (q5)
Step (3.1)
k cliques with k5 vertices k?1,3,5
?6?traitors
k1,3,5
II (q6)
as in Step (3.1)
k cliques with k6 vertices k?1,3,5
?7?traitors
2 subsets with at least 1 and at least 3
traitors
k1,3,5
??basic algorithm? (locate p traitors with 2p1
colors)
??
??????7????? (?????4?disjoint subsets)
2 subsets with at least 1 and at least 3
traitors
4?subset?13??,?? 2 subsets with 1 / 3 traitors
??
??
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• Zone Z1
• ???blocks????(V),?????z1?????block Bi
  ??(V)?,??????????
• Case 1 Block Bi ?????subsets,?????1?traitor???????
  3??
• ????? zone Z4(??(3))
• ?????zone Z1,????block
• Case 2 ????????traitor?2-user subset???????clique
  Q(???2),?????,?????user???????zone????,????
• ??zone Z3???,?clique Q??Z3?
• ??,?Z3????clique???,?Q????,???Z2??????????cliques?
  block???Z3???
• ???Bi??zone Z4(??(3))
• ??????,Bi???subset X,?????traitor,????zone
  Z4?????????,???????X????,???X???????????????vertic
  es,??????????2?clique,??zone Z4??z4z41???z43,??
  Z4?Step(1.2)?
• Zone Z2
• Run the clique algorithm. Z2?clique???????(blocks?
  ??cliques??)?
• ?????vertices??singleton,??????????????(unless we
  locate a traitor)
• ??,?block Bi????clique???????? bi1 ?clique Q
• ? Bi?? ??????K(??? bi1 ?clique
  Q?????bi?traitors),????????zone,????
• ??zone Z3???,?clique Q??Z3?
• ??,?Z3????clique???,?Q????,???Z2??????????cliques?
  block???Z3???

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• Zone Z3 and Z4 (??zone?????)
• ?? zT,?? z3z41 ?????zltT,??z3z42????
• ?Z4?,????????
• (1.1) ?Z4?,run the clique algorithm,??????Z4??
  z43 ?traitors?
• ??????O(logn)??(???????z43)
• (1.2) ? z43 ,??Z4??,?????block??Z1??Zone Z4???,?
  z40,Kempty set.
• ??Z3????,??????
• ??Z3????,?? zltT
• ?Z3?????,??????????
• ??Z3????,?? zT???????,??Z4???
• ????,Z4???,??X1, , Xq??subsets,???qz41???????r
  ound?????q?rounds
• ???? Z3 ??? marked ?? S??round k (k1, , q)
  ?,?????,?Z4??subsets X1, , Xq??,??? S ? Xk
  ?????,???,?Z3???????????????????
• (3.1) If pirate broadcasts the color of Xj, for
  some j?k?Z4?????????
• (3.2) ??pirate broadcast???Xk?S (for k1,,q)
• Mark S(??S???traitor,??Z4?? q
  (z41)?traitors(??z1T1 traitors))
• ??Z3???????marked?,?TT1(??????marks)
• ??q rounds in Z3?,?????
• mark???vertex,???

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Correctness and Efficiency

• Theorem 4.1. The algorithm locates p traitors in
  O(p2plogn) rounds, using p1 colors
• ??????
• z1 z1z2z3z41 if z T.
• z2 z1z2z3z42 if z lt T.
• ????T1 p1 ???

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Correctness and Efficiency-- ????????

• ? sz2z3
• r3bz2z3z4
• (bZ1?blocks??)
• ?????,0 T, s, r p ???
• s?T????(T???traitor??)
• sz2z3?
• Z2????blocks(paired cliques)
• Z3???(???)clique
• Z2??blocks??merge????bi1,??????Z3
• ??Z3????clique,??????clique????,?????clique pair
  ???block,??Z2?
• ??sz2z3????????

vertices??singletons
54
Correctness and Efficiency-- ????????

• s?T????,??????p??
• r3bz2z3z4
• ???Z1????a subset of 2 users contains a
  traitor??????
• b b-1 (Z1?????block Bi)
• z4z41 (???1?traitor?subset X? ??Z4)
• z2z3z2z31 (2-user subset??????Z3 ??Z2)
• r ?? 1 (paper??2,?????1)
• s ?? 1
• ??,r??????2p?

55
Correctness and Efficiency-- ????????

• r??
• ??O(logn) rounds
• ????2p?,?????O(plogn).
• ??rounds?????mark vertex,???cliques of
  singleton??
• unmark vertex?remove edge???T??,????clique merge
• ??s????p?,????O(p)???unmarked,???O(p2) rounds?
• s???,????clique??(p?),?????clique
  merge????????p?????p???1?marked
  vertex????????O(p2)
• Total number of rounds is O(p2plogn)