- Email: [email protected]
A hypergraph approach to digital fingerprinting
A hypergraph approach to digital ngerprinting Gerard Cohen ENST, 46 rue Barrault, 75013 Paris, France, e-mail: [email protected]
Sylvia Encheva HSH, Bj rnsonsg. 45, 5528 Haugesund, Norway, e-mail: [email protected]
Gilles Zemor ENST, 46 rue Barrault, 75013 Paris, France, e-mail: [email protected]
Abstract We present a hypergraph treatment for a problem in digital ngerprinting. Key words: Helly-property, error-correcting codes, digital ngerprinting.
1
Introduction
With digital ngerprinting, a publisher embeds a unique ngerprint pattern into each distributed copy of a document. If an illegal copy is discovered, he may trace to the o ending user. Wen users collude, they can locate the di erences and combine their copies into a new one. Following [2] (see also [4]), we want to prevent forging a copy with no member of the coalition being caught. 2 Partially identifying codes
A subset of GF (q )n, the n-th dimensional vector space over the nite eld with q elements, is called an (n; M )-code when j j = M , and its elements codewords; those in GF (q )n n unregistered words. Suppose C . For any S position i de ne the projection Pi (C ) = a2C ai : We shall say that position i Preprint submitted to Elsevier Preprint
30 May 2000
is undetectable for C if jPi j = 1. De ne the feasible set of C by: F (C ) = fx 2 ( )n : 8i; xi 2 Pi g:
GF q
Two non intersecting coalitions should not produce the same descendant. This motivates the following reworded de nition from [4].
De nition 1 An (n; M )- code
8C C ;
0
is called a t-partially identifying code if, such that jCj t; jC j t and C \C = ;, we have F (C ) \ F (C ) = ;. 0
0
0
Proposition 1 If a code q has minimum distance d satisfying 2 d ((t 1)=t2 )n, then it is t-partially identifying. 3 Hypergraph approach The set of users is now identi ed with a set of vertices V . We consider an unregistered word s (for son) and denote by H(s) the hypergraph whose hyperedges are the coalitions of size at most t able to produce s, i.e, E (s) = fC V; jCj t : s 2 F (C )g. Then De nition 1 reads: A code is partially identifying if, for any s, any two hyperedges of H(s) have a nonempty intersection. The set V is partially identifying if H(s) is partially identifying for all s. We now de ne a stronger property, total identi cation.
De nition 2 A code is totally t-identifying if, for all s, the intersection of all edges of H(s) is nonempty. This means that H(s) is a star, its center belonging to any coalition producing s. Thus, upon seizing s, it is possible to trace a subcoalition having produced it. Recall that a family of sets has the t-Helly property if every t-wise intersecting nite subfamily is a star. We need a result due to Berge and Duchet ([1]):
Theorem 1 A hypergraph has the t-Helly property if and only if, for every set A of t + 1 vertices, all the edges E such that jE \ Aj t share a common vertex.
H(s) has the t-Helly property if and only if it does not contain the complete t-uniform subhypergraph on t + 1 vertices Kt (t + 1) (also called the t- simplex). Corollary 1
Proof: Consider any subset A of size t + 1. Since all hyperedges in H(s) have size at most t, the only relevant ones to apply Berge-Duchet Theorem are those of size t contained in A. Clearly, taking all the e such edges, jE1 \ E2 ::: \ Ee j = t + 1 e. Thus, the induced subhypergraph on A is not complete (e < t + 1)
2
if and only if the previous intersection is nonempty. 4 Back to codes
Again, the set of users is q GF (q )n . The absence of triangle is equivalent to 3-hashing. We now generalize this result. 4.1
Hashing families
A subset q of GF (q )n is said to be t-hashing if any t of its members have t distinct entries in some common coordinate i 2 [1 : : : n]. Proposition 2
q is
(t + 1)-hashing if and only if it has the t-Helly property.
Proof:
If q is (t + 1)-hashing, then for any set A of t + 1 users, there is a coordinate i where A projects injectively on a subset of GF (q ) of size t + 1. On this coordinate, the t +1 possible coalitions of size t cannot have a common component. On the other hand, if q is at most t-hashing, then it is indeed possible to nd a common component. In other words, consider the simplex Kt (t + 1). Assign colours to its t + 1 vertices; then colour the faces (coalitions of maximal size t) by assigning every face the colour of any of its vertices. Then there exists a monochromatic colouring of the t + 1 faces of the simplex if and only if at most t colours were used for the vertices. Again, there is a suÆcient condition for a code to be (t + 1)-hashing, Proposition 3
1
4.2
Let q be a code of minimum Hamming distance d. If 2=t(t + 1), then its elements make up a (t + 1)-hashing family.
d=n >
Coalitions of size at most 3
By possibly including \dummy" members, all coalitions are assumed to have size exactly 3. We search for conditions insuring that, for any s, any subfamily of three hyperedges in H(s) is 3-wise intersecting (a local star). Combined with the Helly property, this will give us that q is totally 3-identifying. Proposition 4
For any
s,
a subhypergraph of three hyperedges in
3-wise intersecting if and only if i) and ii) hold.
3
H( ) s
is
i) Any two hyperedges intersect. ii) No three hyperedges
E; E ; E 0
00
form a triangle.
Condition i) is equivalent to 3-partial identi cation. 5 The star system in the binary world
In this section, V = GF (2) . Here, we cannot hope for totally identifying codes, since the (t + 1)-hashing property already fails for t = 2. Instead, we get upperbounds on the number of suspect coalitions or users if some con gurations (like stars) are avoided. We shall invoke the following result of Deza, slightly reworded. Theorem 2 ([3]) If C1 ; :::; C are coalitions of size at most t, any two of which intersect in exactly elements, then either m < t2 t + 2 or they form a star n
m
with a center of size
.
Upon seizing s, the \police" know H(s), and they are interested in capturing \pirates". This is possible with certainty only if H(s) is a star. If not, our goal is to obtain a subset of suspects with the highest density of pirates. The relevant notion is that of i-hitting set or i-transversal T , de ned as a subset of the vertices intersecting every hyperedge in at least i elements. We are thus searching for := Max fi=jT jg. Clearly 1 , with equality characterizing stars. i
i
i
References [1] C. Berge and P. Duchet, \A generalisation of Gilmore's theorem", Advances in Graph Theory, ed. M. Fiedler (Acad. Praha), pp. 49-55. [2] D. Boneh and J. Shaw, \Collusion-secure ngerprinting for digital data", 963, pp. 452-465, 1995.
Recent
LNCS
[3] M. Deza, \Une propriete extremale des plans projectifs nis dans une classe de codes equidistants", Discrete Math. 6 (1973) 343-352. [4] D.R. Stinson, Tran van Trung and R. Wei, \Secure Frameproof Codes, Key Distribution Patterns, Group Testing Algorithms and Related Structures", J. Stat. Planning and Inference, vol. 86 (2)(2000), pp. 595-617.
4
Sylvia Encheva HSH, Bj rnsonsg. 45, 5528 Haugesund, Norway, e-mail: [email protected]
Gilles Zemor ENST, 46 rue Barrault, 75013 Paris, France, e-mail: [email protected]
Abstract We present a hypergraph treatment for a problem in digital ngerprinting. Key words: Helly-property, error-correcting codes, digital ngerprinting.
1
Introduction
With digital ngerprinting, a publisher embeds a unique ngerprint pattern into each distributed copy of a document. If an illegal copy is discovered, he may trace to the o ending user. Wen users collude, they can locate the di erences and combine their copies into a new one. Following [2] (see also [4]), we want to prevent forging a copy with no member of the coalition being caught. 2 Partially identifying codes
A subset of GF (q )n, the n-th dimensional vector space over the nite eld with q elements, is called an (n; M )-code when j j = M , and its elements codewords; those in GF (q )n n unregistered words. Suppose C . For any S position i de ne the projection Pi (C ) = a2C ai : We shall say that position i Preprint submitted to Elsevier Preprint
30 May 2000
is undetectable for C if jPi j = 1. De ne the feasible set of C by: F (C ) = fx 2 ( )n : 8i; xi 2 Pi g:
GF q
Two non intersecting coalitions should not produce the same descendant. This motivates the following reworded de nition from [4].
De nition 1 An (n; M )- code
8C C ;
0
is called a t-partially identifying code if, such that jCj t; jC j t and C \C = ;, we have F (C ) \ F (C ) = ;. 0
0
0
Proposition 1 If a code q has minimum distance d satisfying 2 d ((t 1)=t2 )n, then it is t-partially identifying. 3 Hypergraph approach The set of users is now identi ed with a set of vertices V . We consider an unregistered word s (for son) and denote by H(s) the hypergraph whose hyperedges are the coalitions of size at most t able to produce s, i.e, E (s) = fC V; jCj t : s 2 F (C )g. Then De nition 1 reads: A code is partially identifying if, for any s, any two hyperedges of H(s) have a nonempty intersection. The set V is partially identifying if H(s) is partially identifying for all s. We now de ne a stronger property, total identi cation.
De nition 2 A code is totally t-identifying if, for all s, the intersection of all edges of H(s) is nonempty. This means that H(s) is a star, its center belonging to any coalition producing s. Thus, upon seizing s, it is possible to trace a subcoalition having produced it. Recall that a family of sets has the t-Helly property if every t-wise intersecting nite subfamily is a star. We need a result due to Berge and Duchet ([1]):
Theorem 1 A hypergraph has the t-Helly property if and only if, for every set A of t + 1 vertices, all the edges E such that jE \ Aj t share a common vertex.
H(s) has the t-Helly property if and only if it does not contain the complete t-uniform subhypergraph on t + 1 vertices Kt (t + 1) (also called the t- simplex). Corollary 1
Proof: Consider any subset A of size t + 1. Since all hyperedges in H(s) have size at most t, the only relevant ones to apply Berge-Duchet Theorem are those of size t contained in A. Clearly, taking all the e such edges, jE1 \ E2 ::: \ Ee j = t + 1 e. Thus, the induced subhypergraph on A is not complete (e < t + 1)
2
if and only if the previous intersection is nonempty. 4 Back to codes
Again, the set of users is q GF (q )n . The absence of triangle is equivalent to 3-hashing. We now generalize this result. 4.1
Hashing families
A subset q of GF (q )n is said to be t-hashing if any t of its members have t distinct entries in some common coordinate i 2 [1 : : : n]. Proposition 2
q is
(t + 1)-hashing if and only if it has the t-Helly property.
Proof:
If q is (t + 1)-hashing, then for any set A of t + 1 users, there is a coordinate i where A projects injectively on a subset of GF (q ) of size t + 1. On this coordinate, the t +1 possible coalitions of size t cannot have a common component. On the other hand, if q is at most t-hashing, then it is indeed possible to nd a common component. In other words, consider the simplex Kt (t + 1). Assign colours to its t + 1 vertices; then colour the faces (coalitions of maximal size t) by assigning every face the colour of any of its vertices. Then there exists a monochromatic colouring of the t + 1 faces of the simplex if and only if at most t colours were used for the vertices. Again, there is a suÆcient condition for a code to be (t + 1)-hashing, Proposition 3
1
4.2
Let q be a code of minimum Hamming distance d. If 2=t(t + 1), then its elements make up a (t + 1)-hashing family.
d=n >
Coalitions of size at most 3
By possibly including \dummy" members, all coalitions are assumed to have size exactly 3. We search for conditions insuring that, for any s, any subfamily of three hyperedges in H(s) is 3-wise intersecting (a local star). Combined with the Helly property, this will give us that q is totally 3-identifying. Proposition 4
For any
s,
a subhypergraph of three hyperedges in
3-wise intersecting if and only if i) and ii) hold.
3
H( ) s
is
i) Any two hyperedges intersect. ii) No three hyperedges
E; E ; E 0
00
form a triangle.
Condition i) is equivalent to 3-partial identi cation. 5 The star system in the binary world
In this section, V = GF (2) . Here, we cannot hope for totally identifying codes, since the (t + 1)-hashing property already fails for t = 2. Instead, we get upperbounds on the number of suspect coalitions or users if some con gurations (like stars) are avoided. We shall invoke the following result of Deza, slightly reworded. Theorem 2 ([3]) If C1 ; :::; C are coalitions of size at most t, any two of which intersect in exactly elements, then either m < t2 t + 2 or they form a star n
m
with a center of size
.
Upon seizing s, the \police" know H(s), and they are interested in capturing \pirates". This is possible with certainty only if H(s) is a star. If not, our goal is to obtain a subset of suspects with the highest density of pirates. The relevant notion is that of i-hitting set or i-transversal T , de ned as a subset of the vertices intersecting every hyperedge in at least i elements. We are thus searching for := Max fi=jT jg. Clearly 1 , with equality characterizing stars. i
i
i
References [1] C. Berge and P. Duchet, \A generalisation of Gilmore's theorem", Advances in Graph Theory, ed. M. Fiedler (Acad. Praha), pp. 49-55. [2] D. Boneh and J. Shaw, \Collusion-secure ngerprinting for digital data", 963, pp. 452-465, 1995.
Recent
LNCS
[3] M. Deza, \Une propriete extremale des plans projectifs nis dans une classe de codes equidistants", Discrete Math. 6 (1973) 343-352. [4] D.R. Stinson, Tran van Trung and R. Wei, \Secure Frameproof Codes, Key Distribution Patterns, Group Testing Algorithms and Related Structures", J. Stat. Planning and Inference, vol. 86 (2)(2000), pp. 595-617.
4