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Parent Identifying Codes The case of multiple parents

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Lemma 4: ft(s(t),q) s(t) q ( ft(s(t),q) = O(q) ) Proof (t is even) ... By Theorem 1 (Real Lemma 4): f2(3,q) 3q. Constructing a 2-IPP code C Q3 of size (3-o(1))q ... – PowerPoint PPT presentation

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Title: Parent Identifying Codes The case of multiple parents


1
Parent Identifying CodesThe case of multiple
parents
  • Noga Alon
  • Uri Stav

2
Definitions
  • An alphabet Q, Qq
  • A code C?Qn , of length n
  • A parameter t
  • A set of codewords P p1,p2, ,pt ? C
  • (called parents)

3
Descendants
  • The descendants of P
  • D(P)

Q0,1,2,3,4,5 t3 n5
4
Parent Identifying Codes (def.)
  • A code C ? Qn has the identifiable parent
    property (IPP) of order t if for any n-word s ?Qn
    either
  • It is not a descendant of any set of t codewords
    in C
  • There exists p?C such that
  • p can be identified from s

5
IPP - Example
  • IPP definition
  • ? s?Qn either
  • Not a descendant
  • ? p?C such that

t 2 n 4 Q 1,2,3,4
? p c4
P1 c1,c4 P2 c2,c3 P3 c1,c2
P1 c1,c4 P2 c3,c4
? Not a descendant
? C does not have 2-IPP
6
IPP - Example 2
  • Any repetition code has IPP of any order t

c1
c2
C
cq
7
Binary IPP codes
  • For t gt1
  • There are no binary t-IPP codes of size gt 2

x
y
z
s Maj(x,y,z)
  • Generally, we require q gt t

8
IPP codes - Motivation
  • Fingerprinting

Distributor
9
Fingerprinting (cont.)
  • At most t traitors
  • The traitors compare their copies
  • Any non-constant symbol can be changed
  • Narrow case
  • The new symbol is taken from one of the parents
  • Zero-Error traitor tracing using IPP codes
  • Wide case
  • No restriction on the new symbol
  • Zero-Error traitor tracing is impossible

10
Maximum cardinality IPP codes
  • Define
  • ft(n,q) max C?Qn C has t-IPP
  • Large Qq

11
Previous Work
  • Chor, Fiat, Naor Traitor tracing (1994)
  • Boneh, Shaw Fingerprinting (1998)
  • Hollman, Van-Lint, Linnartz, Tolhuizen (1998)
  • 2-IPP codes,
  • f2(3,q) (3-o(1))q
  • ?(q1.5) lt f2(4,q) lt O(q2)
  • c(q/4)n/3 f2(n,q) 3q?n/3?

12
Previous Work (cont.)
  • Alon, Fischer, Szegedy (2001)
  • q2-? lt f2(4,q) lt ? q2
  • Barg , Cohen, Encheva, Kabatiansky, Zemor (2001)
    Alon, Cohen, Krivelevich, Litsyn (2002)
  • t gt 2, fixed q large n
  • Partially Hashing
  • Staddon, Stinson, Wei (2001)
  • Traceability codes

13
General bound for ft(n,q)
  • Define
  • s(t)
  • That is
  • Theorem 1
  • There exist two functions c1(t) and c2(t), such
    that for every n,q
  • (c1(t)?q) ? ft(n,q) ? c2(t)?q? ?

14
The lower bound (BCEKZ, 2001)
  • Minimal configuration
  • Let ?(X1,X2,,Xm) be a collection of subsets of
    codewords such that Xi? t and

15
Minimal Configurations
  • Suppose C does not have t-IPP
  • A parent of s cannot be identified
  • There are Pi s such that s ? D(Pi ), ?Pi ??
  • Remove Pi until it is a minimal configuration
  • ? There is a minimal configuration Pi
    consisting of parents sets, with a common
    descendant

16
The lower bound (cont.)
  • Recall
  • Lemma 1
  • Suppose ? is a minimal configuration, then

17
Lemma 1
  • Proof
  • There exist B(X) b1,b2,,bm
  • such that
  • All bi s are different
  • B(X)\ bi ? Xi
  • m 1 ? t

18
Lemma 1 (cont.)
QED Lemma 1
19
Hashing
i
U
  • Definition
  • A code C?Qn is u - hashing if for any subset
    U?C, Uu, there is some coordinate
  • i? 1,,n such that
  • ? x?U, y?U, x?y xi?yi

20
Hashing
i
x
?Pj
  • Lemma 2
  • (s(t)1)- hashing ? t-IPP
  • Proof

The son s
Suppose the code does not have t-IPP
?Contradicting example ( s, Pj s?D(Pj), ?Pj ?
)
?Minimal configuration (? ?Pj ? s(t)1 )
?By (s(t)1) - hashing, there is x??Pj (for
which xi si)
QED Lemma 2
21
The lower bound (cont.)
  • Probabilistic construction
  • QED
  • Pick a code at random
  • Delete one codeword from any set that violates
    the (s(t)1)-hashing property, to obtain a t-IPP
    code
  • Calculate the expectation of the number of such
    sets

22
Theorem 1 - The upper bound
  • We have to show ft(n,q) ? c2(t)?q ? ?
  • Proof idea
  • Reduce the problem to a problem on codes of
    length s(t)
  • Bound the size of such codes
  • ft(n,q) ? ft(s(t)?? ?,q) ?
  • ? ft(s(t), q? ? ) ? c2(t)?q ? ?

Lemma 3
Lemma 4
23
Reducing the problem to ft(s(t),q)
  • Lemma 3
  • For any a,t,n,q ft(na,q) ? ft(n,qa)
  • Proof
  • Every a symbols from Q ? One symbol from Qa
  • The short (n) code with large (qa) alphabet
    also has t-IPP.

n
a
a
a
a
24
Bounding ft(s(t),q)
  • Lemma 4
  • ft(s(t),q) ? s(t) q ( ? ft(s(t),q) O(q)
    )
  • Proof (t is even)
  • Suppose we have a code C of size C gt s(t) q
    that has the t-IPP
  • Stage 1
  • Remove codewords until in each coordinate every
    symbol appears at least 1 times

25
Bounding ft(s(t),q) (cont.)
  • Each coordinate is responsible for deleting at
    most q codewords
  • We remove at most s(t) q codewords

26
Bounding ft(s(t),q) (cont.)
  • Contradict C has t-IPP
  • Pick a set P of 1 distinct parents
  • Construct a descendant s of P s?D(P)
  • Every parent p? P can be replaced by
    codewords
  • ? None of the parents p? P is essential

27
Example (t8 ? s(t)( 1)2-1 24, 1 5)
s
28
Bounding ft(s(t),q)
s
  • Every parent p contributes at most
    coordinates that no other parent does
  • p can be replaced by other codewords
  • (? There is a parents set of size t without p)
  • No essential parent
  • C does not have t-IPP

QED Lemma 4, Theorem 1
29
Bounding ft(s(t),q) - Remarks
  • Similar calculations for odd t
  • It is enough if every symbol appears twice
  • Lemma 4 can be improved
  • Real Lemma 4
  • ft(s(t),q) ? s(t) q

30
The case t 2
  • s(2)3
  • By Theorem 1
  • c(q/4)n/3 f2(n,q) 3q?n/3?
  • ns(2)3 The longest linear size code
  • Hollman, VanLint, Linnartz, Tolhuizen 1998
    f2(3,q) (3-o(1))q

31
f2(3,q)
  • By Theorem 1 (Real Lemma 4) f2(3,q) ? 3q
  • Constructing a 2-IPP code C ? Q3 of size (3-o(1))q
  • Split the alphabet Q
  • Q2, Q3 of size
  • Q1 - the rest of the alphabet (
    )
  • C1 concatenating a symbol from Q1 and a couple
    of symbols from Q2?Q3 (? C1 Q1 )
  • C2, C3 Cyclic shifts of the codewords in C1
  • C C1? C2 ? C3 (? C 3Q1(3-o(1))q)

32
f2(3,q) (cont.)
  • C has 2-IPP
  • We show how to identify s ?Q3 s parent
  • We can tell from which Ci every symbol in s comes
  • If the the symbols came from
  • 1 set ? There is a symbol from Q1
  • This symbol comes from a unique parent
  • This parent can be identified
  • 2 sets ? 2 coordinates from one set
  • Only one parent came from that set
  • This parent can be identified
  • 3 sets ? not a descendant of at most 2 codewords
    in C

33
The case t 3
  • s(3)5
  • By Theorem 1
  • When n s(3) 5
  • f3(5,q) ? (5-o(1))q

34
f3(5,q)
  • Theorem 2
  • f3(5,q) (5-o(1))q
  • Proof idea
  • Similar construction to the one for f2(3,q)
  • Split the alphabet Q into Q1,,Q5
  • Q2,,Q5 are small (q3/4), Q1 is large ( (1-o(1)q
    )
  • Construct a code B ? Q2 ? Q3?Q4 ? Q5 of size
    Q1
  • (with some properties)
  • C1 - Concatenating a symbol from Q1 to a codeword
    from B
  • Take all 5 cyclic shifts

35
The case ns(t)1 (? f2(4,q), f3(6,q))
  • The first super-linear code for t2 is n4
  • Theorem 1 ? (q1.33) f2(4,q) 3q2
  • HLLT(1998) ? (q1.5) lt f2(4,q) lt O(q2)
  • AFS (2001) ????
  • The first super-linear code for t3 is n6
  • Theorem 1
  • Theorem 3
  • (similar technique used by AFS for t2, including
    Szemeredi Regularity Lemma)

36
The case n ? s(t)
  • When n ? t ft(n,q)q
  • When nt1
  • e.g. f3(4,q) ( o(1))q

37
The case t 3
38
Open Problems
  • What is the exact power for n ? 0 (mod s(t))?
  • Is ft(s(t),q) (s(t)-o(1))q ?
  • Exact constants for n lt s(t)
  • Explicit code constructions
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