14 research outputs found
Improved Constructions of Frameproof Codes
Frameproof codes are used to preserve the security in the context of
coalition when fingerprinting digital data. Let be the largest
cardinality of a -ary -frameproof code of length and
. It has
been determined by Blackburn that when ,
when and is even, and . In this paper, we
give a recursive construction for -frameproof codes of length with
respect to the alphabet size . As applications of this construction, we
establish the existence results for -ary -frameproof codes of length
and size for all odd when and for all
when . Furthermore, we show that
meeting the upper bound given by Blackburn, for all integers such that
is a prime power.Comment: 6 pages, to appear in Information Theory, IEEE Transactions o
Linear time Constructions of some -Restriction Problems
We give new linear time globally explicit constructions for perfect hash
families, cover-free families and separating hash functions
Probabilistic Existence Results for Separable Codes
Separable codes were defined by Cheng and Miao in 2011, motivated by
applications to the identification of pirates in a multimedia setting.
Combinatorially, -separable codes lie somewhere between
-frameproof and -frameproof codes: all -frameproof codes are
-separable, and all -separable codes are
-frameproof. Results for frameproof codes show that (when is large)
there are -ary -separable codes of length with
approximately codewords, and that no -ary
-separable codes of length can have more than approximately
codewords.
The paper provides improved probabilistic existence results for
-separable codes when . More precisely, for all and all , there exists a constant (depending only on
and ) such that there exists a -ary -separable code of
length with at least codewords for all sufficiently
large integers . This shows, in particular, that the upper bound (derived
from the bound on -frameproof codes) on the number of codewords in a
-separable code is realistic.
The results above are more surprising after examining the situation when
. Results due to Gao and Ge show that a -ary -separable
code of length can contain at most codewords, and that codes with at
least codewords exist. So optimal -separable
codes behave neither like -frameproof nor -frameproof codes.
Also, the Gao--Ge bound is strengthened to show that a -ary
-separable code of length can have at most
codewords.Comment: 16 pages. Typos corrected and minor changes since last version.
Accepted by IEEE Transactions on Information Theor
Separating hash families with large universe
Separating hash families are useful combinatorial structures which generalize
several well-studied objects in cryptography and coding theory. Let
denote the maximum size of universe for a -perfect hash family of length
over an alphabet of size . In this paper, we show that for all , which answers an open problem about separating
hash families raised by Blackburn et al. in 2008 for certain parameters.
Previously, this result was known only for . Our proof is obtained by
establishing the existence of a large set of integers avoiding nontrivial
solutions to a set of correlated linear equations.Comment: 17 pages, no figur
Fingerprinting Codes and Related Combinatorial Structures
Fingerprinting codes were introduced by Boneh and Shaw in 1998 as a method of copyright control. The desired properties of a good fingerprinting code has been found to have deep connections to combinatorial structures such as error-correcting codes and cover-free families. The particular property that motivated our research is called "frameproof". This has been studied extensively when the alphabet size q is at least as large as the colluder size w. Much less is known about the case q < w, and we prove several interesting properties about the binary case q = 2 in this thesis.
When the length of the code N is relatively small, we have shown that the number of codewords n cannot exceed N, which is a tight bound since the n = N case can be satisfied a trivial construction using permutation matrices. Furthermore, the only possible candidates are equivalent to this trivial construction. Generalization to a restricted parameter set of separating hash families is also given.
As a consequence, the above result motivates the question of when a non-trivial construction can be found, and we give some definitive answers by considering combinatorial designs. In particular, we give a necessary and sufficient condition for a symmetric design to be a binary 3-frameproof code, and provide example classes of symmetric designs that satisfy or fail this condition. Finally, we apply our results to a problem of constructing short binary frameproof codes