# New upper bounds for parent-identifying codes and traceability codes

New upper bounds for parent-identifying codes and traceability codes In the last two decades, parent-identifying codes and traceability codes are introduced to prevent copyrighted digital data from unauthorized use. They have important applications in the scenarios like digital fingerprinting and broadcast encryption schemes. A major open problem in this research area is to determine the upper bounds for the cardinalities of these codes. In this paper we will focus on this theme. Consider a code of length N which is defined over an alphabet of size q. Let $$M_{IPPC}(N,q,t)$$ M I P P C ( N , q , t ) and $$M_{TA}(N,q,t)$$ M T A ( N , q , t ) denote the maximal cardinalities of t-parent-identifying codes and t-traceability codes, respectively, where t is known as the strength of the codes. We show $$M_{IPPC}(N,q,t)\le rq^{\lceil N/(v-1)\rceil }+(v-1-r)q^{\lfloor N/(v-1)\rfloor }$$ M I P P C ( N , q , t ) ≤ r q ⌈ N / ( v - 1 ) ⌉ + ( v - 1 - r ) q ⌊ N / ( v - 1 ) ⌋ , where $$v=\lfloor (t/2+1)^2\rfloor$$ v = ⌊ ( t / 2 + 1 ) 2 ⌋ , $$0\le r\le v-2$$ 0 ≤ r ≤ v - 2 and $$N\equiv r \mod (v-1)$$ N ≡ r mod ( v - 1 ) . This new bound improves two previously known bounds of Blackburn, and Alon and Stav. On the other hand, $$M_{TA}(N,q,t)$$ M T A ( N , q , t ) is still not known for almost all t. In 2010, Blackburn, Etzion and Ng asked whether $$M_{TA}(N,q,t)\le cq^{\lceil N/t^2\rceil }$$ M T A ( N , q , t ) ≤ c q ⌈ N / t 2 ⌉ or not, where c is a constant depending only on N, and they have shown the only known validity of this bound for $$t=2$$ t = 2 . By using some complicated combinatorial counting arguments, we prove this bound for $$t=3$$ t = 3 . This is the first non-trivial upper bound in the literature for traceability codes with strength three. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Designs, Codes and Cryptography Springer Journals

# New upper bounds for parent-identifying codes and traceability codes

, Volume 86 (8) – Oct 16, 2017
11 pages

/lp/springer_journal/new-upper-bounds-for-parent-identifying-codes-and-traceability-codes-Ilk2nU46lP
Publisher
Springer US
Subject
Mathematics; Combinatorics; Coding and Information Theory; Data Structures, Cryptology and Information Theory; Data Encryption; Discrete Mathematics in Computer Science; Information and Communication, Circuits
ISSN
0925-1022
eISSN
1573-7586
D.O.I.
10.1007/s10623-017-0420-y
Publisher site
See Article on Publisher Site

### Abstract

In the last two decades, parent-identifying codes and traceability codes are introduced to prevent copyrighted digital data from unauthorized use. They have important applications in the scenarios like digital fingerprinting and broadcast encryption schemes. A major open problem in this research area is to determine the upper bounds for the cardinalities of these codes. In this paper we will focus on this theme. Consider a code of length N which is defined over an alphabet of size q. Let $$M_{IPPC}(N,q,t)$$ M I P P C ( N , q , t ) and $$M_{TA}(N,q,t)$$ M T A ( N , q , t ) denote the maximal cardinalities of t-parent-identifying codes and t-traceability codes, respectively, where t is known as the strength of the codes. We show $$M_{IPPC}(N,q,t)\le rq^{\lceil N/(v-1)\rceil }+(v-1-r)q^{\lfloor N/(v-1)\rfloor }$$ M I P P C ( N , q , t ) ≤ r q ⌈ N / ( v - 1 ) ⌉ + ( v - 1 - r ) q ⌊ N / ( v - 1 ) ⌋ , where $$v=\lfloor (t/2+1)^2\rfloor$$ v = ⌊ ( t / 2 + 1 ) 2 ⌋ , $$0\le r\le v-2$$ 0 ≤ r ≤ v - 2 and $$N\equiv r \mod (v-1)$$ N ≡ r mod ( v - 1 ) . This new bound improves two previously known bounds of Blackburn, and Alon and Stav. On the other hand, $$M_{TA}(N,q,t)$$ M T A ( N , q , t ) is still not known for almost all t. In 2010, Blackburn, Etzion and Ng asked whether $$M_{TA}(N,q,t)\le cq^{\lceil N/t^2\rceil }$$ M T A ( N , q , t ) ≤ c q ⌈ N / t 2 ⌉ or not, where c is a constant depending only on N, and they have shown the only known validity of this bound for $$t=2$$ t = 2 . By using some complicated combinatorial counting arguments, we prove this bound for $$t=3$$ t = 3 . This is the first non-trivial upper bound in the literature for traceability codes with strength three.

### Journal

Designs, Codes and CryptographySpringer Journals

Published: Oct 16, 2017

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly ### Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. ### Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations