New Bounds on 2-Frameproof Codes of Length 4

Penying Rochanakul

doi:10.1155/2020/4879108

Abstract

Frameproof codes were first introduced by Boneh and Shaw in 1998 in the context of digital fingerprinting to protect copyrighted materials. These digital fingerprints are generally denoted as codewords in Qn, where Q is an alphabet of size q and n is a positive integer. A 2-frameproof code is a code C such that any 2 codewords in C cannot form a new codeword under a particular rule. Thus, no pair of users can frame a user who is not a member of the coalition. This paper concentrates on the upper bound for the size of a q-ary 2-frameproof code of length 4. Our new upper bound shows that C≤2q2−2q+1 when q is odd and q>10.

Highlights

In order to protect a digital content, a distributor marks each copy with a codeword. is marking discourages users from releasing an unauthorized copy, since a mark allows the distributor to detect any unauthorized copy and trace it back to the user
A coalition of users may detect some of the marks, namely, the ones where their copies differ. us, they can forge a new copy by changing these marks arbitrarily
We show that |C| ≤ 2q2 − 2q + 1 for odd q > 10. us, 2q2 − 4q + 3 ≤ M4,2(q) ≤ 2q2 − 2q + 1

Summary

Introduction

In order to protect a digital content, a distributor marks each copy with a codeword. is marking discourages users from releasing an unauthorized copy, since a mark allows the distributor to detect any unauthorized copy and trace it back to the user. For 2-frameproof codes of length 4, eorem 3 only gives lower bound for odd q. In 2003, Blackburn proved the following upper bound of w-frameproof codes. In 2019, Cheng et al proved the following theorem, which is the best previously known result on the upper bound of 2-frameproof code of length 4. For any positive integer q > 48, if C is a q-ary 2-frameproof code of length 4, . For any odd positive integer q > 10, if C is a qary 2-frameproof code of length 4, . E gap between the lower bound of odd and even q in Corollary 1 is the key motivation for proving the main result.

Preliminaries

Main Results

Conclusion