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Quantitative Frameproof Codes and Hypergraphs

Wenjie Zhong, Xinqi Huang, Xiande Zhang

TL;DR

This work introduces quantitative frameproof codes and hypergraphs, unifying frameproof and focal-free regimes through a generalized Erdős matching number framework. It derives asymptotically optimal bounds for the maximum sizes $f_{c,s}(n,k)$ and $f_{c,s}^{q}(n)$ via a generalized matching number $m(n,t,\lambda;k_1,k_2)$ and packing techniques, with exact values for infinitely many parameter sets. The paper also defines a critical variant and analyzes the corresponding extremal quantities $g_{c,s}(n,k)$ and $g^{q}_{c,s}(n)$, together with detailed bounds on $m(n,t,\lambda;s_1+1,s_2+1)$. By connecting hypergraph packings, designs, and coding-theoretic constructions, it advances the combinatorial theory of secure codes and provides a unified lens for frameproof and focal-free problems, with potential implications for digital fingerprinting and traitor-tracing schemes.

Abstract

Frameproof codes are a class of secure codes introduced by Boneh and Shaw in the context of digital fingerprinting, and have been widely studied from a combinatorial point of view. In this paper, we study a quantitative extension of frameproof codes and hypergraphs, referred to as {\it quantitative frameproof codes and hypergraphs}. We give asymptotically optimal bounds on the maximum sizes of these structures and determine their exact sizes for a broad range of parameters. In particular, we introduce a generalized version of the Erdős matching number in our proof and derive relevant estimates for it.

Quantitative Frameproof Codes and Hypergraphs

TL;DR

This work introduces quantitative frameproof codes and hypergraphs, unifying frameproof and focal-free regimes through a generalized Erdős matching number framework. It derives asymptotically optimal bounds for the maximum sizes and via a generalized matching number and packing techniques, with exact values for infinitely many parameter sets. The paper also defines a critical variant and analyzes the corresponding extremal quantities and , together with detailed bounds on . By connecting hypergraph packings, designs, and coding-theoretic constructions, it advances the combinatorial theory of secure codes and provides a unified lens for frameproof and focal-free problems, with potential implications for digital fingerprinting and traitor-tracing schemes.

Abstract

Frameproof codes are a class of secure codes introduced by Boneh and Shaw in the context of digital fingerprinting, and have been widely studied from a combinatorial point of view. In this paper, we study a quantitative extension of frameproof codes and hypergraphs, referred to as {\it quantitative frameproof codes and hypergraphs}. We give asymptotically optimal bounds on the maximum sizes of these structures and determine their exact sizes for a broad range of parameters. In particular, we introduce a generalized version of the Erdős matching number in our proof and derive relevant estimates for it.

Paper Structure

This paper contains 17 sections, 22 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Outline of paper
  4. Quantitative frameproof hypergraphs
  5. The upper bounds of $f_{c,s}(n,k)$
  6. The lower bounds of $f_{c,s}(n,k)$
  7. Quantitative frameproof codes
  8. Upper and lower bounds of $f_{c,s}^q(n)$
  9. Exact values of $f_{c,s}^q(n)$
  10. Critical quantitative frameproof codes and hypergraphs
  11. Estimating generalized matching numbers
  12. An Upper Bound of $m(n,t,\lambda;s_1+1,s_2+1)$
  13. A Lower Bound of $m(n,t,\lambda;s_1+1,s_2+1)$
  14. Concluding remarks
  15. Proof of Theorem \ref{['mainthm:code']}
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $c$, $s$ and $k$ be fixed integers satisfying $c,k\ge 2$ and $1\le s \le c-1$. We have where $t=\lceil sk/c\rceil$ and $\lambda\in [c]$ is the integer that satisfies $\lambda\equiv sk \pmod{c}$.

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3
  • Definition 1.4: Generalized matching number
  • Proposition 1.5
  • Theorem 1.6
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Claim 2.4
  • ...and 32 more