[Paper Review] A Code Equivalence between Secure Network and Index Coding

Lawrence Ong, Jörg Kliewer+2 more|arXiv (Cornell University)|Apr 26, 2018

Cooperative Communication and Network Coding8 references3 citations

TL;DR

This paper establishes a code equivalence between secure index coding and secure network coding, demonstrating that any secure index code can be mapped to a secure network code (and vice versa) while preserving both decoding error performance and information-theoretic security. The key contribution is a formal mapping that maintains security under eavesdropping, including for non-zero error and randomized encoding scenarios.

ABSTRACT

A code equivalence between index coding and network coding was established, which shows that any index-coding instance can be mapped to a network-coding instance, for which any index code can be translated to a network code with the same decoding-error performance, and vice versa. Also, any network-coding instance can be mapped to an index-coding instance with a similar code translation. In this paper, we extend the equivalence to secure index coding and secure network coding, where eavesdroppers are present in the networks, and any code construction needs to guarantee security constraints in addition to decoding-error performance.

Motivation & Objective

• To extend the known equivalence between index coding and network coding to the secure setting, where eavesdroppers attempt to access messages.
• To address the challenge that non-secure code equivalences do not directly apply to secure variants due to differing eavesdropper models and the need for randomized encoding.
• To establish bidirectional mappings between secure index-coding and secure network-coding instances that preserve both decoding error probability and information-theoretic security.
• To analyze the security and decoding performance trade-offs when translating codes between the two models, especially for linear and non-zero error cases.
• To show that strongly-secure index codes map to weakly-secure network codes, and vice versa, under specific conditions on codelength and error criteria.

Proposed method

• Proposes a mapping between secure index-coding and secure network-coding configurations, specifying eavesdropper access and target decoding sets in both models.
• Extends the non-secure code equivalence framework to include security constraints by aligning eavesdropper capabilities: in index coding, eavesdroppers observe messages; in network coding, they monitor specific links.
• Uses probabilistic analysis and mutual information bounds to evaluate the decoding error and security performance of mapped codes.
• Applies the mapping to both directions: from secure index coding to secure network coding and vice versa, ensuring equivalence in error and security metrics.
• Analyzes the impact of codelength and error probability on security, showing that for linear codes with non-zero error, security degrades linearly with codelength.
• Employs entropy and information-theoretic measures (e.g., mutual information, binary entropy) to derive upper bounds on information leakage under different conditions.

Experimental results

Research questions

• RQ1 Can the known code equivalence between index coding and network coding be extended to the secure setting with eavesdroppers?
• RQ2 How do the differing eavesdropper models in secure index coding (message-level knowledge) and secure network coding (link-level monitoring) affect code equivalence?
• RQ3 Does the equivalence hold when randomized encoding is required, as in some secure network-coding constructions?
• RQ4 What is the relationship between security levels in the two models—specifically, do strongly-secure index codes map to weakly-secure network codes?
• RQ5 How do decoding error probability and security performance scale when translating codes between the two models?

Key findings

• Any secure index-coding instance can be mapped to a secure network-coding instance such that any code for the former translates to a code for the latter with identical decoding error and security performance.
• Any secure network-coding instance can be mapped to a secure index-coding instance, preserving decoding error and security criteria, except in the case of non-zero error linear codes, where security degrades linearly with codelength.
• Strongly-secure index codes map to weakly-secure network codes, indicating a one-way security trade-off in the translation process.
• For non-zero error linear codes, the security criterion grows linearly with codelength, while the decoding error criterion remains bounded and independent of codelength.
• The probability of decoding error in the mapped network code is bounded above by $ | abla| u + heta $, which is derived from the original index code's error and security parameters.
• The information leakage in the mapped network code is bounded by an expression involving $ | abla|( u + H_b( u)) + heta $, with additional terms accounting for error and codelength, ensuring information-theoretic security under the derived bounds.