[Paper Review] Secret Key and Private Key Constructions for Simple Multiterminal Source Models

Chunxuan Ye, Prakash Narayan|arXiv (Cornell University)|Aug 12, 2010

Wireless Communication Security Techniques31 references3 citations

TL;DR

This paper proposes a novel secret and private key construction framework for multiterminal source models using Slepian-Wolf coding principles, leveraging correlated source observations and public communication. It establishes that secret key capacity equals joint entropy minus minimal Slepian-Wolf compression rate, and demonstrates the effectiveness of LDPC codes in achieving near-optimal key rates with low error probability under binary symmetric channels.

ABSTRACT

We propose an approach for constructing secret and private keys based on the long-known Slepian-Wolf code, due to Wyner, for correlated sources connected by a virtual additive noise channel. Our work is motivated by results of Csiszár and Narayan which highlight innate connections between secrecy generation by multiple terminals that observe correlated source signals and Slepian-Wolf near-lossless data compression. Explicit procedures for such constructions and their substantiation are provided. The performance of low density parity check channel codes in devising a new class of secret keys is examined.

Motivation & Objective

• To develop practical constructions of secret and private keys from correlated source signals observed by multiple terminals.
• To establish a direct link between secret key generation and Slepian-Wolf near-lossless compression in multiterminal source models.
• To demonstrate the feasibility and performance of low-density parity-check (LDPC) codes in constructing secret keys with high reliability and secrecy.
• To provide explicit, provably secure key generation procedures that achieve the theoretical secret key capacity.

Proposed method

• Uses the Slepian-Wolf coding framework to model interterminal communication required for omniscience, where terminals reconstruct all source components from their observations and public messages.
• Applies the minimal total communication rate $ R_{\text{min}} $ from Slepian-Wolf coding as a lower bound on public communication needed to achieve secret key generation.
• Derives secret key capacity $ C_S = H(X_1,\ldots,X_d) - R_{\text{min}} $, showing it equals the joint entropy minus the minimal communication rate.
• Adapts the framework to private key generation for a subset $ A $ of terminals, with capacity $ C_P(A) = H(X_A|X_{A^c}) - R_{\text{min}}(A) $, ensuring secrecy from both eavesdroppers and cooperating terminals in $ A^c $.
• Employs maximum likelihood decoding over binary symmetric channels (BSC) and leverages quasiadmissible sets to bound error probabilities.
• Evaluates performance using LDPC codes, showing that error probabilities decay exponentially with block length $ n $, enabling high-rate, secure key generation.

Experimental results

Research questions

• RQ1 Can secret key generation in multiterminal source models be systematically constructed using Slepian-Wolf coding principles?
• RQ2 What is the precise relationship between Slepian-Wolf compression rates and the achievable secret key capacity in correlated source models?
• RQ3 How do LDPC codes perform in constructing secret keys under binary symmetric channel conditions with respect to error probability and key rate?
• RQ4 Can private key generation be achieved such that cooperating terminals in $ A^c $ remain ignorant of the key generated by terminals in $ A $, even when they reveal their observations?
• RQ5 What is the asymptotic behavior of error probability in maximum likelihood decoding for LDPC codes used in secret key construction?

Key findings

• The secret key capacity $ C_S $ is exactly equal to the joint entropy of all source components minus the minimal total communication rate $ R_{\text{min}} $ required for omniscience, as per Slepian-Wolf coding.
• The private key capacity $ C_P(A) $ is given by $ H(X_A|X_{A^c}) - R_{\text{min}}(A) $, where $ R_{\text{min}}(A) $ is the minimal rate for terminals in $ A $ to achieve omniscience under conditioning on $ X_{A^c} $.
• For any $ \delta > 0 $, the probability that the reconstructed source sequence deviates from its typical set is bounded by $ (n+1)^{|\mathcal{X}|} \cdot 2^{-n \delta^2 / (2\ln 2)} $, ensuring typicality with high probability.
• Maximum likelihood decoding error probability for LDPC codes on a BSC decreases monotonically with decreasing crossover probability $ p $, implying better performance on less noisy channels.
• The error probability for decoding any terminal’s source sequence along a path from source to target terminal decays exponentially as $ d \cdot 2^{-n\eta} $ for some $ \eta > 0 $, ensuring reliable reconstruction.
• LDPC codes are shown to be effective in achieving near-optimal secret key rates with exponentially small error probabilities, validating their use in practical secret key generation systems.