[Paper Review] General non-asymptotic and asymptotic formulas in channel resolvability and identification capacity and their application to wire-tap channel

Masahito Hayashi|arXiv (Cornell University)|Mar 31, 2005

Wireless Communication Security Techniques7 references9 citations

TL;DR

This paper establishes general non-asymptotic and asymptotic formulas for channel resolvability and identification capacity in arbitrary channels, without assuming memoryless or stationary properties. It resolves an open problem by proving the achievable rate of channel resolvability for general sequences of channels and derives explicit lower bounds on error and wiretapper information exponents in wire-tap channels under the stationary memoryless setting.

ABSTRACT

Several non-asymptotic formulas are established in channel resolvability and identification capacity, and they are applied to wire-tap channel. By using these formulas, the $ε$ capacities of the above three problems are considered in the most general setting, where no structural assumptions such as the stationary memoryless property are made on a channel. As a result, we solve an open problem proposed in Han & Verdu and Han. Moreover, we obtain lower bounds of the exponents of error probability and the wire-tapper's information in wire-tap channel.

Motivation & Objective

• To resolve an open problem in channel resolvability by establishing the achievable rate for general sequences of channels without assuming memoryless or stationary properties.
• To derive non-asymptotic and asymptotic formulas for identification capacity and channel resolvability in the most general setting.
• To apply these formulas to wire-tap channels and obtain lower bounds on error probability and wiretapper’s information exponents.
• To correct and extend Han & Verdú’s earlier work by providing a valid converse proof for channel resolvability even without the strong converse property.

Proposed method

• Derives upper bounds on the average variational distance and Kullback-Leibler divergence between output distributions when M input elements are randomly selected under a given input distribution p.
• Uses the information spectrum method and likelihood-type random variables to characterize performance in non-asymptotic settings.
• Applies the duality between identification codes and channel resolvability to derive capacity formulas for general channel sequences.
• Introduces and analyzes the functions ψ(s|W,p) and φ(t|W,p) to compute divergence rates and exponent bounds.
• Applies the formulas to wire-tap channels by modeling the eavesdropper’s information as a resolvability problem.
• Uses Taylor expansions to approximate the exponent expressions and derive closed-form lower bounds.

Experimental results

Research questions

• RQ1 What is the achievable rate of channel resolvability for general sequences of channels without assuming memoryless or stationary properties?
• RQ2 How can non-asymptotic formulas for channel resolvability be used to bound the error exponent and wiretapper’s information in wire-tap channels?
• RQ3 Can the converse proof for channel resolvability be validated in general channels, even when the strong converse property does not hold?
• RQ4 What are the tightest lower bounds on the exponents of error probability and wiretapper’s information in stationary memoryless wire-tap channels?

Key findings

• The paper proves the achievable rate of channel resolvability for general sequences of channels, resolving an open problem posed by Han & Verdú.
• It establishes that the lower bound on the exponent of the worst-case variational distance is approximately Δ²/(4J(p;W)) for general channels.
• For stationary memoryless channels, the lower bound on the error exponent is Δ²/(4(J(p₀;W) + Eₚ₀H(Wₓ))) when R is close to the channel capacity.
• The lower bound on the wiretapper’s information exponent is approximately Δ²/(8J(p₀;W)) under the same conditions.
• The method provides a valid converse proof for channel resolvability even in the absence of the strong converse property.
• The results are extended to wire-tap channels, yielding explicit lower bounds on the exponents of error probability and eavesdropper’s information.