Reliability and Secrecy Functions of the Wiretap Channel Under Cost Constraint

Abstract

The wiretap channel has been devised and studied first by Wyner, and subsequently extended to the case with nondegraded general wiretap channels by Csiszár and Körner. Focusing mainly on the stationary memoryless channel with cost constraint, we newly introduce the notion of reliability and secrecy functions as a fundamental tool to analyze and/or design the performance of an efficient wiretap channel system, including binary symmetric wiretap channels, Poisson wiretap channels, and Gaussian wiretap channels. Compact formulas for those functions are explicitly given for stationary memoryless wiretap channels. It is also demonstrated that, based on such a pair of reliability and secrecy functions, we can control the tradeoff between reliability and secrecy (usually conflicting), both with exponentially decreasing rates as block length n becomes large. Four ways to do so are given on the basis of rate shifting, rate exchange, concatenation, and change of cost constraint. In addition, the notion of the δ secrecy capacity is defined and shown to attain the strongest secrecy standard among others. The maximized versus averaged secrecy measures is also discussed.Index Terms-Reliability function, secrecy function, secrecy measures, Poisson wiretap channel, cost constraint, Gaussian wiretap channel, binary symmetric wiretap channel, tradeoff between reliability and secrecy, concatenation, rate shifting, rate exchange, change of cost constraint.

Exact Random Coding Secrecy Exponents for the Wiretap Channel

“…That (a lower bound to) the resolvability exponent, lower-bounds the secrecy exponent is already used in [7], [10], [11]. Theorem 1 complements this result by showing that the exact resolvability exponent equals the exact secrecy exponent.…”

“…Let C n be a random code of block-length n and rate R constructed by sampling M = ⌊exp(nR)⌋ codewords independently from the distribution P X n ∈ P(X n ) (see (10)). Let W : X → Z be a discrete memoryless channel and P Cn be the (random) output distribution of W n when a uniformly chosen codeword from C n is transmitted via n independent uses of W (see (8)).…”

“…This technique is known as privacy amplification. More recently, it was shown (see special cases of [9,Theorem 2], [10,Theorem 3.1], or the proof given in [11]) that privacy amplification is unnecessary and the exponent derived in [8] lower-bounds the exponential decay rate of the ensemble average of the information leaked to Eve when a wiretap channel code constructed from the ensemble of i.i.d. random codes is used for communication.…”

“…Han and Verdú [24] and Hayashi [7] developed this theory further by replacing the measure of approximation by normalized ℓ 1 distance and unnormalized divergence, respectively, and showed first, that the same limits on the code size hold in these cases and, second, that the distance between the output distribution and the target distribution P n Z vanishes exponentially fast as the block-length increases (similar results are derived in [11], [14], [25] as well). In particular, in [7], [10], [11], [15], the exponential decay of the informational divergence is leveraged to establish an exponentially decaying upper bound on the information leaked to the eavesdropper in wiretap channel's model.…”

“…where C n is a random code of size M = ⌊exp(nR)⌋ distributed according to (10) and the sequence of target distributions {P Z n ∈ P(Z n )} n∈N is defined as…”