Construction of Codes for the Wiretap Channel and the Secret Key Agreement From Correlated Source Outputs Based on the Hash Property

Abstract

The aim of this paper is to prove coding theorems for the wiretap channel and secret key agreement based on the the notion of a hash property for an ensemble of functions. These theorems imply that codes using sparse matrices can achieve the optimal rate. Furthermore, fixed-rate universal coding theorems for a wiretap channel and a secret key agreement are also proved. Index TermsShannon theory, hash property, linear codes, sparse matrix, maximum-likelihood decoding, minimum-divergence encoding, minimum-entropy decoding, secret key agreement from correlated source outputs, wiretap channel, universal codes I. INTRODUCTIONThe aim of this paper is to prove the coding theorems for the wiretap channel (Fig. 1) introduced in [23] and secret key agreement problem (Fig. 2) introduced in [12][1]. The proof of theorems is based on the notion of a hash property for an ensemble of functions introduced in [18][19]. This notion provides a sufficient condition for the achievability of coding theorems. Since an ensemble of sparse matrices has a hash property, we can construct codes by using sparse matrices where the rate of codes is close to the optimal rate. In the construction of codes, we use minimum-divergence encoding, maximum-likelihood decoding, and minimum-entropy decoding, where we can use the approximation methods introduced in [9][5] to realize these operations.Wiretap channel coding using a sparse matrices is studied in [21] for binary erasure wiretap channels. On the other hand, our construction can be applied to any stationary memoryless channel. It should be noted here that the encoder design is based on the standard channel code presented in [14][18][19] [13]. Furthermore, we prove the fixed-rate universal coding theorem for a wiretap channel, where our construction is reliable and secure for any channel under some conditions specified by the encoding rate. Universality is not considered in [23] [21].

Secure Multiplex Coding With Dependent and Non-Uniform Multiple Messages

“…Channel-universal coding for BCC had not been studied before [19], and coding for BCC can be regarded as a special case of SMC while Muramatsu et al [29] treat channeluniversal coding for wire-tap channel independently of [19]. In Section XII and [19] we consider SMC universal to channel, but its universality to the source is not considered.…”

“…for any s p,n ∈ S p,n , s c,n ∈ S c,n and any W ∈ W(V, Y× Z), where the exponentsẼ (29) and (30), respectively.…”

Secret-Key Distribution Based on Bounded Observability

“…Proposition 4 asserts that IðX; YÞ À IðX; ZÞ is achievable by using a sparse matrix (LDPC code) for the information 14 In [32] and [33], the weak security conditions HðKÞ=n ! R À " and IðK; Z n ; C t Þ=n " are considered.…”

“…The strong security condition (3) for a general correlated source is considered in [34]. It should be noted that Z and Y are allowed to be continuous variables in [34] while they were assumed to be discrete variables in [32] and [33]. inspired by [21].…”

“…This means that the desired matrices A and B in Propositions 2-4 may depend on the joint probability distribution of ðX; Y; ZÞ. However, it is shown in [32] and [33] that for given ðr; RÞ, " > 0, and sufficiently large n, there is a pair ðA; BÞ satisfying (1)-(3) for all ðX; Y; ZÞ satisfying (8) and (9). This fact means that it is unnecessary for Alice and Bob to know the joint probability distribution of ðX; Y; ZÞ and it is sufficient to know HðXjYÞ and HðXjZÞ.…”