Identification via wiretap channels

The wiretap channel can be viewed as a probabilistic model for cryptography. The channel has two outputs. One is for the legitimate receiver and the other is for the wiretapper. The goal of communication is to send messages to the legitimate receiver while the wiretapper must be kept ignorant. A wiretap channel is a quintuple (X,W(y|x),V(x|x),Y,Z), where X is the input alphabet, Y is the output alphabet for the legitimate receiver, Z is the output alphabet for the wiretapper, W(y|x) is the channel transition matrix, whose output is available to the legitimate receiver, and V(x|x) is the channel transition matrix, whose output is available to the wiretapper. The channel is assumed to be memoryless. In the classical transmission problem, an (n,M,ε)-code for the wiretap channel is defined as a system {(c_i,D_i)|1⩽i⩽M}, where, for all i,c _i∈Xⁿ are the codewords and D_i⊂yⁿ are the disjoint decoding sets. It is required that for any i λ_i=^defWⁿ(D_i^c |c_i)⩽ε, and if Xⁿ has uniform distribution over {c_i|⩽i⩽M}, then 1/nI(Xⁿ;Zⁿ)⩽ε. The secret capacity of the wiretap channel is defined as the maximum rate of any code which satisfies these conditions. Formally, let M(n,ε)=max{M:∃a(n,M,ε) code}, then the secret capacity of the wiretap channel is defined as C_s=max{R:∀ε>0,∃n such that M(n,C)⩾ ^nR. The secret capacity of the wiretap channel can then be determined. The problem of identification via this channel is then formulated