Perfect Secrecy Over Binary Erasure Wiretap Channel of Type II

We introduce a binary erasure wiretap channel of type II in which the number of eavesdropped bits μ becomes available a posteriori. We aim at achieving perfect secrecy over such a channel model. The most appropriate application is a secret key agreement scheme. We present a secret key agreement scheme that adopts the formulation S = HX of Wyner-Ozarows´s linear coset coding. The scheme is based on the following simple observation: even if some information on a secret message leaked out, I(S; X_μ) >; 0, where X_μ is a binary sequence of length μ, it is still possible to have perfect secrecy I(S_J; X_μ) = 0 for some subsequence S_J of S. Our secret key agreement scheme achieves perfect secrecy by taking only those subsequences S_J that are independent of the eavesdropped bits X_μ. Our secret key agreement scheme naturally leads to defining a security measure D_H(μ) for parity-check matrices such that the eavesdropper gets zero information on S_J as long as the length of S_J is less than D_H(μ). We study basic properties of D_H(μ) and prove the perfect secrecy of our key agreement scheme. For parity-check matrices of small sizes, we perform an exhaustive search for matrices maximizing D_H(μ).