Perfect Secrecy Over Binary Erasure Wiretap Channel of Type II

Abstract

We introduce a binary erasure wiretap channel of type II in which the number of eavesdropped bits mu 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 S leaked out, I(S; X-mu) > 0, where X-mu is a binary sequence of length mu, it is still possible to have perfect secrecy I(S-J; X-mu) 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-mu. Our secret key agreement scheme naturally leads to defining a security measure D-H(mu) for parity-check matrices H such that the eavesdropper gets zero information on S-J as long as the length of S-J is less than D-H(mu). We study basic properties of D-H(mu) 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(mu).