Expurgation exponent of leaked information in privacy amplification for binary sources

We investigate the privacy amplification problem in which Eve can observe the uniform binary source through a binary erasure channel (BEC) or a binary symmetric channel (BSC). For this problem, we derive the so-called expurgation exponent of the information leaked to Eve. The exponent is derived by relating the leaked information to the error probability of the linear code that is generated by the linear hash function used in the privacy amplification, which is also interesting in its own right. The derived exponent is larger than state-of-the-art exponent recently derived by Hayashi at low rate.