Strongly Secure Privacy Amplification Cannot Be Obtained by Encoder of Slepian-Wolf Code

The privacy amplification is a technique to distill a secret key from a random variable by a function so that the distilled key and eavesdropper's random variable are statistically independent. There are three kinds of security criteria for the key distilled by the privacy amplification: the normalized divergence criterion, which is also known as the weak security criterion, the variational distance criterion, and the divergence criterion, which is also known as the strong security criterion. As a technique to distill a secret key, it is known that the encoder of a Slepian-Wolf (the source coding with full side-information at the decoder) code can be used as a function for the privacy amplification if we employ the weak security criterion. In this paper, we show that the encoder of a Slepian-Wolf code cannot be used as a function for the privacy amplification if we employ the criteria other than the weak one.