Strong security and separated code constructions for the broadcast channels with confidential messages

The strong security criterion in the information theoretic security requires the mutual information between the secret information and the eavesdropped information converges to zero as the code length goes to the infinity, even when the mutual information is not divided by the code length, while the weak security criterion requires only the mutual information divided by the code length converges to zero. The capacity region of the broadcast channels with confidential messages remains unknown when the strong security is required. We prove that the capacity region under the strong security criterion is the same as that under the weak one. Our proof technique attaches inverses of hash functions to a random coding argument for the broadcast channel with degraded message sets, whose analysis on the decoding error probability is carried over to our proof without change. Thus, our proof technique separates the analysis of secrecy from that of decoding error probability, and an advance in the upper bound on the decoding error probability over the broadcast channels with degraded message sets automatically also advances the decoding probability analysis of our proof technique. Although the above argument separates the analysis of the decoding error probability and the mutual information, it does not separate the construction of a code for error correction and provision of secrecy. We introduce another form of the privacy amplification theorem so that we can separate the code constructions for error correction and secrecy, then we clarify which rate pairs can be achieved by our separated code construction.