Unconditional information-theoretic security of secret keys in binary channels

We consider the problem of the secret key rate in the modified Maurer’s model. The modified model accounts for the noise generated by the source, aligning more closely with realistic scenarios. All receivers receive broadcast signals through binary channels. Unlike Maurer’s model, the bit error rate (BER) of the binary channels in the modified model is correlated because the channels share a partial common noise source. First, we demonstrate that in the case of using the repeat-code protocol, regardless of the channel quality of the eavesdropper Eve, as long as the block length is large enough, the legitimate receivers Alice and Bob can achieve a positive secret key rate. In addition, the block length required to achieve a positive secret key rate is determined, and the lower bound of the secret key rate is given. Next, the upper bound of the secret key rate, i.e., conditional mutual information, is analyzed with the change in Eve’s channel quality. We show that the conditional mutual information decreases by $1/\left(1+s_Z\right)$ as Eve’s signal-to-noise ratio $s_Z$ increases, but it remains positive even though the channel noise of Eve is zero.