In this paper, we investigate relationships between the following formalizations of information-theoretic security for key agreement protocols which may have agreement-errors: formalizations extended (or relaxed) from Shannon's perfect secrecy by using mutual information and statistical distance; and the ones of composable security by Maurer et al. and Canetti. Then, we explicitly show that those are essentially equivalent. We also derive lower bounds on the adversary's (or distinguisher's) advantage and the size of a correlated randomness resource required under all of the above formalizations at once through our relationships. In addition, we observe impossibility results which easily follow from the lower bounds.
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