Coding theorems for a (2, 2)-threshold scheme secure against impersonation by an opponent

In this paper, we focus on a (2,2)-threshold scheme in the presence of an opponent who impersonates one of the two participants. We consider an asymptotic setting where two shares are generated by an encoder blockwisely from an n-tuple of secrets generated from a stationary memoryless source and a uniform random number available only to the encoder. We introduce a notion of correlation level of the two shares and give coding theorems on the rates of the shares and the uniform random number. It is shown that, for any (2,2)-threshold scheme with correlation level r, none of the rates can be less than H(S) + r, where H(S) denotes the entropy of the source. We also show that the impersonation by the opponent is successful with probability at least 2^-nr+o(n). In addition, we prove the existence of an encoder and a decoder of the (2, 2)-threshold scheme that asymptotically achieve all the bounds on the rates and the success probability of the impersonation.