Coding Theorems for a $(2,2)$ -Threshold Scheme With Detectability of Impersonation Attacks

Coding theorems on a $(2,2)$ -threshold scheme with an opponent are discussed in an asymptotic setup, where the opponent tries to impersonate one of the two participants. A situation is considered where $n$ secrets $S^{n}$ from a memoryless source is blockwisely encoded to two shares and the two shares are decoded to $S^{n}$ with permitting negligible decoding error. We introduce correlation level of the two shares and characterize the minimum attainable rates of the shares and a uniform random number for realizing a $(2, 2)$ -threshold scheme that is secure against the impersonation attack by the opponent. It is shown that if the correlation level between the two shares equals to $\ell \geq 0$ , the minimum attainable rates coincide with $H(S)+\ell$ , where $H(S)$ denotes the entropy of the source, and the maximum attainable exponent of the success probability of the impersonation attack equals to $\ell$ . It is also shown that a simple scheme using an ordinary $(2,2)$ -threshold scheme attains all the bounds as well.