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

In this paper, we discuss coding theorems on a $(2, 2)$--threshold scheme in the presence of an opponent who impersonates one of the two shareholders in an asymptotic setup. We consider a situation 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 an opponent. It is shown that, if the correlation level between the two shares equals to an $\ell \ge 0$, 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 $\ell$. We also give a simple construction of an encoder and a decoder using an ordinary $(2,2)$--threshold scheme where the two shares are correlated and attains all the bounds.