  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.
