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.