Two generalizations of a coding theorem for a (2, 2)-threshold scheme with a cheater

This paper presents two new coding theorems on a (2, 2)-threshold scheme with an opponent who impersonates one of the shareholders. In the (2, 2)-threshold scheme an encoder blockwisely generates two shares X_n and Y_n from n secrets Sⁿ and a uniform random number E_n, where Sⁿ is generated from a general source. There are three kinds of inputs to a decoder, (N_n., Y_n), (X̅_n, X_n) and (X_n, Y̅_n), where X̅_n and Y̅_n are fraudulent shares generated by the opponent. The decoder judges whether the input is legitimate or not under negligible decoding error probability that vanishes as n → ∞. The two coding theorems given in this paper characterize the minimum attainable rates of X_n, Y_n and E_n and the maximum attainable exponent of the probability of the successful impersonation attack. It turns out that the (2, 2)-threshold scheme with a cheater is related to not only hypothesis testing but also optimistic coding of general sources and channels.