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

Abstract

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 ), (X n , X n ) and (X n , Y n ), where X n and Y 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.