Coding theorems for a (2,2)-threshold scheme with detectability of impersonation attacks

Abstract

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 ℓ ≥ 0, the minimum attainable rates coincide with H(S)+ℓ , where H(S) denotes the entropy of the source, and the maximum attainable exponent of the success probability of the impersonation attack equals to ℓ . It is also shown that a simple scheme using an ordinary (2,2)-threshold scheme attains all the bounds as well. © 1963-2012 IEEE.