This paper gives definitions,constructions and applications of signature schemes,which are selectively unforgeable but existentially forgeable(SUEF).We formalize the special notion under conditions of chosen message attack(CMA) and known message attack(KMA).Then two general constructions are presented with two concrete examples.We apply the SUEF secure signature schemes to an elementary authentication mechanism,the challenge-response mechanism,to obtain a basic deniable authentication method.The method is presented as a deniable message transmission(MT) authenticator in the extension framework of Raimondo. This paper gives definitions, constructions and applications of signature schemes, which are selectively unforgeable but existentially forgeable (SUEF) .We formalize the special notion under conditions of chosen message attack (CMA) and known message attack (KMA) .Then two general constructions are presented with two concrete examples. We apply the SUEF secure signature schemes to an elementary authentication mechanism, the challenge-response mechanism, to obtain a basic deniable authentication method. The method is presented as a deniable message transmission (MT) authenticator in the extension framework of Raimondo.