Double-authentication-preventing signature (DAPS) is a novel signature notion proposed at ESORICS 2014. The double-authentication-preventing property means that any pair of signatures on two different messages with the same subject will result in an immediate collapse of the signature system. A few potential applications of DAPS have been discussed by its inventors, such as providing a kind of self-enforcement to discourage certificate authority (CA) from misbehaving in public key infrastructure and offering CA some cryptographic arguments to resist legal coercion. In this study, we focus on some fundamental issues on DAPS. We propose a new definition, which is slightly weakened but still reasonable and strong enough to capture the DAPS concept. We develop the new notion of invertible chameleon hash functions with key exposure. Then we propose a generic DAPS scheme, which is provably secure if the underlying invertible chameleon hash function with key exposure is secure. We instantiate this general construction to obtain the DAPS schemes respectively based on the well-known assumptions of integer factorization, Rivest-Shamir-Adleman (RSA), and computational Diffie-Hellman (CDH). They are more efficient than previous DAPS schemes. Furthermore, unlike previous constructions, the trusted setup condition is not needed by our DAPS schemes based on RSA and CDH.