Security of Pon-Lu-Jeng's Meta-He digital signature schemes

In 2001 He proposed a signature scheme that has the previously described advantages, notably: (1) It is based on two hard problems; (2) it uses the same modulus; and (3) it only requires one pair of public and private keys. Soon, Pon et al. generalized He's scheme in order to get efficient digital signature schemes with high security (their schemes are claimed to be based on two cryptographic assumptions simultaneously). In this paper, we shall point out that He and Pon et al.'s signature schemes are not based on two cryptographic assumptions simultaneously. So all their signature schemes in fact have not any efficiency advantage if only based on one cryptographic assumptions. Meanwhile, the task of designing efficient digital signature schemes based on two cryptographic assumptions simultaneously is still an open problem.