Nowadays, digital signature schemes are widely used in electronic commerce. Thus, various digital signature schemes are proposed to make the electronic transactions more convenient. In 2007, Chung et al. proposed an ID-based digital signature scheme on elliptic curve cryptosystem (ECC). They claimed that their scheme is secure because it is based upon the difficulty of elliptic curve discrete logarithm problem (ECDLP). However, we find that their scheme has a security flaw, and thus we propose a feasible attack on Chung et al.psilas scheme in this paper. The proposed attack is based on the technique for solving the linear Diophantine equation. Using the proposed technique, an attacker can easily obtain the signer's secret key without facing the difficulty of ECDLP. Therefore, Chung et al.'s scheme cannot prevent from our attack such that it is not secure.