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Digital signature schemes based on public-key cryptosystems are important building blocks of key agreement protocols. They are vulnerable by existential forgery attack. A usual method to prevent existential forgery attack is on-way hash function and message redundancy. In 2004, Chang and Chang proposed a new digital signature scheme. The scheme is without using one-way hash function and any redundancy padding, which is very interesting to many designers. Recently, Chien found an existential forgery attack. In this paper, we show another forgery attack on it and propose an improved scheme without oracle, which is secure against existential forgery attacks. Then we proposed a key agreement protocol by amending the signature slightly.