The paper introduces a new short signature scheme that is existentially unforgeable under a chosen message attack without relying on random oracles. The security of this scheme is based on a new complexity assumption called the Strong Diffie-Hellman (SDH) assumption, which is similar to the Strong RSA assumption. The SDH assumption states that it is difficult to solve a specific problem involving bilinear groups. The scheme constructs signatures that are as short as those in the Boneh-Lynn-Shacham (BLS) signature scheme but do not require random oracles. The paper also discusses the properties of the SDH assumption, including its relation to the Strong RSA assumption and its application in traitor tracing systems. Additionally, the paper presents a weakly secure signature scheme and demonstrates how it can be extended to a fully secure scheme. The scheme supports limited message recovery, allowing for further reduction in the total length of signed messages. The paper also shows how to use random oracles to achieve even shorter signatures and provides a generic security analysis of the SDH assumption in the generic group model.The paper introduces a new short signature scheme that is existentially unforgeable under a chosen message attack without relying on random oracles. The security of this scheme is based on a new complexity assumption called the Strong Diffie-Hellman (SDH) assumption, which is similar to the Strong RSA assumption. The SDH assumption states that it is difficult to solve a specific problem involving bilinear groups. The scheme constructs signatures that are as short as those in the Boneh-Lynn-Shacham (BLS) signature scheme but do not require random oracles. The paper also discusses the properties of the SDH assumption, including its relation to the Strong RSA assumption and its application in traitor tracing systems. Additionally, the paper presents a weakly secure signature scheme and demonstrates how it can be extended to a fully secure scheme. The scheme supports limited message recovery, allowing for further reduction in the total length of signed messages. The paper also shows how to use random oracles to achieve even shorter signatures and provides a generic security analysis of the SDH assumption in the generic group model.