This paper presents a short signature scheme that is existentially unforgeable under a chosen message attack without using random oracles. The scheme is based on a new complexity assumption called the Strong Diffie-Hellman (SDH) assumption. The SDH assumption is similar to the Strong RSA assumption and has properties that make it useful for constructing cryptographic systems. The proposed signature scheme is as short as DSA signatures and is provably secure in the absence of random oracles. It also supports limited message recovery, which allows for further reduction in the total length of a message/signature pair. The scheme is shown to be secure under a chosen message attack, and its security is proven using the SDH assumption. The paper also discusses the relationship between the SDH assumption and other cryptographic assumptions, and provides a lower bound on the computational complexity of solving the SDH problem in a generic group model. The results show that the SDH assumption is a strong and practical foundation for constructing secure cryptographic systems.This paper presents a short signature scheme that is existentially unforgeable under a chosen message attack without using random oracles. The scheme is based on a new complexity assumption called the Strong Diffie-Hellman (SDH) assumption. The SDH assumption is similar to the Strong RSA assumption and has properties that make it useful for constructing cryptographic systems. The proposed signature scheme is as short as DSA signatures and is provably secure in the absence of random oracles. It also supports limited message recovery, which allows for further reduction in the total length of a message/signature pair. The scheme is shown to be secure under a chosen message attack, and its security is proven using the SDH assumption. The paper also discusses the relationship between the SDH assumption and other cryptographic assumptions, and provides a lower bound on the computational complexity of solving the SDH problem in a generic group model. The results show that the SDH assumption is a strong and practical foundation for constructing secure cryptographic systems.