The information security has become more and more crucial with the development of computer and network technologies. The digital signature is one of key techniques in information security, especially in the authentication, data integrity, and non-repudiation. It has been widely used in military, communication, e-commerce and e-government, etc., and will become more and more popular after the e-signature law is put in practice.The main interest of this dissertation is on the theory and method of digital signature and its application. The research focuses on some key problems of digital signature.Based on the characteristics of various signing equation, problems such as constructing signature scheme and chooing parameter of signing equation are fully investigated in this dissertation. Some significant results are obtained, including: 1)a method of choosing the parmeters of signing equation to generate ElGamal signing equation, this method extends the available range of choosing signature. 2)a new elliptic curve authenticated encryption scheme with message recovery. It is pointed out that there are two forgery attacks with known plaintext under the one-way function is homostasis function, all of these indicate some signature schemes satisfying certain conditions can avoid the forgery attack. 3)a new elliptic curve authenticated encryption scheme with message linkage recovery, which solves the problems such as message encryption and authentication, message linkage recovery and load of transmitting data.4)an elliptic curve signature scheme with message recovery, and the generalized forms of constructing signing eqauation and methods of choosing parameteres of signing equation.With the consideration of whether the owner of message is persuing by signer in blind signature, this dissertation proposes a method to generate ElGamal weakly-blind signature scheme, which contains almost all of the known type of weakly-blind signature scheme. Using multi-linear transform formula to describe the relationships among the variables held by user and proxy signer, this dissertation proposes a proxy blind signature based on elliptic curve cryptosystem.Using the bilinear theory of Weil pairing defined on elliptic curve, a new ID-based blind signature scheme is proposed. In this scheme, ID-based public key is not the public key stored in certificate. This scheme can omit the process of getting public key from the system in verification phase, therefore decrease interaction time and reduce the store space.Based on self-certificated public key, an authenticated encryption scheme with message recovery is proposed. In this scheme, both sides of communication can self verify