Abstract: For the research of elliptic curve cryptography (ECC). both the theory of ECC and the standardization and industrialization of ECC are mature. In the design of ECC, the selection of a secure elliptic curve is the foundation of ECC implementation, and is also important to ensure its safety. At present, the method of random selection is considered as one of the best methods for finding a security elliptic curve. The core idea of finding security elliptic curve is to compute the order of Jacobian group of the random elliptic curve over finite fields. This paper mainly introduces several kinds of classic algorithms to compute the order of Jacobian group of the elliptic curves: Schoof algorithm, SEA algorithm, Satoh algorithm, and AGM algorithm. For the Schoof algorithm, the improved algorithms based on discrete logarithm problem are put forward: the improved algorithms of Kangaroo algorithm and Big step gain step (BSGS) algorithm, and the experimental results illustrate the accelerated algorithms improve the running time. For SEA algorithm, this paper also presents its BSGS improved algorithm, analyzing and comparing the efficiency of original SEA algorithm and BSGS improved algorithm through an example. For Satoh algorithm and AGM algorithm, the paper introduce the theoretical basis and the concrete implementation of the algorithms, comparing their advantages and applicable conditions.