Hyperelliptic Curve Cryptosystem

The hyperelliptic curve is a kind of special algebra curve, it can be regarded as the popularization of the elliptic curve, elliptic curve is a kind of special hyperelliptic curve whose genus equals to 1. Compared with Elliptic Curve Cryptosystems (ECC), Hyperelliptic Curve Cryptosystems(HCC) has obvious security advantages , therefore the theory of HCC has caused the crypto circle's extensive attention in recent years. The study on it mainly remains at theory stage at present, and a large amount of problems await to solve further. This paper regards the Hyperelliptic curve based on the binary finite field as the main research object, has carried on deep research to this kind of cryptosystems from two respects which are the mathematics foundation and agreement of HCC.The basis of building secure hyperelliptic curve cryptosystems is constructing secure hyperelliptic curve over finite field, therefore firstly must select an hyperelliptic curve suitable to build cryptosystem. Research on hand shows that the genus of an secure hyperelliptic curve should be small and the cardinality of its Jacobian group should contain a 160bit big prime factor at least. Proceeding from this safe prerequisite, this paper presents a method to construct safe curve, it is an improvement to Weil algorithm. In order to improve the HCC speed of realization, the fast problem of implementation which are connected with basis operation on the finite field and the jacobian group is studied, and the corresponding algorithms are presents in this paper. In addition, some important problems, such as the choice of a base point and the embedding problem of plain text etc, also are studied in this paper. In the course of study on the agreements, this paper provides a kind of hyperelliptic cryptosystem scheme based on public key certificate, this scheme solves some safe problems in the traditional cryptosystem effectively, have the small communication amount, safe intensity and computation speed, especially suitable for solving the security problem existing in some systems with limited resource. Except for some basic functions, such as data encrypting and figures signing, this cryptosystems still can be used in the two-way authentication between the system and the users. But because of the relation on time, it fails to can be realized.