The Research And Application Of Digital Signature Schemes Based On Elliptic Curves

With the development and application of information technology, the security problem of electronic information becomes more and more important. The RSA cryptosystem, which is a public-key cryptosystem being used widely nowadays, seems to have difficulties to meet the users' need of higher security. So far, Elliptic Curve Cryptosystem provides the highest strength-per-bit of any public-key cryptosystems. In addition to its higher security, Elliptic Curve Cryptosystem also has many other advantages, such as the less computation overheads, shorter key size, considerable bandwidth savings and so on. All of these merits have made it the best public-key cryptosystem that is suitable for use in the future.The theory and application of Elliptic Curve Cryptosystem are studied as follow.Firstly, basic arithmetic operations of the finite field F_p(p>3) and F_2^m are discussed respectively, then the theory of Elliptic Curve Cryptosystem based on the finite field F_p and F_2^m are studied. Furthermore, we analyze the Elliptic Curve Discrete Logarithm Problem and several attack algorithms on the Elliptic Curve Cryptosystem. Herein below, in accordance with the geometric figures, we bring forward a new mathematical description of point addition operation and point doubling operation on the elliptic curve based on the field F_p . By the mathematical description, we can clearlyunderstand point generation process and point computation process over the elliptic curve and the computation problem of the order of the base point over the elliptic curve.Secondly, the scalar multiplication is described and the choice of secure elliptic curves is analyzed. Then we compile a program on choosing the security elliptic curve base on the field Fp where p is a great prime number more than 2160. We can find security elliptic curves conveniently and fast by using the program. We hope that the program results will serve as a benchmark for practitioners of Elliptic Curve Cryptosystem.Thirdly, the Elliptic Curve Digital Signature Algorithm is described and the improved algorithm is put forward. Moreover, we present a new two-way authentication scheme based on ECDSA. In this scheme, two-way authentication may be done between arbitrary two users without exposing their secret information.Finally, two kinds of new message recovery digital signature schemes based on the elliptic curve are introduced. One kind of the schemes is a message recovery digital signature scheme with a redundant function, and the other is the one with a Hash function.