Z <sub> 2 ~ E </ Sub> Recursive Sequences, Computing With Elliptic Curves

It is a basic work and one of the most important needs, to the sequences cryptography design, to find a new sequences generator, since the importance and the usefulness of the sequences generators,.In this paper, we find a new sequences generators, which is a kind of linear recurring sequences on elliptic curves over the finite field F_2^e. In chapter one, we propose the concept of the cryptography symbol constants and compare the operations over field F_2^e with those over the ringZ/(2^e). We get a computing formula of algebraic degrees and algebraic term numbers of polynomials. In chapter two, we give a kind of linear recurring sequences on elliptic curves over the finite fieldF_2^e. We also present the definition and the number of primitive polynomial over a cyclic sub-group H of an elliptic curve. Moreover, we present the period of a maximal length sequence, the number of a original status and the number of a maximal length circle.