Fast Arithmetic in Generalized Optimal Extension Fields

In Elliptic Curves Cryptosystems (ECC), the optimal extension fields is preferable to others method, whether concerns performance or memory request. But, it is very difficult to choose pseudo-Mersenne prime numbers, and satisfy the condition that p just is presented by a register of processor. This paper replaces pseudo-Mersenne numbers by generalized Mersenne numbers, provides a new notation-Generalized Optimal Extension Fields(GOEFs), and studies the fast arithmetic about multiplication and modular reduction in GOEFs, finally, deduces common formulas for multiplication and some more general formulas for modular reduction in GOEFs. The results in this paper extend the corresponding work on arithmetic of Bailey, Mihǎilescu, and Woodbury in OEFs.