Researchers Of The Algorithms On ECC Over Finite Field GF(2～n)

Because of the popularization of the Internet, its application involves many realms such as government, military, culture and education, business, finance and so on. In thus an information society, especially the computing speed of computers and network transmission speed raising continuously, public-key cryptosystems will play more and more important roles in the world.Elliptic curve cryptosystem (ECC) was proposed independently by Koblitz and Miller in 1985. Elliptic curve cryptosystem is one of the most important cryptosystems, which is obtained by substituting finite group of elliptic curve on finite field for finite group of cryptosystem based on discrete logarithm problem. Because of its many special advantages, it has attracted more attention and been researched as an independent part. It is generally considered that the elliptic curve cryptosystem will become the most important public-key cryptosystem in the 21th century.For a long time, how to implement elliptic curve cryptosystem much more quickly is a problem, which people starve for solving. And how to efficiently implement the multiplication over finite field GF(2~n) is the key to optimize the bottom algorithms of elliptic curve cryptosystem over finite field GF(2~n).Some investigations and improvements of the multiplication over finite field GF(2~n) is given in the paper. First of all, four kinds of representative algorithms of the multiplication over finite field GF(2~n) are reviewed and analyzed. Then, three kinds of sliding-window algorithms used in the scalar multiplication of elliptic curve cryptosystem are comprehensively summarized. Finally, a totally new multiplication algorithms in finite field GF(2~n), which is designed according to the sliding-window algorithms used in scalar multiplication in ECC. After analysis, it shows that the efficiency of the new algorithm is 20%~25% more than the efficiency of the fixed-window algorithm. And it does not need more saving space.