Kwang Ho Kim
New Fast Algorithms for Arithmetic on Elliptic Curves over Fields of Characteristic Three
In previous works on ECC(Elliptic Curve Cryptography), the case of characteristic three has been considered relatively less than cases of fields of even characteristic and large prime fields. To the best of our knowledge, for point multiplication on ordinary elliptic curves over fields of characteristic three the most efficient way is known as one shown by N.P. Smart et al.(cf. ). In first portion of this paper we propose new fast algorithms for arithmetic on Hessian elliptic curves over finite field of characteristic three, which reduce costs of a doubling and a mixed point addition from 3M+3C and 10M (cf. ) to 3M+2C and 9M+1C, respectively. These algorithms can realize fast point multiplication nearly comparable with the case of even characteristic, on ordinary elliptic curves over finite field of characteristic three. In next portion we propose a kind of projective coordinates we call ML coordinates and new algorithms for arithmetic on Weierstrass elliptic curve in it, which reduce costs of a tripling and a mixed point addition from 7M+4C and 10M+2C (cf. ) to 6M+6C and 8M+2C, respectively. In conclusion, we can say that ternary elliptic curves are another alternative to existing technology for elliptic curve cryptosystems.
A New Method for Speeding Up Arithmetic on Elliptic Curves over Binary Fields
Now, It is believed that the best costs of a point doubling and addition on elliptic curves over binary fields are 4M+5S(namely, four finite field multiplications and five field squarings) and 8M+5S, respectively. In this paper we reduce the costs to less than 3M+3S and 8M+1S, respectively, by using a new projective coordinates we call PL-coordinates and rewriting the point doubling formula. Combining some programming skills, the method can speed up a elliptic curve scalar multiplication by about 15?20 percent in practice.
A Note on Point Multiplication on Supersingular Elliptic Curves over Ternary Fields
Recently, the supersingular elliptic curves over ternary fields are widely used in pairing based crypto-applications since they achieve the best possible ratio between security level and space requirement. We propose new algorithms for projective arithmetic on the curves, where the point tripling is field multiplication free, and point addition and point doubling requires one field multiplication less than the known best algorithms, respectively. The algorithms combined with DBNS can lead to apparently speed up scalar multiplications on the curves.