Abstract: To raise the efficiency of field operation on elliptic curve, based on the idea of trading multiplications for squares, two modified algorithms are proposed to compute 4P and 5P directly over prime field [FP] in terms of affine coordinates, their computational complexity are[(3k-1)M+(5k+3)S] and[(6k-1)M+(9k+3)S] respectively, which are improved to 6.25% and 5% respectively than those of Dimitroy’s and Zhou meng’s method. Moreover, using the same idea, an improved method is given to compute [3kP] directly in terms of affine coordinates, its computational complexity is [I+(6k+1)M+(9k+1)S], and the efficiency of the new method is improved to 3.4% and 24% respectively than those of Zhong meng’s and Yin xin-chun’s method.

Key words: elliptic curve cryptosystem, scalar multiplication, field operation, affine coordinate, jacobian coordinate

摘要： 为了提高椭圆曲线底层域运算的效率，基于将乘法转换为平方运算的思想，提出在素数域[FP]上用雅克比坐标直接计算[2kP]和[3kP]的改进算法，其运算量分别为[(3k-1)M+(5k+3)S]和[(6k-1)M+(9k+3)S]，与DIMITROY和周梦等人所提的算法相比，算法效率分别提升了6.25%和5%。另外，利用相同的原理，给出了素数域[FP]上用在仿射坐标系直接计算[3kP]的改进算法，其运算量为[I+(6k+1)M+(9k+1)S]，与周梦和殷新春等人所提的算法相比，效率分别提升了3.4%和24%。

关键词: 椭圆曲线密码体制, 标量乘法, 底层域运算, 仿射坐标, 雅克比坐标