To raise the efficiency of field operations on elliptic curve, based on the idea of trading multiplications for squares, two improved algorithms are proposed to compute 7P and 7k P directly over GFP in terms of affine coordinates, their computational complexity is I+18M+12S and I+(17k+2)M+(14k+1)S respectively, and the new algorithm's efficiency is improved by 8.3% and 13.5% respectively compared with the best algorithms at present. In addition, based on the same idea, a modified method is given to compute 5k P directly over GFP in terms of affine coordinates, its com-putational complexity is I+(9k+2)M+(14k+1)S , and the efficiency of the new method is improved by 17.2%and 35.7%respectively compared with Xu Kaiping's and MISHRA's method.%为了提高椭圆曲线底层域运算的效率,基于将乘法运算转换为平方运算的思想,提出在素数域GFP上用仿射坐标直接计算7P和7kP的改进算法,其运算量分别为I+18M+12S和I+(17k+2)M+(14k+1)S,与已有的最好算法相比,效率分别提升了8.3%和10.3%.另外,基于相同的思想给出了素数域GFP上用仿射坐标系直接计算5kP的改进算法,其运算量为I+(9k+2)M+(14k+1)S,与徐凯平和Mishra等人所提的算法相比,效率分别提升了17.2%和35.7%.
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