为实现快速的数据加密 ,Koblitz首先引入了一类定义在有限域F2 上的椭圆曲线 ,并利用Frobenius映射给出了计算椭圆数乘法的一类快速算法 .接着Solinas严格定义了TNAF概念 ,从而完善和推广了Koblitz的想法 ,并从密度方面断言TNAF能显著地减少Hamming重量 .本文类比标准二进制的有关结果 ,进一步证明 :对于Z[τ]中的任何元素 ,其TNAF的Hamming重量在它的所有广义τ adic展式中是最小的 ;在此意义下 ,Koblitz曲线上的TNAF算法已达最优 .同时 ,证明的过程构造了一个把广义τ adic展式转化为TNAF的具体算法 In order to achieve fast data encryption, Koblitz first introduced a class of elliptic curves defined in the finite field F2 and used Frobenius mapping to give a fast algorithm to calculate the elliptic multiplication. Then Solinas strictly defined the concept of TNAF, And popularized the idea of ​​Koblitz and asserted from the density that TNAF can significantly reduce Hamming weight.The relevant results of this paper over standard binary further prove that the Hamming weight of TNAF for any element in Z [τ] In this sense, the TNAF algorithm on the Koblitz curve has been optimized. At the same time, the proof process constructs a concrete algorithm to transform the generalized τ adic expansions into TNAF