Abstract: Koblitz elliptic curves implement scalar multiplication without point-doubling operation by Frobenius map，this method improves the efficiency of scalar multiplication greatly.Koblitz curves over a finite field of characteristic 2 and characteristic 3 disassembles k using prime element of simple Euclidean domain.To further improve the efficiency，presents an assumption of disassembling k based on prime element of quadratic Euclidean domain.To prove this assumption，this paper interprets the feasibility by math analysis，gives out corresponding algorithms，and finally proves that the method can improve the efficiency evidently.

Key words: Koblitz elliptic curves, scalar multiplication, quadratic Euclidean domain, Frobenius map

摘要： Koblitz椭圆曲线通过Frobenius 映射实现了不需要倍点运算的标量乘法，很大程度上提高了标量乘法的效率。特征2和特征3的这类Koblitz曲线是以一次欧式环的素元来分解k，对此设想是否可以二次欧式环的素元为基底来二次分解k从而进一步提高效率。基于这一设想，通过数学分析诠释了上述设想的可行性，并给出了相应算法，效率明显提高。

关键词: Koblitz椭圆曲线, 标量乘法, 二次欧式环, Frobenius映射