In previous research, quantum resources were concretely estimated for solving Elliptic Curve Discrete Logarithm Problem(ECDLP). In [1], the quantum algorithm was optimized for the binary elliptic curves and the main optimization target was the number of the logical qubits. The division algorithm was mainly optimized in [1] since every ancillary qubit is used in the division algorithm. In this paper, we suggest a new quantum division algorithm on the binary field which uses a smaller number of qubits. For elements in a field of $2^n$, we can save $\lceil n/2 \rceil - 1$ qubits instead of using $8n^2+4n-12+(16n-8)\lfloor\log(n)\rfloor$ more Toffoli gates, which leads to a more space-efficient quantum algorithm for binary elliptic curves.

翻译：先前的研究中，针对求解椭圆曲线离散对数问题（ECDLP）的量子资源进行了具体估算。文献[1]对二进制椭圆曲线的量子算法进行了优化，主要优化目标为逻辑量子比特数量。由于所有辅助量子比特均用于除法算法，文献[1]主要优化了除法算法。本文提出一种二进制域上使用更少量子比特的新型量子除法算法。对于$2^n$域中的元素，我们可节省$\lceil n/2 \rceil - 1$个量子比特，但需额外使用$8n^2+4n-12+(16n-8)\lfloor\log(n)\rfloor$个Toffoli门，从而得到更空间高效的二进制椭圆曲线量子算法。