In the past years, research on Shor's algorithm for solving elliptic curves for discrete logarithm problems (Shor's ECDLP), the basis for cracking elliptic curve-based cryptosystems (ECC), has started to garner more significant interest. To achieve this, most works focus on quantum point addition subroutines to realize the double scalar multiplication circuit, an essential part of Shor's ECDLP, whereas the point doubling subroutines are often overlooked. In this paper, we investigate the quantum point doubling circuit for the stricter assumption of Shor's algorithm when doubling a point should also be taken into consideration. In particular, we analyze the challenges on implementing the circuit and provide the solution. Subsequently, we design and optimize the corresponding quantum circuit, and analyze the high-level quantum resource cost of the circuit. Additionally, we discuss the implications of our findings, including the concerns for its integration with point addition for a complete double scalar multiplication circuit and the potential opportunities resulting from its implementation. Our work lays the foundation for further evaluation of Shor's ECDLP.

翻译：过去几年中，针对求解椭圆曲线离散对数问题的Shor算法（Shor's ECDLP）的研究——这是破解基于椭圆曲线的密码系统（ECC）的基础——开始受到越来越多的关注。为实现这一目标，大多数工作侧重于通过量子点加子程序来构建双标量乘法电路（Shor's ECDLP的关键组成部分），而点倍子程序则常被忽略。本文针对Shor算法中需考虑点倍乘的更严格假设，研究了量子点倍乘电路。我们特别分析了实现该电路的挑战并提供了解决方案。随后，我们设计并优化了相应的量子电路，分析了电路的高层级量子资源成本。此外，我们还讨论了研究发现的启示，包括其与点加子程序集成构成完整双标量乘法电路的考量，以及实现该电路所带来的潜在机遇。我们的工作为后续评估Shor's ECDLP奠定了基础。