  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.
