Quantum Circuit Designs of Point Doubling Operation for Binary Elliptic Curves

In the past years, research on Shor’s algorithm for solving the elliptic curve discrete logarithm problem (Shor’s ECDLP) as the basis for cracking elliptic curve-based cryptosystems (ECC) has started to garner significant interest. To achieve this, existing works put their focus on quantum point addition subroutines to realize the double scalar multiplication circuit essential to Shor’s ECDLP. In contrast, the quantum point doubling subroutines have often been overlooked. In this paper, we bridge this gap by investigating the quantum point doubling operation to be used for the stricter assumption of Shor’s algorithm, i.e., when doubling a point should also be taken into consideration. In particular, we analyze the challenges on implementing the circuit and provide the probable solution. Subsequently, we design and optimize the corresponding quantum circuit, then analyze the high-level quantum resource cost. 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 advancing the evaluation of Shor’s ECDLP.