Minimizing CNOT-count in quantum circuit of the extended Shor's algorithm for ECDLP

Abstract: Since the elliptic curve discrete logarithms problem (ECDLP) was proposed, it has been widely used in cryptosystem because of its strong security. Although the proposal of the extended Shor's algorithm offers hope for cracking ECDLP, it is debatable whether the algorithm can actually pose a threat in practice. From the perspective of the quantum circuit of the algorithm, we analyze the feasibility of cracking ECDLP with improved quantum circuits using an ion trap quantum computer. We give precise quantum circuits for extended Shor's algorithm to calculate discrete logarithms on elliptic curves over prime fields, including modulus subtraction, three different modulus multiplication, modulus inverse, and windowed arithmetic. Whereas previous studies mostly focused on minimizing the number of qubits or the depth of the circuit, we minimize the number of CNOTs, which greatly affects the time to run the algorithm on an ion trap quantum computer. First, we give the implementation of the basic arithmetic with the lowest known number of CNOTs and the construction of an improved modular inverse, point addition, and the windowing technique. Then, we precisely estimate the number of improved quantum circuits needed to perform the extended Shor's algorithm for factoring an n-bit integer. We analyze the running time and feasibility of the extended Shor's algorithm on an ion trap quantum computer according to the number of CNOTs. Finally, we discussed the lower bound of the number of CNOTs needed to implement the extended Shor's algorithm.