Depth-Optimization of Quantum Cryptanalysis on Binary Elliptic Curves

This paper presents quantum cryptanalysis for binary elliptic curves from a time-efficient implementation perspective (i.e., reducing the circuit depth), complementing the previous research that focuses on the space-efficiency perspective (i.e., reducing the circuit width). To achieve depth optimization, we propose an improvement to the existing circuit implementation of the Karatsuba multiplier and FLT-based inversion, then construct and analyze the resource in the Qiskit quantum computer simulator. The proposed multiplier architecture, which improves the quantum Karatsuba multiplier from the previous study, reduces the depth and yields a lower number of CNOT gates that bound to
$O(n^{log_{2}(3)})$
while maintaining a similar number of Toffoli gates and qubits. Furthermore, our improved FLT-based inversion reduces CNOT count and overall depth, with a tradeoff of higher qubit size. Finally, we employ the proposed multiplier and FLT-based inversion for performing quantum cryptanalysis of binary point addition as well as the complete Shor’s algorithm for the elliptic curve discrete logarithm problem (ECDLP). As a result, apart from depth reduction, we are also able to reduce up to 90% of the Toffoli gates required in a single-step point addition compared to prior work, leading to significant improvements and giving new insights on quantum cryptanalysis for a depth-optimized implementation.