Toffoli gate count optimized space-efficient quantum circuit for binary field multiplication

Shor’s algorithm solves the elliptic curve discrete logarithm problem (ECDLP) in polynomial time. To optimize Shor’s algorithm for binary elliptic curves, reducing the cost of binary field multiplication is essential because it is the most cost-critical arithmetic operation. In this paper, we propose Toffoli gate count-optimized, space-efficient (i.e., no ancilla qubits are used) quantum circuits for binary field ( \((\mathbb {F}_{2^{n}})\) $({\mathbb{F}}_{2^n})$ ) multiplication. To achieve this, we leverage the Karatsuba-like formulae and demonstrate that its application can be implemented without the need for ancillary qubits. We optimize these circuits in terms of CNOT gate count and depth. Building upon the Karatsuba-like formulae, we develop a space-efficient CRT-based multiplication technique utilizing two types of out-of-place multiplication algorithms to reduce the CNOT gate count. Our quantum circuits exhibit an extremely low Toffoli gate count of \(O(n2^{\log {2}^{*}n})\) $O(n2_{}^{log2_{}^{\ast }n})$ , where \(\log _{2}^{*}\) $log_2^{\ast }$ represents the iterative logarithmic function that grows very slowly. When compared to recent Karatsuba-based space-efficient quantum circuit, our approach requires only (10–25 %) of the Toffoli gate count and Toffoli depth for cryptographic field sizes in the range of n = 233–571. To the best of our knowledge, this represents the first successful utilization of the Karatsuba-like formulae and CRT-based multiplication in quantum circuits.