The Shor's algorithm can find solutions to the discrete logarithm problem on binary elliptic curves in polynomial time. A major challenge in implementing Shor's algorithm is the overhead of representing and performing arithmetic on binary elliptic curves using quantum circuits. Multiplication of binary fields is one of the critical operations in the context of elliptic curve arithmetic, and it is especially costly in the quantum setting. Our goal in this paper is to optimize quantum multiplication in the binary field. In the past, efforts to optimize quantum multiplication have centred on reducing the Toffoli gate count or qubits required. However, despite the fact that circuit depth is an important metric for indicating the performance of a quantum circuit, previous studies have lacked sufficient consideration for reducing circuit depth. Our approach to optimizing quantum multiplication differs from previous work in that we aim at reducing the Toffoli depth and full depth. To optimize quantum multiplication, we adopt the Karatsuba multiplication method which is based on the divide-and-conquer approach. In summary, we present an optimized quantum multiplication that has a Toffoli depth of one. Additionally, the full depth of the quantum circuit is also reduced thanks to our Toffoli depth optimization strategy. To demonstrate the effectiveness of our proposed method, we evaluate its performance using various metrics such as the qubit count, quantum gates, and circuit depth, as well as the qubits-depth product. These metrics provide insight into the resource requirements and complexity of the method. Our work achieves the lowest Toffoli depth, full depth, and the best trade-off performance for quantum multiplication. Further, our multiplication is more effective when not used in stand-alone cases. We show this effectiveness by using our multiplication to the Itoh-Tsujii algorithm-based inversion of F(x8+x4+x3+x+1).
Multiplication over binary fields is a crucial operation in quantum algorithms designed to solve the discrete logarithm problem for elliptic curve defined over
GF(2n). In this paper, we present an algorithm for constructing quantum circuits that perform multiplication over
GF(2n) with
O(nlog2(3)) Toffoli gates. We propose a variant of our construction that achieves linear depth by using
O(nlog2(n)) ancillary qubits. This approach provides the best known space-time trade-off for binary field multiplication with a subquadratic number of Toffoli gates. Additionally, we demonstrate that for some particular families of primitive polynomials, such as trinomials, the multiplication can be done in logarithmic depth and with
O(nlog2(3)) gates.
This paper provides a detailed analysis and estimation of the quantum resources required for depth-optimized elliptic curve point multiplication (ECPM), with a focus on qubits, Toffoli gates, CNOT gates, and circuit depth. The proposed ECPM quantum circuit, which is tailored for binary elliptic curve cryptanalysis using Shor’s algorithm, incorporates improved arithmetic operations into optimized point addition (PA) and point doubling (PD) processes. The quantum resources required for single-step and full-step ECPM implementations (comprising
2nPD +2 PA) were systematically evaluated to enable in-depth cryptanalysis of binary elliptic curve cryptography using Shor’s algorithm. The proposed approach offers a significant improvement in resource efficiency compared to circuits that rely solely on the existing PA, which typically require (
2n+2)PA operations. The execution of a full iteration of quantum cryptanalysis using the Shor circuit was estimated to require
2n3+2n(log3+1)+32n2logn+8n2+O(n(log3+1))Toffoli gates, approximately
2n+7nlog6n+7qubits, and a circuit depth of approximately
7+nlog39. Comparisons with leading Toffoli gate-level analysis results validate our findings, which demonstrate reduced quantum resource usage. In addition, key findings related to techniques for solving discrete logarithm problems are summarized, drawing from influential previous studies and current advancements.