Abstract
Shor’s algorithm models discrete logarithms on binary elliptic curves and provides polynomial-time solutions. One of major overheads in applying Shor’s algorithm is implementing binary elliptic curve arithmetic in quantum circuits. Among operations of elliptic curves over binary fields, the multiplication is essential and cost-critical even in the quantum field.
In this paper, we aim to optimize quantum binary field multiplication. Previous works on quantum multiplication focused on minimizing the number of Toffoli gates or qubits. In contrast, our work presents strategies for optimizing Toffoli depth and full depth, which are key factors in the Noisy Intermediate-Scale Quantum (NISQ) era. To achieve our goal, Karatsuba multiplication using divide-and-conquer approach is adopted. In a nutshell, we present an optimized quantum multiplication with Toffoli depth one. Furthermore, under the influence of the optimized Toffoli depth, the full depth is naturally reduced.
In order to show the effectiveness of proposed method, the performance is evaluated by various metrics, such as, qubits, quantum gates, depth, and qubits-depth product. To the best of our knowledge, this is the first study on quantum multiplication that optimizes Toffoli depth and full depth.
This work was partly supported by Institute for Information & communications Technology Promotion(IITP) grant funded by the Korea government(MSIT) (No. 2018-0-00264, Research on Blockchain Security Technology for IoT Services, 50%) and this work was partly supported by Institute for Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government(MSIT) (<Q|Crypton>, No. 2019-0-00033, Study on Quantum Security Evaluation of Cryptography based on Computational Quantum Complexity, 50%).
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Jang, K., Kim, W., Lim, S., Kang, Y., Yang, Y., Seo, H. (2023). Optimized Implementation of Quantum Binary Field Multiplication with Toffoli Depth One. In: You, I., Youn, TY. (eds) Information Security Applications. WISA 2022. Lecture Notes in Computer Science, vol 13720. Springer, Cham. https://doi.org/10.1007/978-3-031-25659-2_18
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