Abstract
The inversion circuit based on the Itoh-Tsujii algorithm, used for many cryptography functions, requires a number of multiplication and squaring operations in circuits. In the past, the optimized inversion implementation has been actively studied in modern computers. However, there are very few works to optimize the inversion on the quantum computer. In this paper, we present the optimized implementation of binary field inversion in quantum circuits. Reversible and non-reversible multiplication circuits are finely combined to reduce the number of CNOT gate. In particular, we optimized the reversible circuit for \(A\cdot B\) and \(A \cdot C\) case in the inversion operation. Afterward, the multiplication and squaring routine efficiently initializes some of the qubits used for the routine into zero value. Lastly, the-state-of-art multiplication and squaring implementation techniques, such as Karatsuba algorithm and shift-based squaring are utilized to obtain the optimal performance. In order to show the effectiveness of the proposed implementation, the inversion is applied to the substitute layer of AES block cipher. Furthermore, the proposed method can be applied to other cryptographic functions, such as binary field inversion for public key cryptography (i.e. Elliptic Curve Cryptography).
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Acknowledgement
This work was 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). This work of Zhi Hu is supported by the Natural Science Foundation of China (Grants No. 61972420, 61602526) and Hunan Provincial Natural Science Foundation of China (Grants No. 2019JJ50827, 2020JJ3050).
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Jang, K., Choi, S.J., Kwon, H., Hu, Z., Seo, H. (2020). Impact of Optimized Operations \(A\cdot B\), \(A\cdot C\) for Binary Field Inversion on Quantum Computers. In: You, I. (eds) Information Security Applications. WISA 2020. Lecture Notes in Computer Science(), vol 12583. Springer, Cham. https://doi.org/10.1007/978-3-030-65299-9_12
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DOI: https://doi.org/10.1007/978-3-030-65299-9_12
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