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).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Hankerson, D., Menezes, A.J., Vanstone, S.: Guide to Elliptic Curve Cryptography. Springer, New York (2006)
Abdul-Aziz Gutub, A., Tenca, A.F., Savaş, E., Koç, R.C.C.K.: Scalable and unified hardware to compute Montgomery inverse in GF(p) and GF(2\(^{n}\)). In: Kaliski, B.S., Koç, K., Paar, C. (eds.) CHES 2002. LNCS, vol. 2523, pp. 484–499. Springer, Heidelberg (2003). https://doi.org/10.1007/3-540-36400-5_35
Yen, S.: Improved normal basis inversion in GF(\(2^m\)). IEE Electron. Lett. 33, 196–197 (1997)
Hasan, M.A.: Efficient computation of multiplicative inverses for cryptographic applications. In: 15th IEEE Symposium on Computer Arithmetic (2001)
Itoh, T., Tsujii, S.: A fast algorithm for computing multiplicative inverses in GF(\(2^m\)) using normal basis. Inf. Comput. 78, 171–177 (1988)
Kaminski, M., Bshouty, N.H.: Multiplicative complexity of polynomial multiplication over finite fields. J. ACM 36(3), 150–170 (1989)
O. Karatsuba, “Multiplication of multidigit numbers on automata,” in Soviet physics doklady, pp. 595–596, Springer, 1963
Bernstein, D.J.: Batch binary Edwards. In: Halevi, S. (ed.) CRYPTO 2009. LNCS, vol. 5677, pp. 317–336. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-03356-8_19
Cook, S.: On the minimum computation time of functions. Phd thesis, Harvard University (1966)
Schwabe, P., Westerbaan, B.: Solving binary \(\cal{MQ}\) with grover’s algorithm. In: Carlet, C., Hasan, M.A., Saraswat, V. (eds.) SPACE 2016. LNCS, vol. 10076, pp. 303–322. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-49445-6_17
Kepley, S., Steinwandt, R.: Quantum circuits for \({\mathbb{F}}_{2^{n}}\)-multiplication with subquadratic gate count. Quantum Inf. Process. 14(7), 2373–2386 (2015). https://doi.org/10.1007/s11128-015-0993-1
Li, X., et al.: An all-optical quantum gate in a semiconductor quantum dot. Science 301(5634), 809–811 (2003)
Muñoz-Coreas, E., Thapliyal, H.: Design of quantum circuits for Galois field squaring and exponentiation. In: 2017 IEEE Computer Society Annual Symposium on VLSI (ISVLSI), pp. 68–73 (2017)
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).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/978-3-030-65299-9_12
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-65298-2
Online ISBN: 978-3-030-65299-9
eBook Packages: Computer ScienceComputer Science (R0)