https://doi.org/10.1140/epjqt/s40507-022-00144-z
Research
Quantum reversible circuits for
multiplicative inverse
1
Department of Computer Science and Technology, School of Information Engineering, Hubei Minzu University, 44500, Enshi, China
2
School of Information and Software Engineering, University of Electronic Science and Technology of China, 610054, Chengdu, China
3
Big data research Center & School of Computer Science and Engineering, University of Electronic Science and Technology of China, 611731, Chengdu, China
4
Advanced Cryptography and System Security Key Laboratory of Sichuan Province, Chengdu University of Information Technology, 610225, Chengdu, Sichuan, China
a
qingbinluo@126.com
b
guowu@uestc.edu.cn
Received:
10
February
2022
Accepted:
21
September
2022
Published online:
7
October
2022
The synthesis of quantum circuits for multiplicative inverse over are discussed in this paper. We first convert the multiplicative inverse operation in
to arithmetic operations in the composite field
, and then discuss the expressions of the square calculation, the inversion calculation and the multiplication calculation separately in the finite field
, where the expressions of multiplication calculation in
are given directly in
and given through being transformed into the composite field
. Then the quantum circuits of these calculations are realized one by one. Finally, two quantum circuits for multiplicative inverse over
are synthesized. They both use 21 qubits, the first quantum circuit uses 55 Toffoli gates and 107 CNOT gates and the second one uses 37 Toffoli gates and 209 CNOT gates. As an example of the application of multiplication inverse, we apply these quantum circuits to the implementations of the S-box quantum circuit of the AES cryptographic algorithm. Two quantum circuits for implementing the S-box of the AES cryptographic algorithm are presented. The first quantum circuit uses 21 qubits, 55 Toffoli gates, 131 CNOT gates and 4 NOT gates and the second one uses 21 qubits, 37 Toffoli gates, 233 CNOT gates and 4 NOT gates. Through the evaluation of quantum cost, the two quantum circuits of the S-box of AES cryptographic algorithm use less quantum resources than the existing schemes.
Key words: Quantum circuit / Composite field / Multiplicative inverse / S-box / AES
© The Author(s) 2022
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.