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Quantum reversible circuits for \({\text {GF}}(2^8)\) multiplication based on composite field arithmetic operations

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Abstract

In this paper, we mainly discuss the quantum reversible circuits of multiplication over \({\text {GF}}(2^8)\), which has many applications in modern cryptography. The quantum circuits of multiplication over \({\text {GF}}(2^8)\) implemented by using the existing methods need at least 64 Toffoli gates without auxiliary qubits. However, Toffoli gates need a lot of quantum resources in physical implementation. Therefore, we try to construct the quantum circuits with as few Toffoli gates as possible. We first convert multiplication over \({\text {GF}}(2^8)\) into multiplication over composite field \({\text {GF}}((2^4)^2)\), and then realize the quantum circuits of multiplication over \({\text {GF}}(2^4)\) by means of product matrix and converting the multiplication into composite field \({\text {GF}}((2^2)^2)\), respectively. In addition, we also discuss the case where the initial output qubits of the product are not \(|0\rangle \)s, and give the quantum circuit of multiplication over \({\text {GF}}(2^4)\) in this case according to the principle of minimizing the number of Toffoli gates. Finally, according to the calculation formula of multiplication over composite field \({\text {GF}}((2^4)^2)\) and the isomorphic mappings between \({\text {GF}}(2^8)\) and \({\text {GF}}((2^4)^2)\), the quantum circuits of multiplication over \({\text {GF}}(2^8)\) are realized. These quantum circuits without auxiliary qubits only needs 42 Toffoli gates, which are 22 less than the quantum circuits realized by the existing methods. Specifically, we give the specific quantum circuits with irreducible polynomials \(f(x)=x^8+x^4+x^3+x+1\) and \(f(x)=x^8+x^4+x^3+x^2+1\), respectively.

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Acknowledgements

This work is supported by the Natural Sciences Foundation of Hubei Province (Grant No. 2020CFB326), the Natural Science Foundation of Fujian Province (Grant No. 2020J01812), the National Natural Sciences Foundation of China (Grant No. 62172075), the National Key R &D Program of China (Grant No. 2018YFA0306703), Chengdu Innovation and Technology Project (Grant No. 2021-YF05-02414-GX).

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Authors and Affiliations

  1. School of Information and Software Engineering, University of Electronic Science and Technology of China, Chengdu, 610054, China

    Qing-bin Luo & Xiao-yu Li

  2. Department of Computer Science and Technology, School of Information Engineering, Hubei Minzu University, Enshi, 44500, China

    Qing-bin Luo & Qiang Li

  3. Big Data Research Center & School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu, 611731, China

    Guo-wu Yang

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  1. Qing-bin Luo
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  2. Xiao-yu Li
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  4. Qiang Li
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Correspondence to Qing-bin Luo.

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Appendices

A The quantum circuit of multiplication over \({\text {GF}}(2^8)\) with the irreducible polynomial \(f(x)=x^8+x^4+x^3+x+1\)

figure a

B The quantum circuit that restores input for \({\text {GF}}(2^8)\) multiplication with the irreducible polynomial \(f(x)=x^8+x^4+x^3+x+1\)

figure b

C The quantum circuit of multiplication over \({\text {GF}}(2^8)\) with the irreducible polynomial \(f(x)=x^8+x^4+x^3+x^2+1\)

figure c

D The quantum circuit that restores input for \({\text {GF}}(2^8)\) multiplication with the irreducible polynomial \(f(x)=x^8+x^4+x^3+x^2+1\)

figure d

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Luo, Qb., Li, Xy., Yang, Gw. et al. Quantum reversible circuits for \({\text {GF}}(2^8)\) multiplication based on composite field arithmetic operations. Quantum Inf Process 22, 58 (2023). https://doi.org/10.1007/s11128-022-03799-w

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