<p>In order to reflect the nonlinear dynamics of quantum game map more precisely and improve the performance of current cryptosystem, in this paper, we first obtain the fractional four-dimensional quantum game chaotic map according to the fractional calculus. We then proceed to obtain the trajectory, the maximum Lyapunov index, the bifurcation diagram of the aforementioned map, and compare its dynamical behaviours with those of the four-dimensional quantum game chaotic map. Subsequently, we design a synchronisation control system for the proposed fractional system and apply the two systems to the field of information security. Finally, we undertake a comprehensive analysis of the encryption system, examining it from five distinct perspectives. As a result, the key space of proposed algorithm equals to <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10489_2025_6281_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">$6.7 \times 10^{165}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>6.7</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>165</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> and the encrypted image’s information entropy is 7.9997. The NPCR and UACI for the difference attack analysis is <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10489_2025_6281_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$99.61\%$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>99.61</mn> <mo>%</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10489_2025_6281_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$33.44\%$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>33.44</mn> <mo>%</mo> </mrow> </math></EquationSource> </InlineEquation> respectively accompanied with that the correlation coefficients is near to 0. The result indicates that the proposed algorithm can defense both known plaintext and chosen plaintext attacks which means that it is superior to other algorithms in most of the above aspects.</p>

The synchronisation control of fractional 4-D quantum game chaotic map with its application in image encryption

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发表日期：

2025年1月18日

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Zeyu Liu,
Binshuai Feng,
Yuxin Yao,
Xujing Wang

摘要

In order to reflect the nonlinear dynamics of quantum game map more precisely and improve the performance of current cryptosystem, in this paper, we first obtain the fractional four-dimensional quantum game chaotic map according to the fractional calculus. We then proceed to obtain the trajectory, the maximum Lyapunov index, the bifurcation diagram of the aforementioned map, and compare its dynamical behaviours with those of the four-dimensional quantum game chaotic map. Subsequently, we design a synchronisation control system for the proposed fractional system and apply the two systems to the field of information security. Finally, we undertake a comprehensive analysis of the encryption system, examining it from five distinct perspectives. As a result, the key space of proposed algorithm equals to $6.7 \times 10^{165}$ $6.7 \times 10^{165}$ and the encrypted image’s information entropy is 7.9997. The NPCR and UACI for the difference attack analysis is $99.61\%$ $99.61 %$ and $33.44\%$ $33.44 %$ respectively accompanied with that the correlation coefficients is near to 0. The result indicates that the proposed algorithm can defense both known plaintext and chosen plaintext attacks which means that it is superior to other algorithms in most of the above aspects.