<p>Existing chaotic maps for image encryption often exhibit limited dynamic complexity, which can potentially compromise security. To address this limitation, this work introduces a novel four-dimensional Hyperchaotic Memristive Log-Logistic Sine Map (4D-HMLLSM). Its architecture integrates memristive dynamics with coupled logarithmic and sinusoidal nonlinearities, a design that generates complex hyperchaotic behaviour, supports a large key space, and promotes high unpredictability. An encryption scheme is presented that couples the 4D-HMLLSM with histogram equalisation of its chaotic sequences. In this approach, the 4D-HMLLSM drives permutation and diffusion, while histogram equalisation flattens the statistical signature of the key stream. The resulting encrypted images exhibit high information entropy, with average values for <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11071_2025_11694_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="86" /> </InlineMediaObject> <EquationSource Format="TEX">$1024 \times 1024$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1024</mn> <mo>×</mo> <mn>1024</mn> </mrow> </math></EquationSource> </InlineEquation> images 7.9998. The scheme’s parameters provide a key space <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11071_2025_11694_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">$2^{278}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>2</mn> <mn>278</mn> </msup> </math></EquationSource> </InlineEquation> for effective resistance to brute-force attempts, while the average encryption time for a <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11071_2025_11694_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="86" /> </InlineMediaObject> <EquationSource Format="TEX">$1024 \times 1024$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1024</mn> <mo>×</mo> <mn>1024</mn> </mrow> </math></EquationSource> </InlineEquation> image is approximately 0.7612 s. Security evaluations confirm the algorithm’s robust performance, achieving a number of pixels change rate value of an average 99.6083% and a unified average changing intensity value of an average 33.4683, indicating a strong diffusion characteristic. The system also demonstrates near-zero inter-pixel correlation, 1-bit change key sensitivity, and resilience to noise and cropping attacks.</p>

A novel hyperchaotic four-dimensional memristive log-logistic sine map with histogram equalisation for image encryption

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发表日期：

2025年8月18日

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Karim H. Moussa

摘要

Existing chaotic maps for image encryption often exhibit limited dynamic complexity, which can potentially compromise security. To address this limitation, this work introduces a novel four-dimensional Hyperchaotic Memristive Log-Logistic Sine Map (4D-HMLLSM). Its architecture integrates memristive dynamics with coupled logarithmic and sinusoidal nonlinearities, a design that generates complex hyperchaotic behaviour, supports a large key space, and promotes high unpredictability. An encryption scheme is presented that couples the 4D-HMLLSM with histogram equalisation of its chaotic sequences. In this approach, the 4D-HMLLSM drives permutation and diffusion, while histogram equalisation flattens the statistical signature of the key stream. The resulting encrypted images exhibit high information entropy, with average values for $1024 \times 1024$ $1024 \times 1024$ images 7.9998. The scheme’s parameters provide a key space $2^{278}$ $2^{278}$ for effective resistance to brute-force attempts, while the average encryption time for a $1024 \times 1024$ $1024 \times 1024$ image is approximately 0.7612 s. Security evaluations confirm the algorithm’s robust performance, achieving a number of pixels change rate value of an average 99.6083% and a unified average changing intensity value of an average 33.4683, indicating a strong diffusion characteristic. The system also demonstrates near-zero inter-pixel correlation, 1-bit change key sensitivity, and resilience to noise and cropping attacks.