🌀 Chaos-Based Image Encryption: Arnold Cat & Henon Map

Harnessing nonlinear dynamical systems (Arnold Cat Map & Henon Map) to achieve confusion and diffusion for robust image encryption.

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❗ Problem Statement

Modern multimedia systems demand strong protection of visual data. Conventional ciphers struggle with large, high-correlation image datasets. This project applies chaos theory to craft encryption that is:

• Highly sensitive to initial conditions (keys)
• Pseudo-random and structure-destroying
• Resistant to statistical, brute-force, and differential attacks

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This project uses the Arnold Cat Map transformation defined as:

$$ \begin{aligned} x' = (x + y)\ mod N \\ y' = (x + 2y)\ mod N \end{aligned} $$

and the Henon map:

$$ \begin{aligned} x_{n+1} &= 1 - a x_n^2 + y_n \\ y_{n+1} &= b x_n \end{aligned} $$

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🧮 Methodology Snapshot

1️⃣ Arnold Cat Map Workflow

1. Load square image (pad if needed).
2. Apply matrix transform iteratively (key-driven iteration count).
3. Output permuted image (cipher stage 1).
4. Decrypt by inverse iteration (same count).

2️⃣ Henon Map Workflow

1. Initialize (x_0, y_0, a, b) ⇒ secret key set.
2. Iterate to produce chaotic float sequence.
3. Scale & quantize to 8-bit mask array matching image dimensions.
4. XOR original / permuted image with mask ⇒ ciphertext.
5. Regenerate mask with identical key to reverse XOR.

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📊 Comparative Feature Grid

Technique	Confusion	Diffusion	Key Sensitivity	Histogram Resistance	Speed
Arnold Cat Map	✅ High	⚠️ Low	✅ High	⚠️ Moderate	🚀 Fast
Henon Map	✅ High	✅ High	✅ High	✅ Strong	⏱️ Medium

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🔍 Observations

Note

Combining both (permutation + masking) forms a hybrid cipher with layered security.

• Arnold Cat alone keeps pixel values intact ⇒ vulnerable to statistical analysis if iteration count is small.
• Henon-based XOR alters value distribution ⇒ stronger entropy & flatter histograms.
• Parameter/key perturbations (1e-12 scale) yield dramatically different outputs.

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🧠 Key Insights

• Chaotic dynamics naturally embed unpredictability + sensitivity ⇒ ideal cryptographic primitives.
• Multi-map or multi-phase designs (e.g., Cat → Henon → Diffusion pass) can mitigate singular weaknesses.
• Practical for real-time (low-latency) encryption in surveillance / telemedicine when optimized.

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🔑 Parameter & Key Considerations

Map	Primary Secret Components	Expansion Strategy
Cat	Iteration count k, image size N	Combine with dynamic iteration schedule
Henon	a, b, x0, y0, sequence length	Derive per-session seeds via KDF

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Made with chaos theory 🔁 for secure pixels