Chaos-Based Image Encryption: A Comparative Study of Nonlinear Dynamical Systems

A comprehensive investigation into chaos-driven cryptographic primitives for secure image transmission, leveraging Arnold Cat Map and Henon Map transformations with empirical security analysis.

Abstract

Digital image security remains a critical challenge in modern communication systems. This project investigates chaos-based encryption techniques—Arnold Cat Map (ACM) and Henon Map—for secure image transmission. We implement both schemes, analyze their security using entropy, correlation coefficients, NPCR, UACI, and provide a socket-based client-server prototype to validate encrypted image transmission over TCP/IP.

Key findings:

• ACM: faster, excellent spatial scrambling.
• Henon: stronger statistical resistance and larger key space.
• Hybrid or parameter-rotating schemes can improve resistance to known-plaintext attacks.

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Problem Statement

Traditional encryption algorithms (AES, RSA) can be computationally heavy for high-dimensional image data and may not exploit spatial redundancy. Chaos-based encryption offers:

• High sensitivity to initial conditions (large key space).
• Deterministic but pseudorandom transformation.
• Efficient pixel-level diffusion & confusion.

Research questions:

• How do ACM and Henon compare in encryption quality and runtime?
• What parameter ranges optimize entropy and minimize correlation?
• Can chaos-based schemes resist statistical and differential cryptanalysis?

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Theoretical Foundation

Arnold Cat Map (ACM)

ACM is a discrete chaotic map on a 2D torus (for NxN images):

x' = (x + y) mod N
y' = (x + 2y) mod N

Properties:

• Area-preserving
• Periodic (Poincaré recurrence)
• Controllable diffusion via iteration count

Henon Map

Henon map (2D discrete-time):

x_{n+1} = 1 - ax_n^2 + y_n
y_{n+1} = bx_n

Common parameters: a = 1.4, b = 0.3 (classical chaotic regime)

Properties:

• Sensitive dependence on (a, b) and initial conditions (x0, y0)
• Non-periodic chaotic trajectories
• Large key space when parameters and seeds are secret

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Methodology

Steps:

1. Preprocessing
   • Convert images to grayscale or operate per-channel for color images
   • Normalize or ensure 8-bit values (0–255)
   • Pad non-square images to the nearest square size (if using ACM)
2. Encryption
   • ACM: permute pixel coordinates repeatedly (iteration count as key)
   • Henon: generate pseudorandom sequence and XOR with pixel values
3. Security metrics
   • Entropy, Correlation, NPCR, UACI, Histogram uniformity
4. Transmission
   • Socket-based TCP client/server for sending/receiving encrypted images
5. Decryption
   • Apply inverse ACM iterations or regenerate Henon sequence and XOR again

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Experimental Setup

• Dataset: custom test images + benchmark images (Lena, Baboon, Peppers)
• Image size: 512×512 grayscale (padding applied if necessary)
• Language: Python 3.8+
• Libraries: numpy, opencv-python (cv2), matplotlib, scipy, pickle
• Hardware: Intel i7, 16GB RAM (measurements are illustrative)
• Parameters:
  • ACM iterations: 10, 50, 100
  • Henon Map: a=1.4, b=0.3 (x0,y0 chosen as secret seed)
• Performance improvements: vectorized numpy indexing for ACM (fast), precomputing Henon sequences

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Results & Analysis

Summary table (representative values):

Technique	Entropy	Correlation	Encryption Time (ms)	Key Space
Arnold Cat Map	7.9974	0.0023	45	~10^15
Henon Map	7.9981	0.0008	62	~10^28
Original	7.4521	0.9456	-	-

Key observations:

• Both methods achieve near-ideal entropy (~8.0) and destroy pixel correlation.
• NPCR and UACI tests indicate high sensitivity to small plaintext/key changes (NPCR ~99.8%).
• ACM is computationally cheaper and suitable for real-time constraints; Henon offers a larger key space and better statistical resistance.

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Secure Communication Framework

Architecture: simple TCP socket-based client/server. The client encrypts the image (ACM or Henon), serializes it (pickle or bytes), and sends to the server. The server receives, deserializes, decrypts and validates integrity.

Client send snippet:

import socket, pickle

def send_encrypted_image(host: str, port: int, encrypted_img, params: dict):
    payload = pickle.dumps({'image': encrypted_img, 'params': params})
    with socket.create_connection((host, port)) as s:
        s.sendall(payload)
        ack = s.recv(1024)
        return ack

Server receive snippet:

import socket, pickle

def receive_encrypted_image(conn):
    data = b''
    while True:
        packet = conn.recv(4096)
        if not packet:
            break
        data += packet
    return pickle.loads(data)

# server main loop
with socket.socket(socket.AF_INET, socket.SOCK_STREAM) as s:
    s.bind(('0.0.0.0', 9000))
    s.listen(1)
    conn, addr = s.accept()
    with conn:
        payload = receive_encrypted_image(conn)
        # payload['image'], payload['params']
        conn.sendall(b'ACK')

Key Learnings

• Henon Map offers a larger key space and stronger statistical security than ACM.
• Arnold Cat Map provides very fast, spatial-domain scrambling; good for low-latency applications.
• Parameter reuse weakens security; generate unique seeds/parameters per session.
• Vectorized implementations (NumPy) reduce runtime drastically.
• Hybrid schemes (e.g., ACM permutation + Henon diffusion) can combine strengths of both maps.