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The Fourier series is a powerful mathematical tool for representing periodic signals as an infinite sum of complex exponentials. In practice, this infinite series is truncated to a finite number of terms, yielding a partial sum. This truncation makes the approximation of the signal feasible but introduces certain challenges, particularly near discontinuities, known as the Gibbs phenomenon.
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The Fast Fourier Transform (FFT) is a computational algorithm designed to compute the Discrete Fourier Transform (DFT) efficiently. By breaking down the calculations into smaller, manageable sections, the FFT significantly reduces the computational complexity involved. Direct computation of an N-point DFT requires N2 complex multiplications, whereas the FFT algorithm needs only (N/2)log⁡2N multiplications, offering a much faster performance.
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The Discrete Fourier Transform (DFT) is a fundamental tool in signal processing, extending the discrete-time Fourier transform by evaluating discrete signals at uniformly spaced frequency intervals. This transformation converts a finite sequence of time-domain samples into frequency components, each representing complex sinusoids ordered by frequency. The DFT translates these sequences into the frequency domain, effectively indicating the magnitude and phase of each frequency component present...
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Deconvolution, also known as inverse filtering, is the process of extracting the impulse response from known input and output signals. This technique is vital in scenarios where the system's characteristics are unknown, and they must be inferred from the observable signals.
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Time scaling of signals is a crucial concept in signal processing that affects the Fourier series representation without altering its coefficients. The process modifies the fundamental frequency, thereby changing how the series represents the signal over time. This principle is essential in various applications, including audio and image processing, where signal manipulation is frequent. Understanding function symmetries is fundamental to simplifying the Fourier series.
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Updated: Jun 25, 2025

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Enhancing image security via chaotic maps, Fibonacci, Tribonacci transformations, and DWT diffusion: a robust data

Mohammad Mazyad Hazzazi1, Mujeeb Ur Rehman2, Arslan Shafique3

  • 1Department of Mathematics, College of Science, King Khalid University, 61413, Abha, Saudi Arabia.

Scientific Reports
|May 28, 2024
PubMed
Summary
This summary is machine-generated.

This study introduces a novel grayscale image encryption method combining chaotic maps, Fibonacci Transformation, Tribonacci Transformation, and Discrete Wavelet Transform for robust security and fast processing. The new scheme offers strong protection against cyberattacks with minimal processing time, outperforming existing techniques.

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Area of Science:

  • Computer Science
  • Cybersecurity
  • Information Security

Background:

  • Existing image encryption schemes face challenges in balancing robust security with real-time application efficiency.
  • Many methods excel in either security or speed, but not both, creating a need for improved approaches.

Purpose of the Study:

  • To propose a robust and efficient grayscale image encryption framework.
  • To enhance data security for digital images through a multi-phase encryption process.
  • To achieve high security without compromising processing speed for practical applications.

Main Methods:

  • A five-phase encryption framework involving secret key generation using chaotic maps.
  • Integration of Fibonacci Transformation (FT) and Tribonacci Transformation (TT) for pixel scrambling and diffusion.
  • Multi-level diffusion using bit-plane decomposition, XOR operations, and Discrete Wavelet Transform (DWT) with substitution box (S-box).

Main Results:

  • The proposed encryption framework demonstrated high robustness, validated by statistical tests (entropy 7.999, correlation 0.0001, histogram variance 6458).
  • Achieved processing time under one second, indicating suitability for real-time applications.
  • Comparative analysis showed superior performance over existing chaos-based, DWT, TT, and FT encryption methods.

Conclusions:

  • The proposed encryption scheme effectively balances robust security and real-time processing requirements.
  • Its multi-layered approach provides strong resistance against various cyberattacks.
  • The method is a promising solution for secure grayscale image transmission and storage.